Physics for IIT - JEE 2012-13 mechanics I

B.M. Sharma

Chapter 7

Newton's Laws of Motion - all with Video Answers

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Chapter Questions

Problem 1

When a body is stationary
a. there is no force acting on it
b. the forces acting on its are not in contact with it
c. the combination of forces acting on it balance each other
d. the body is in vacuum

Ajay Singhal

Ajay Singhal

Numerade Educator

Problem 2

A block of metal weighing $2 \mathrm{~kg}$ is resting on a frictionless plane. It is struck by a jet releasing water at a rate of 1 $\mathrm{kg} / \mathrm{s}$ and a speed of $5 \mathrm{~m} / \mathrm{s}$. The initial acceleration of the block will be
a. $2.5 \mathrm{~m} / \mathrm{s}^{2}$
b. $5 \mathrm{~m} / \mathrm{s}^{2}$
c. $10 \mathrm{~m} / \mathrm{s}^{2}$
d. $20 \mathrm{~m} / \mathrm{s}^{2}$

Ajay Singhal

Numerade Educator

Problem 3

Two persons are holding a rope of negligible weight tightly at its ends so that it is horizontal. A $15 \mathrm{~kg}$ weight is attached to the rope at the mid point which how no. longer remains horizontal. The minimum tension required to completely straighten the rope is
a. $15 \mathrm{~kg}$
b. $15 / 2 \mathrm{~kg}$
c. $5 \mathrm{~kg}$
d. Infinitely large

Ajay Singhal

Numerade Educator

Problem 4

Threc equal weights $A, B$, and $C$ of mass $2 \mathrm{~kg}$ each are hanging on a string passing over a fixed frictionless pulley as shown in the Fig. $7.308$. The tension in the string connecting weights $B$ and $C$ is
a. zero
b. $13 \mathrm{~N}$
c. $3.3 \mathrm{~N}$
d. $19.6 \mathrm{~N}$

Ajay Singhal

Numerade Educator

Problem 5

A block of mass $M$ is pulled along a horizontal frictionless surface by a rope of mass $m$. Force $P$ is applied at one end of rope. The force which the rope exerts on the block $\mathrm{s}$
a. $\frac{P}{(M-m)}$
b. $\frac{P}{M(m+M)}$
c. $\frac{P M}{(m+M)}$
d. $\frac{P M}{(M-m)}$

Ajay Singhal

Numerade Educator

Problem 6

The elevator shown in Fig. $7.309$ is descending with an acceleration of $2 \mathrm{~ms}^{-2}$. The mass of the block $A=0.5 \mathrm{~kg}$. The force exerted by the block $A$ on the block $B$ is (take $\left.g=10 \mathrm{~m} / \mathrm{s}^{2}\right)$
a. $2 \mathrm{~N}$
b. $4 \mathrm{~N}$
c. $6 \mathrm{~N}$
d. $8 \mathrm{~N}$

Ajay Singhal

Numerade Educator

Problem 7

A mass $M$ is suspended by a rope from a rigid support at $A$ as shown in Fig. $7.310 .$ Another rope is tied at the end $B$ and it is pulled horizontally with a force $F$. If the rope $A B$ make an angle $\theta$ with the vertical, then the tension in the string $A B$ is
a. $F \sin \theta$
b. $F / \sin \theta$
c. $F \cos \theta$
d. $F / \cos \theta$

Ajay Singhal

Numerade Educator

Problem 8

Two bodies of mass $4 \mathrm{~kg}$ and $6 \mathrm{~kg}$ are attached to the ends of a string passing over a pulley (see Fig. 7,311 ). The 4 $\mathrm{kg}$ mass is attached to the table top by another string. The tension in this string $T_{1}$ is equal to (take $g=10 \mathrm{~m} / \mathrm{s}^{2}$ )
a. $20 \mathrm{~N}$
b. $25 \mathrm{~N}$
c. $10.6 \mathrm{~N}$
d. $10 \mathrm{~N}$

Ajay Singhal

Numerade Educator

Problem 9

In the Fig. $7.312$, the pulley $P_{1}$ is fixed and the pulley $P_{2}$ is movable. If $W_{1}=W_{2}=100 \mathrm{~N}$, what is the angle $A P_{2} P_{1}$ ? The pulleys are frictionless
a. $30^{\circ}$
b. $60^{\circ}$
c. $90^{\circ}$
d. $120^{\circ}$

Ajay Singhal

Numerade Educator

Problem 10

A man sits on a chair supported by a rope passing over a frictionless fixed pulley. The man who weighs $1.000 \mathrm{~N}$ exerts a force of $450 \mathrm{~N}$ on the chair downwards while puiling the rope on the other side. If the chair weighs $250 \mathrm{~N}$, then the acceleration of the chair is
a. $0.45 \mathrm{~m} / \mathrm{s}^{2}$
b. 0
c. $2 \mathrm{~m} / \mathrm{s}^{2}$
d. $9 / 25 \mathrm{~m} / \mathrm{s}^{2}$

Ajay Singhal

Numerade Educator

Problem 11

In the Fig. $7.313$, the ball $A$ is released from rest, when the spring is at its natural (unstretched) length. For the block $B$ of mass $M$ to leave contact with ground at some stage. the minimum mass of $A$ must be
a. $2 M$
b. $M$
c. $M / 2$
d. $M / 4$

Ajay Singhal

Numerade Educator

Problem 12

Two skaters weighing in the ratio $4: 5$ and $9 \mathrm{~m}$ apart are skating on a smooth frictionless surface. They pull on a rope stretched between them. The ratio of the distance covered by them when they meet each other will be
a. $5: 4$
b. $4: 5$
c. $25: 16$
d. $16: 25$

Ajay Singhal

Numerade Educator

Problem 13

Three forces are acting on a particle of mass $m$ initially in equilibrium. If the first 2 forces $\left(R_{1}\right.$ and $\left.R_{2}\right)$ are perpendicular to each other and suddenly the third force $\left(R_{3}\right)$ is removed, then the acceleration of the particle is
a. $\frac{R_{3}}{m}$
b. $\frac{R_{1}+R_{2}}{m}$
c. $\frac{R_{1}-R_{2}}{m}$
d. $\frac{R_{1}}{m}$

Nishant Kumar

Numerade Educator

Problem 14

$n$ balls each of mass $m$ impinge elastically each second on a surface with velocity $u$. The average force experienced by the surface will be
a. $\mathrm{mu}$
b. $2 \mathrm{mnu}$
c. $4 \mathrm{~mm}$
d. mnu/2

Ajay Singhal

Numerade Educator

Problem 15

A ball of mass $m$ moving with a velocity $u$ rebounds from a wall. The collision is assumed to be elastic and the force of interaction between the ball and wall varies as shown in the Fig. 7.314. Then the value of $F_{0}$ is
a. $m u / T$
b. $2 m u / T$
c. $4 m u / T$
d. $m u / 2 T$

Ajay Singhal

Numerade Educator

Problem 16

A unidirectional force $F$ varying with time $t$ as shown in the Fig. $7.315$ acts on a body initially at rest for a short duration $2 \mathrm{~T}$. Then the velocity acquired by the body is
a. $\frac{\pi F_{0} T}{4 m}$
b. $\frac{\pi F_{0} T}{2 m}$
c. $\frac{F_{0} T}{4 m}$
d. zero

Ajay Singhal

Numerade Educator

Problem 17

A body of mass $2 \mathrm{~kg}$ has an initial velocity of $3 \mathrm{~m} / \mathrm{s}$ along $O E$ and it is subjected to a force of $4 \mathrm{~N}$ in a direction perpendicular to $O E$ (see Fig. 7.316). The distance of body from $O$ after $4 \mathrm{~s}$ will be
a. $12 \mathrm{~m}$
b. $20 \mathrm{~m}$
c. $8 \mathrm{~m}$
d. $48 \mathrm{~m}$

Ajay Singhal

Numerade Educator

Problem 18

In order to raise a mass of $100 \mathrm{~kg}$ a man of mass $60 \mathrm{~kg}$ fastens a rope to it and passes the rope over a smooth pulley. He climbs the rope with acceleration $5 \mathrm{~g} / 4$ relative to the rope (sec Fig. 7.317). The tension in the rope is (take $g=10 \mathrm{~m} / \mathrm{s}^{2}$ ).
a. $1432 \mathrm{~N}$
b. $928 \mathrm{~N}$
c. $1219 \mathrm{~N}$
d. $642 \mathrm{~N}$

Ajay Singhal

Numerade Educator

Problem 19

A plumb bob is hung from the ceiling of a train compartment. The train moves on an inclined track of inclination $30^{\circ}$ with horizontal. Acceleration of train up the plane is $a=g / 2$. The angle which the string supporting the bob makes with normal to the ceiling in equilibrium is
a. $30^{\circ}$
b. $\tan ^{-1}(2 / \sqrt{3})$
c. $\tan ^{-1}(\sqrt{3} / 2)$
d. $\tan ^{-1}(2)$

Narayan Hari

Numerade Educator

Problem 20

Two particles $A$ and $B$, each of mass $m$, are kept stationary by applying a horizontal force $F=m g$ on particle $B$ as shown in Fig. $7.318$. Then
a. $2 \tan \beta=\tan \alpha$
b. $2 T_{1}=5 T_{2}$
c. $T_{1}=T_{2}$
d. None of these.

Dheeraj Sharma

Numerade Educator

Problem 21

A lift is going up, the total mass of the lift and the passengers is $1500 \mathrm{~kg}$. The variation in the speed of lift is shown in Fig. 7.319. Then the tension in the rope at $t=1 \mathrm{~s}$ will be
a. $17400 \mathrm{~N}$
b. $14700 \mathrm{~N}$
c. $12000 \mathrm{~N}$
d. None of the above

Ajay Singhal

Numerade Educator

Problem 22

In the above problem the tension in the rope will be least at
a. $t=1 \mathrm{~s}$
b. $t=4 \mathrm{~s}$
c. $t=9 \mathrm{~s}$
d. $t=11 \mathrm{~s}$

Ajay Singhal

Numerade Educator

Problem 23

A block is placed on a rough horizontal plane attached with an elastic spring as shown in Fig. $7.320$Initially spring is unscratched. If the plane is gradually lifted from $\theta=0^{\circ}$ to $\theta=90^{\circ}$, then the graph showing extension in the spring $(x)$ versus angle $(\theta)$ is
a.
b.
c.
d.

Narayan Hari

Numerade Educator

Problem 24

A balloon of mass $M$ is descending at a constant acceleration $\alpha$. When a mass $m$ is released from the balloon it starts rising with the same acceleration $a$. Assuming that its volume does not change, what is the value of $m$ ?
a. $\frac{\alpha}{\alpha+g} M$
b. $\frac{2 \alpha}{\alpha+g} M$
c. $\frac{\alpha+g}{\alpha} M$
d. $\frac{\alpha+g}{2 \alpha} M$

Ajay Singhal

Numerade Educator

Problem 25

The system shown in Fig. $7.321$ is released from rest. The spring gets elongated
a. if $M>m$
b. if $M>2 m$
c. if $M>m / 2$
d. For any value of $M$ (Neglect friction and masses of pulley, string and spring)

Narayan Hari

Numerade Educator

Problem 26

A trolley $T$ of mass $5 \mathrm{~kg}$ on a horizontal smooth surface is pulled by a load of $2 \mathrm{~kg}$ through a uniform rope $A B C$ of length $2 \mathrm{~m}$ and mass $1 \mathrm{~kg}$ (see Fig. 7.322). As the load falls from $B C=0$ to $B C=2 \mathrm{~m}$, its acceleration (in $\mathrm{m} / \mathrm{s}^{2}$ ) changes from
a. $\frac{20}{6}$ to $\frac{30}{6}$
b. $\frac{20}{8}$ to $\frac{30}{8}$
c. $\frac{20}{5}$ to $\frac{30}{6}$
d. none of these

Narayan Hari

Numerade Educator

Problem 27

Two wooden blocks are moving on a smooth horizontal surface such that the mass $m$ remains stationary with respect to block of mass $M$ as shown in the Fig. $7.323$. The magnitude of force $P$ is
a. $(M+m) g \tan \beta$
b. $g \tan \beta$
c. $m g \cos \beta$
d. $(M+m) g \operatorname{cosec} \beta$

Narayan Hari

Numerade Educator

Problem 28

A bead of mass $m$ is attached to one end of a spring of natural length $R$ and spring constant $K=\frac{(\sqrt{3}+1) m g}{R}$. The other end of the spring is fixed at a point $A$ on a smooth vertical ring of radius $R$ as shown in the Fig. $7.324$. The normal reaction at $B$ just after it is released to move is
a. $m g / 2$
b. $\sqrt{3} m g$
c. $3 \sqrt{3} m g$
d. $\frac{3 \sqrt{3} m g}{2}$

Narayan Hari

Numerade Educator

Problem 29

An inclined plane makes an angle $30^{\circ}$ with the horizontal. $A$ groove $(O A)$ of length $5 \mathrm{~m}$ cut, in the plane makes an angle $30^{\circ}$ with $O X$. A short smooth cylinder is free to slide down the influence of gravity. The time taken by the cylinder to reach from $A$ to $O$ is $\left(g=10 \mathrm{~m} / \mathrm{s}^{2}\right)$
a. $4 \mathrm{~s}$
b. $2 \mathrm{~s}$
c. $2 \mathrm{~s}$
d. $1 \mathrm{~s}$

Ajay Singhal

Numerade Educator

Problem 30

A man is raising himself and the crate on which he stands with an acceleration of $5 \mathrm{~m} / \mathrm{s}^{2}$ by a massless rope-andpulley arrangement. Mass of the man is $100 \mathrm{~kg}$ and that of the crate is $50 \mathrm{~kg}$. If $g=10 \mathrm{~m} / \mathrm{s}^{2}$, then the tension in the rope is
a. $2250 \mathrm{~N}$
b. $1125 \mathrm{~N}$
c. $750 \mathrm{~N}$
d. $375 \mathrm{~N}$

Ajay Singhal

Numerade Educator

Problem 31

In question 30, contact force between man and the crate is
a. $2250 \mathrm{~N}$.
b. $1125 \mathrm{~N}$
c. $750 \mathrm{~N}$
d. $375 \mathrm{~N}$

Ajay Singhal

Numerade Educator

Problem 32

Two objects $A$ and $B$ each of mass $m$ are connected by a light inextensible string. They are restricted to move on a frictionless ring of radius $R$ in a vertical plane (as shown in Fig. 7.327). The objects are released from rest at the position shown. Then, the tension in the cord just after release is
a. 0
b. $m g$
c. $\sqrt{2} m g$
d. $m g / \sqrt{2}$

Ajay Singhal

Numerade Educator

Problem 33

Blocks $A$ and $C$ start from rest and move to the right with acceleration $a_{A}=12 t \mathrm{~m} / \mathrm{s}^{2}$ and $a_{C}=3 \mathrm{~m} / \mathrm{s}^{2} .$ Here $t$ is in seconds. The time when block $B$ again comes to rest is
a. $2 \mathrm{~s}$
b. $1 \mathrm{~s}$
c. $3 / 2 \mathrm{~s}$
d. $1 / 2 \mathrm{~s}$

Ajay Singhal

Numerade Educator

Problem 34

In the given Fig. $7.329$ the mass $m_{2}$ starts with velocity $t_{0}$ and moves with constant velocity on the surface. During motion the normal reaction between the horizontal surface and fixed triangle block $m_{1}$ is $\mathrm{N}$. Then during motion
a. $N=\left(m_{1}+m_{2}\right) g$
b. $N=m_{1} g$
c. $N<\left(m_{1}+m_{2}\right) g$
d. $N>\left(m_{1}+m_{2}\right) g$

Ajay Singhal

Numerade Educator

Problem 35

Figure $7.330$ shows an arrangement in which three identical blocks are joined together with an inextensible string. All the surfaces are smooth and pulleys are massless. If $a_{A}$, $a_{B}$, and $a_{C}$ are the respective accelerations of the blocks $A, B$, and $C$, then the value of $a_{B}$ in terms of $a_{A}$ and $a_{C}$ is
a. $a_{A} t a_{C}$
b. $\frac{a_{A}-a_{C}}{2}$
c. $\frac{a_{A}+a_{C}}{2}$
d. $a_{A}+a_{C}$

Ajay Singhal

Numerade Educator

Problem 36

In the Fig. $7.331$ shown, blocks $A$ and $B$ move with velocities $v_{1}$ and $v_{2}$ along horizontal direction. The ratio of $\frac{v_{1}}{v_{2}}$ is
a. $\frac{\sin \theta_{1}}{\sin \theta_{2}}$
b. $\frac{\sin \theta_{2}}{\sin \theta_{1}}$
c. $\frac{\cos \theta_{2}}{\cos \theta_{1}}$
d. $\frac{\cos \theta_{1}}{\cos \theta_{2}}$

Ajay Singhal

Numerade Educator

Problem 37

Assuming that the block is always remains horizontal, hence the acceleration of $B$ is
a. $6 \mathrm{~m} / \mathrm{s}^{2}$
b. $2 \mathrm{~m} / \mathrm{s}^{2}$
c. $4 \mathrm{~m} / \mathrm{s}^{2}$
d. None of these

Dheeraj Sharma

Numerade Educator

Problem 38

If the block $B$ moves towards right with acceleration $b$ then the net acceleration of block $A$ is
a. $b \hat{i}+4 b \hat{j}$
b. $b \hat{i}+b \hat{j}$.
c. $b \hat{i}+2 b \hat{j}$
d. None of these

Dheeraj Sharma

Numerade Educator

Problem 39

If the blocks $A$ and $B$ are moving towards each other with acceleration $a$ and $b$ as shown in the Fig. $7.334$. Find the net acceleration of block $C$.
a. $a \hat{i}-2(a+b) \hat{j}$
b. $-(a+b) \hat{j}$
c. $a \hat{i}-(a+b) \hat{j}$
d. None of these

Prem Bijarniya

Numerade Educator

Problem 40

The small marble is projected with a velocity of $10 \mathrm{~m} / \mathrm{s}$ in a direction $45^{\circ}$ from the horizontal $y$ -direction on the smooth inclined plane. Calculate the magnitude $v$ of its velocity after $2 \mathrm{~s}$.
a. $10 \sqrt{2} \mathrm{~m} / \mathrm{s}$
b. $5 \mathrm{~m} / \mathrm{s}$
c. $10 \mathrm{~m} / \mathrm{s}$
d. $5 \sqrt{2} \mathrm{~m} / \mathrm{s}$

Ajay Singhal

Numerade Educator

Problem 41

Two masses each equal to $m$ are lying on $X$ -axis at $(-a, 0)$ and $(+a, 0)$, respectively. as shown in Fig. $7.335 .$ They are connected by a light string. $A$ force $F$ is applied at the origin along vertical direction. As a result, the masses move towards each other without loosing contact with ground. What is the acceleration of each mass? Assume the instantaneous position of the masses as $(-x, 0)$ and $(x, 0)$, respectively
a. $\frac{2 F}{m} \frac{\sqrt{\left(a^{2}-x^{2}\right)}}{x}$
b. $\frac{2 F}{m} \frac{x}{\sqrt{\left(a^{2}-x^{2}\right)}}$
c. $\frac{F}{2 m} \frac{x}{\sqrt{\left(a^{2}-x^{2}\right)}}$
d. $\frac{F}{m} \frac{x}{\sqrt{\left(a^{2}-x^{2}\right)}}$

Dheeraj Sharma

Numerade Educator

Problem 42

A light string passing over a smooth light pulley connects two blocks of masses $m_{1}$ and $m_{2}$ (vertically). If the acceleration of the system is $(g / 8)$, then the ratio of masses is
a. $5: 3$
b. $4: 3$
c. $9: 7$
d. $8: 1$

Ajay Singhal

Numerade Educator

Problem 43

A lift is moving down with an acceleration $a .$ A man in the lift drops a ball inside the lift. The acceleration of the ball as observed by the man in the lift, and a man standing stationary on the ground are, respectively
a. $a, g$
b. $(g-a) ; g$
c. $a_{1} a$
d. $g, g$

Ajay Singhal

Numerade Educator

Problem 44

Block $B$ has a mass $m$ and is released from rest when it is on top of wedge $A$, which has a mass $3 \mathrm{~m}$ (see Fig. 7,336 ). Determine the tension in cord $C D$ needed to hold the wedge from moving while $B$ is sliding down $A$. Neglect friction.
a. $2 m g \cos \theta$
b. $\frac{m g}{2} \cos \theta$
c. $\frac{m g}{2} \sin 2 \theta$
d. $m g \sin 2 \theta$

Ajay Singhal

Numerade Educator

Problem 45

A particle of mass $2 \mathrm{~kg}$ moves with an initial velocity of $v=(4 \hat{i}+4 \hat{j}) \mathrm{m} / \mathrm{s}$. A constant force of $F=-20 \hat{j} \mathrm{~N}$ is
applied on the particle. Initially, the particle was at (0,
0). The $x$ -coordinate of the particle when its $y$ -coordinate again becomes zero is given by
a. $1.2 \mathrm{~m}$
b. $4.8 \mathrm{~m}$
c. $6.0 \mathrm{~m}$
d. $3.2 \mathrm{~m}$

Dheeraj Sharma

Numerade Educator

Problem 46

Three blocks $A, B$, and $C$ are suspended as shown in Fig. 7.337. Mass of each of blocks $A$ and $B$ is $m$. If system is in equilibrium, and mass of $C$ is $M$ then
a. $M>2 m$
b. $M=2 m$
c. $M<2 m$
d. None of these

Ajay Singhal

Numerade Educator

Problem 47

A balloon with mass $M$ is descending down with an acceleration $a(a<g) .$ What mass $m$ be detached from it, so that it starts moving up with an acceleration $a$.
a. $\frac{M a}{g+a}$
b. $\frac{2 M a}{g+a}$
c. $\frac{2 M g}{a}$
d. $\frac{2 \mathrm{Ma}}{g}$

Ajay Singhal

Numerade Educator

Problem 48

A block of mass $m$ is placed on a smooth inclined plane of inclination $\theta$ with the horizontal. The force exerted by the plane on the block has magnitude
a. $m g \tan \theta$
b. $m g \cos \theta$
c. $m g / \cos \theta$
d. $m g$

Ajay Singhal

Numerade Educator

Problem 49

A wooden block of mass $M$ resting on a rough horizontal floor is pulled with a force $F$ at an angle $\phi$ with the horizontal. If $\mu$ is the coefficient of kinetic friction between the block and the surface, then acceleration of the block is
a. $\frac{F}{M} \sin \phi$
b. $\frac{F}{M}(\cos \phi+\mu \sin \phi)-\mu g$
c. $\frac{\mu F}{M} \cos \phi$
d. $\frac{F}{M}(\cos \phi-\mu \sin \phi)-\mu g$

Ajay Singhal

Numerade Educator

Problem 50

A particle of small mass $m$ is joined to a very heavy body by a light string passing over a light pulley. Both bodies are free to move. The total downward force on the pulley is
a. $>>m g$
b. $4 \mathrm{mg}$
c. $2 \mathrm{mg}$
d. $\overline{m g}$

Ajay Singhal

Numerade Educator

Problem 51

A light string passing over a smooth light pulley connects two blocks of masses $m_{1}$ and $m_{2}$ (vertically). If the acceleration of the system is $(g / 8)$, then the ratio of masses is
a. $8: 1$
b. $9: 7$
c. $4: 3$
d. $5: 3$

Ajay Singhal

Numerade Educator

Problem 52

An object is suspended from a spring balance in a lift. The reading is $240 \mathrm{~N}$ when the lift is at rest. If the spring balance reading now changes to $220 \mathrm{~N}$, then the lift is moving
a. downward with constant speed
b. downward with decreasing speed
c. downward with increasing speed
d. upward with increasing speed

Ajay Singhal

Numerade Educator

Problem 53

When force $F_{1}, F_{2}$, and $F_{3}$ are acting on a particle of mass $m$ such that $F_{2}$ and $F_{3}$ are mutually perpendicular, then the particle remains stationary. If the force $F_{1}$ is now removed, then the acceleration of the particle is
a. $\frac{F_{1}}{m}$
b. $\frac{F_{2}}{m}$
c. $\frac{F_{3}}{m}$
d. $\frac{F_{2}+F_{1}}{m}$

Ajay Singhal

Numerade Educator

Problem 54

In Fig. $7.338$, the system is initially at rest. A $5 \mathrm{~kg}$ block is now released. Assuming the pulleys and string to be massless and smooth, the acceleration of block $C$ will be
a. zero
b. $2.5 \mathrm{~m} / \mathrm{s}^{2}$
c. $10 / 7 \mathrm{~m} / \mathrm{s}^{2}$
d. $5 / 7 \mathrm{~m} / \mathrm{s}^{2}$

Ajay Singhal

Numerade Educator

Problem 55

As shown in the Fig. $7.339$, if acceleration of $M$ with respect to ground is $2 \mathrm{~m} / \mathrm{s}^{2}$, then
a. Acceleration of $m$ with respect to $M$ is $5 \mathrm{~m} / \mathrm{s}^{2}$.
b. Acceleration of $m$ with respect to ground is $5 \mathrm{~m} / \mathrm{s}^{2} .$
c. Acceleration of $m$ with respect $M$ is $2 \mathrm{~m} / \mathrm{s}^{2} .$
d. Acceleration of $m$ with respect to ground is $10 \mathrm{~m} / \mathrm{s}^{2}$.

Ajay Singhal

Numerade Educator

Problem 56

A force-time graph for the motion of a body is shown in the Fig. $7.340 .$ The change in the momentum of the body between zero and $10 \mathrm{~s}$ is
a. $15 \mathrm{~kg} \mathrm{~m} / \mathrm{s}$
b. $4 \mathrm{~kg} \mathrm{~m} / \mathrm{s}$
c. $3 \mathrm{~kg} \mathrm{~m} / \mathrm{s}$
d. $5 \mathrm{~kg} \mathrm{~m} / \mathrm{s}$

Ajay Singhal

Numerade Educator

Problem 57

A block $A$ has a velocity of $0.6 \mathrm{~m} / \mathrm{s}$ to the right, determine the velocity of cylinder $B$.
a. $1.2 \mathrm{~m} / \mathrm{s}$
b. $2.4 \mathrm{~m} / \mathrm{s}$
c. $1.8 \mathrm{~m} / \mathrm{s}$
d. $3.6 \mathrm{~m} / \mathrm{s}$

Ajay Singhal

Numerade Educator

Problem 58

For the pulley system shown in Fig. $7.342$, each of the cables at $A$ and $B$ is given a velocity of $2 \mathrm{~m} / \mathrm{s}$ in the direction of the arrow. Determine the upward velocity $v$ of the load $m$.
a. $1.5 \mathrm{~m} / \mathrm{s}$
b. $3 \mathrm{~m} / \mathrm{s}$
c. $6 \mathrm{~m} / \mathrm{s}$
d. $4.5 \mathrm{~m} / \mathrm{s}$

Ajay Singhal

Numerade Educator

Problem 59

A man pulls himself up the $30^{\circ}$ incline by the method shown in Fig. 7.343. If the combined mass of the man and cart is $100 \mathrm{~kg}$, determine the acceleration of the cart if the man exerts a pull of $250 \mathrm{~N}$ on the rope. Neglect all friction and the mass of the rope, pulleys and wheels.
a. $4.5 \mathrm{~m} / \mathrm{s}^{2}$
b. $2.5 \mathrm{~m} / \mathrm{s}^{2}$
c. $3.5 \mathrm{~m} / \mathrm{s}^{2}$
d. $1.5 \mathrm{~m} / \mathrm{s}^{2}$

Ajay Singhal

Numerade Educator

Problem 60

A painter of mass $M$ stands on a platform of mass $m$ and pulls himself up by two ropes which hang over pulley as shown in Fig. $7.344$. He pulls each rope with force $F$ and moves upward with a uniform acceleration $a$. Find $a$ neglecting the fact that no one could do this for long time.
a. $\frac{4 F+(2 M+m) g}{M+2 m}$
b. $\frac{4 F+(M+m) g}{M+2 m}$
c. $\frac{4 F-(M+m) g}{M+m}$
d. $\frac{4 F-(M+m) g}{2 M+m}$

Ajay Singhal

Numerade Educator

Problem 61

An object is resting at the bottom of the two strings which are inclined at an angle of $120^{\circ}$ with each other. Each string can withstand a tension of $20 \mathrm{~N}$. The maximum weight of the object that can be sustained without breaking the string is
a. $10 \mathrm{~N}$
b. $20 \mathrm{~N}$
c. $20 \sqrt{2} N$
d. $40 \mathrm{~N}$

Ajay Singhal

Numerade Educator

Problem 62

A block is lying on the horizontal frictionless surface. One end of a uniform rope is fixed to the block which is pulled in the horizontal direction by applying a force $F$ at the other end. If the mass of the rope is half the mass of the block, the tension in the middle of the rope will be
a. $F$
b. $2 F / 3$
c. $3 F / 5$
d. $5 F / 6$

Ajay Singhal

Numerade Educator

Problem 63

A $60 \mathrm{~kg}$ man stands on a spring scale in a lift. At some instant, he finds that the scale reading has changed from $60 \mathrm{~kg}$ to $50 \mathrm{~kg}$ for a while and then comes back to original mark. What should be concluded?
a. The lift was in constant motion upwards.
b. The lift was in constant motion downwards.
c. The lift while in downward motion suddenly stopped.
d. The lift while in upward motion suddenly stopped.

Ajay Singhal

Numerade Educator

Problem 64

The Fig. represents a light inextensible string $A B C D E$ in which $A B=B C=C D=D E$ and to which are attached masses $M, m$ and $M$ at the points $B, C$ and $D$, respectively. The system hangs freely in equilibrium with ends $A$ and $E$ of the string fixed in the same horizontal line (see Fig. 7.345), It is given that tan $\alpha=3 / 4$ and $\tan \beta=12 / 5$. Then the tension in the string $B C$ is
a. $2 \mathrm{mg}$
b. $(13 / 10) \mathrm{mg}$
c. $(3 / 10) \mathrm{mg}$
d. $(20 / 11) \mathrm{mg}$

Ajay Singhal

Numerade Educator

Problem 65

A monkey of mass $40 \mathrm{~kg}$ climbs on a massless rope of breaking strength $600 \mathrm{~N}$. The rope will break if the monkey
a. Climbs up with a uniform speed of $5 \mathrm{~m} / \mathrm{s}$.
b. Climbs up with an acceleration of $6 \mathrm{~m} / \mathrm{s}^{2}$.
c. Climbs down with an acceleration of $4 \mathrm{~m} / \mathrm{s}^{2}$.
d. Climbs down with a uniform speed of $5 \mathrm{~m} / \mathrm{s}$.

Ajay Singhal

Numerade Educator

Problem 66

Three light strings are connected at the point $P .$ A weight $W$ is suspended from one of the strings. End $A$ of string AP and end $B$ of string $P B$ are fixed as shown. In equilibrium $P B$ is horizontal and $P A$ makes an angle of $60^{\circ}$ with the horizontal. If the tension in $P B$ is $30 \mathrm{~N}$ then the tension in $P A$ and weight $W$ are respectively given by
a. $60 \mathrm{~N} ; 30 \mathrm{~N}$
b. $60 / \sqrt{3} \mathrm{~N} ; 30 / \sqrt{3} \mathrm{~N}$
c. $60 \mathrm{~N} ; 30 \sqrt{3} \mathrm{~N}$
d. $60 \sqrt{3} \mathrm{~N}: 30 \sqrt{3} \mathrm{~N}$

Ajay Singhal

Numerade Educator

Problem 67

Two persons are holding a rope of negligible weight tightly at its ends so that it is horizontal. A $15 \mathrm{~kg}$ weight in attached to rope at the mid point which now no more remains horizontal. The minimum tension required to completely straighten the rope is
a. $150 \mathrm{~N}$
b. $75 \mathrm{~N}$
c. $50 \mathrm{~N}$
d. infinitely large

Ajay Singhal

Numerade Educator

Problem 68

A balloon of fixed volume containing mass $M$ is coming down with an acceleration of a towards earth. How much mass should be released from the balloon so that it starts rising with acceleration $a$.
a. $\frac{2 M a}{g-a}$
b. $\frac{M a}{g+a}$
c. $\frac{M a}{g-a}$
d. $\frac{2 M a}{g+a}$

Ajay Singhal

Numerade Educator

Problem 69

If block is moving with an acceleration of $5 \mathrm{~m} / \mathrm{s}^{2}$ (see Fig. 7,346 ), the acceleration of $B$ w.r.t. ground is
a. $5 \mathrm{~m} / \mathrm{s}^{2}$
b. $5 \sqrt{2} \mathrm{~m} / \mathrm{s}^{2}$
c. $5 \sqrt{5} \mathrm{~m} / \mathrm{s}^{2}$
d. $10 \mathrm{~m} / \mathrm{s}^{2}$

Ajay Singhal

Numerade Educator

Problem 70

Two particles $A$ and $B$, each of mass $m$, are kept stationary by applying a horizontal force $F=m g$ on particle $B$ as shown in Fig. 7.347. Then $\beta \alpha$
a. $2 \tan \beta=\tan \alpha$
b. $2 T_{1}=5 T_{2}$
c. $T_{1} \sqrt{2}=T_{2} \sqrt{5}$
d. None of these

Ajay Singhal

Numerade Educator

Problem 71

A block placed on a horizontal surface is being pushed by a force $F$ making an angle $\theta$ with the vertical. The coefficient of friction between block and surface is $\mu$. The force required to slide the block with uniform velocity on the floor is
a. $\frac{\mu m g}{(\sin \theta-\mu \cos \theta)}$
b. $\frac{(\sin \theta-\mu \cos \theta)}{\mu m g}$
c. $\mu m g$
d. None of these

Ajay Singhal

Numerade Educator

Problem 72

A block slides with velocity of $10 \mathrm{~m} / \mathrm{s}$ on a rough horizontal surface. It comes to rest after covering a distance of $50 \mathrm{~m}$. If $g$ is $10 \mathrm{~m} / \mathrm{s}^{2}$, then the coefficient of dynamic friction between the block and the surface is
a. $0.1$
b. 1
c. 10
d. 5

Ajay Singhal

Numerade Educator

Problem 73

A block of mass $1 \mathrm{~kg}$ is at rest on a horizontal table. The coefficient of static friction between the block and the table is $0.50$. If $g=10 \mathrm{~ms}^{-2}$, then the magnitude of a force acting upwards at an angle of $60^{\circ}$ from the horizontal that will just start the block moving is:
a. $5 \mathrm{~N}$
b. $5.36 \mathrm{~N}$
c. $74.6 \mathrm{~N}$
d. $10 \mathrm{~N}$

Ajay Singhal

Numerade Educator

Problem 74

A heavy uniform chain lies on a horizontal table top. If the coefficient of friction between the chain and the table surface is $0.25$, then the maximum fraction of the length of the chain, that can hang over one edge of the table is
a. $20 \%$
b. $25 \%$
c. $35 \%$
d. $15 \%$

Narayan Hari

Numerade Educator

Problem 75

In the Fig. $7.348$, a block of weight $60 \mathrm{~N}$ is placed on rough surface. The coefficient of friction between the block and the surfaces is $0.5$. What should be the weight $W$ such that the block does not slip on the surface?
a. $60 \mathrm{~N}$
b. $\frac{60}{\sqrt{2}} \mathrm{~N}$
c. $30 \mathrm{~N}$
d. $\frac{30}{\sqrt{2}} \mathrm{~N}$

Ajay Singhal

Numerade Educator

Problem 76

A suitcase is gently dropped on a conveyor belt moving at a velocity of $3 \mathrm{~m} / \mathrm{s}$. If the coefficient of friction between the belt and the suitcase is $0.5$, find the displacement of the suitcase relative to conveyor belt before the slipping between the two is stopped $\left(g=10 \mathrm{~m} / \mathrm{s}^{2}\right)$
a. $2.7 \mathrm{~m}$
b. $1.8 \mathrm{~m}$
c. $0.9 \mathrm{~m}$
d. $1.2 \mathrm{~m}$

Ajay Singhal

Numerade Educator

Problem 77

A horizontal force of $10 \mathrm{~N}$ is necessary to just hold a block stationary against a wall. The coefficient of friction between the block and the wall is $0.2$ (Fig. 7.349). The weight of the block is
a. $2 \mathrm{~N}$
b. $20 \mathrm{~N}$
c. $50 \mathrm{~N}$
d. $100 \mathrm{~N}$

Dheeraj Sharma

Numerade Educator

Problem 78

Two blocks of mass $M_{1}$ and $M_{2}$ are connected with a string which passes over a smooth pulley. The mass $\mathrm{M}_{1}$ is placed on a rough incline plane as shown in the Fig. $7.350$. The coefficient of friction between the block and the inclined plane is $\mu$. What should be the minimum mass $M_{2}$ so that the block $M_{1}$ shides upwards?
a. $M_{2}=M_{1}(\sin \theta+\mu \cos \theta)$
b. $M_{2}=M_{1}(\sin \theta-\mu \cos \theta)$
c. $M_{2}=\frac{M_{1}}{\sin \theta+\mu \cos \theta}$
d. $M_{2}=\frac{M_{1}}{\sin \theta-\mu \cos \theta}$

Ajay Singhal

Numerade Educator

Problem 79

A box of mass $8 \mathrm{~kg}$ is placed on a rough inclined plane of inclination $\theta$. Its downward motion can be prevented by applying an upward pull $F$ and it can be made to slide upwards by applying a force $2 F$. The coefficient of friction between the box and the inclined plane is
a. $(\tan \theta) / 3$
1. $3 \tan \theta$
c. $(\tan \theta) / 2$
d. $2 \tan \theta$

Ajay Singhal

Numerade Educator

Problem 80

A block of mass $15 \mathrm{~kg}$ is resting on a rough inclined plane as shown in Fig. 7.351. The block is tied by a horizontal string which has a tension of $50 \mathrm{~N}$. The coefficient of friction between the surfaces of contact is
a. $1 / 2$
b. $2 / 3$
c. $3 / 4$
d. $1 / 4$

Ajay Singhal

Numerade Educator

Problem 81

A horizontal force, just sufficient to move a body of mass $4 \mathrm{~kg}$ lying on a rough horizontal surface, is applied on it. The coefficient of static and kinetic friction between the body and the surface are $0.8$ and $0.6$, respectively. If the force continues to act even after the block has started moving, the acceleration of the block in $\mathrm{m} / \mathrm{s}^{2}$ is $(g=10$ $\mathrm{m} / \mathrm{s}^{2}$ )
a. $1 / 4$
b. $1 / 2$
c. 2
d. 4

Ajay Singhal

Numerade Educator

Problem 82

Blocks $A$ and $B$ in the Fig. $7.352$ are connected by a bar of negligible weight. Mass of each block is $170 \mathrm{~kg}$ and $\mu_{A}$ $=0.2$ and $\mu_{B}=0.4$, where $\mu_{A}$ and $\mu_{B}$ are the coefficients of limiting friction between blocks and plane. Calculate the force developed in the bar $\left(g=10 \mathrm{~m} / \mathrm{sec}^{2}\right)$.
a. $150 \mathrm{~N}$
b. $75 \mathrm{~N}$
c. $200 \mathrm{~N}$
d. $250 \mathrm{~N}$

Dheeraj Sharma

Numerade Educator

Problem 83

The upper half of an inclined plane with inclination $\phi$ is perfectly smooth while the lower half is rough. A body starting from rest at the top will again come to rest at the bottom if the coefficient of friction for the lower half is given by
a. $2 \tan \phi$
b. $\tan \phi$
c. $2 \sin \phi$
d. $2 \cos \phi$

Dheeraj Sharma

Numerade Educator

Problem 84

A block of mass $m$ is placed on another block of mass $M$ which itself is lying on a horizontal surface (see Fig. 7.353). The coefficient of friction between two block is $\mu_{1}$ and that between the block of block $M$ and horizontal surface is $\mu_{2}$. What maximum horizontal force can be applied to the lower block so that the two blocks move without separation?
$\mathbf{a}_{-}(M+m)\left(\mu_{2}-\mu_{1}\right) g$
b. $(M-m)\left(\mu_{2}-\mu_{1}\right) g$
c. $(M-m)\left(\mu_{2}+\mu_{1}\right) g$
d. $(M+m)\left(\mu_{2}+\mu_{1}\right) g$

Ajay Singhal

Numerade Educator

Problem 85

Two blocks $A$ and $B$ of masses $6 \mathrm{~kg}$ and $3 \mathrm{~kg}$ rest on a smooth horizontal surface as shown in the Fig. $7.354$. If coefficient of friction between $A$ and $B$ is $0.4$, the maximum horizontal force which can make them without separation is
a. $72 \mathrm{~N}$
b. $40 \mathrm{~N}$
c. $36 \mathrm{~N}$
d. $20 \mathrm{~N}$

Ajay Singhal

Numerade Educator

Problem 86

Two blocks of masses $M_{1}$ and $M_{2}$ are connected with a string passing over a pulley as shown in Fig. $7.355$. The block $M_{1}$ lies on a horizontal surface. The coefficient of friction between the block $M_{1}$ and the horizontal surface is $\mu$. The system accelerates. What additional mass $m$ should be placed on the block $M_{1}$ so that the system does not accelerate?
a. $\frac{M_{2}-M_{1}}{\mu}$
b. $\frac{M_{2}}{\mu}-M_{1}$
c. $M_{2}-\frac{M_{1}}{\mu}$
d. $\left(M_{2}-M_{1}\right) \mu$

Ajay Singhal

Numerade Educator

Problem 87

A block of mass $m$ is placed on the top of another block of mass $M$ as shown in the Fig. 7.356. The coefficient of friction between them is $\mu .$ What is the maximum acceleration with which the block $M$ may move so that $m$ also moves along with it?
a. $\mu \mathrm{g}$
b. $\mathrm{g} / \mu$
c. $\mu^{2} / \mathrm{g}$
d. $\mathrm{g} / \mu^{2}$

Ajay Singhal

Numerade Educator

Problem 88

The system is pushed by a force $F$ as shown in Fig. $7.357$. All surfaces are smooth except between $B$ and $C$. Friction coefficient between $B$ and $C$ is $\mu .$ Minimum value of $F$ to prevent block $B$ from down ward slipping is
a. $\left(\frac{3}{2 \mu}\right) m g$
b. $\left(\frac{5}{2 \mu}\right) m g$
c. $\left(\frac{5}{2}\right) \mu m g$
d. $\left(\frac{3}{2}\right) \mu m g$

Ajay Singhal

Numerade Educator

Problem 89

A body of mass $M$ is resting on a rough horizontal plane surface, the coefficient of friction being equal to $\mu .$ At $t=0$ a horizontal force $F=F_{0} t$ starts acting on it, where $F_{0}$ is a constant. Find the time $T$ at which the motion starts?
a. $\mu M g / F_{0}$
b. $M g / \mu F_{0}$
c. $\mu F_{0} / M g$
d. None of these

Ajay Singhal

Numerade Educator

Problem 90

The maximum value of mass of block $C$ so that neither $A$ nor $B$ moves is (see Fig. $7.358$ ) (Given that mass of $A$ is $100 \mathrm{~kg}$ and that of $B$ is $140 \mathrm{~kg} .$ Pulleys are smooth and friction coefficient between $A$ and $B$ and between $B$ and horizontal surface is $\mu=0.3 .$ ) Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$.
a. $210 \mathrm{~kg}$
b. $190 \mathrm{~kg}$
c. $185 \mathrm{~kg}$
d. $162 \mathrm{~kg}$

Prem Bijarniya

Numerade Educator

Problem 91

A block $A$ of mass $2 \mathrm{~kg}$ is placed over another block $B$ of mass $4 \mathrm{~kg}$ which is placed over a smooth horizontal ffoor. The coefficient of friction between $A$ and $B$ is $0.4$. When a horizontal force of magnitude $10 \mathrm{~N}$ is applied on $A$, the acceleration of blocks $A$ and $B$ are
a. $1 \mathrm{~ms}^{-2}$ and $2 \mathrm{~ms}^{-2}$, respectively.
b. $5 \mathrm{~ms}^{-2}$ and $2.5 \mathrm{~ms}^{-2}$, respectively.
c. Both the blocks will moves together with acceleration $1 / 3 \mathrm{~ms}^{-2}$
d. Both the blocks will move together with acceleration $5 / 3 \mathrm{~ms}^{-2}$

Ajay Singhal

Numerade Educator

Problem 92

Two blocks $m$ and $M$ tied together with an inextensible string are placed at rest on a rough horizontal surface with coefficient of friction $\mu$. The block $m$ is pulled with a variable force $F$ at a varying angle $\theta$ with the horizontal. The value of $\theta$ at which the least value of $F$ is required to move the blocks is given by
a. $\theta=\tan ^{-1} \mu$
b. $\theta>\tan ^{-1} \mu$
c. $\theta<\tan ^{-1} \mu$
d. Insufficient data

Ajay Singhal

Numerade Educator

Problem 93

A trolley $A$ has a simple pendulum suspended from a frame fixed to its desk. A block $B$ is in contact on its vertical slide. The trolley is on horizontal rails and accelerates towards the right such that the block is just prevented from falling. The value of coefficient of friction between $A$ and $B$ is $0.5$ (see Fig. 7.361). The inclination of the pendulum to the vertical is
a. $\tan ^{-1}\left(\frac{1}{2}\right)$
b. $\tan ^{-1}(3)$
c. $\tan ^{-1}(\sqrt{2})$
d. $\tan ^{-1}(2)$

Ajay Singhal

Numerade Educator

Problem 94

Two block $M$ and $m$ are arranged as shown in the Fig. 7.362. The coefficient of friction between the blocks $\mu_{1}=0.25$ and between the ground and $M$ be $\mu_{2}=\frac{1}{3}$. If $M=8 \mathrm{~kg}$ then find the value of $m$ so that the system will remain at rest.
a. $\frac{4}{3} \mathrm{~kg}$
b. $\frac{8}{9} \mathrm{~kg}$
c. $1 \mathrm{~kg}$
d. $\frac{8}{5} \mathrm{~kg}$

Prem Bijarniya

Numerade Educator

Problem 95

Find the minimum force required to pull the lower block. If the coefficient of friction between the blocks is $0.1$ and between the ground and $2 \mathrm{~kg}$ block is $0.2$.
a. $1 \mathrm{~N}$
b. $5 \mathrm{~N}$
c. $7 \mathrm{~N}$
d. $10 \mathrm{~N}$

Ajay Singhal

Numerade Educator

Problem 96

A body of mass $m$ is launched up on a rough inclined plane making an angle $45^{\circ}$ with horizontal. If the time of ascent is half of the time of descent, the frictional coefficient between plane and body is
a. $\frac{2}{5}$
b. $\frac{3}{5}$
c. $\frac{3}{4}$
d. $\frac{4}{5}$

Ajay Singhal

Numerade Educator

Problem 97

A wooden block of mass $M$ resting on a rough horizontal floor is pulled with a force $F$ at an angle $\phi$ with the horizontal. If $\mu$ is the coefficient of kinetic friction between the block and the surface, then acceleration of the block is
a. $\frac{F}{M}(\cos \phi-\mu \sin \phi)-\mu g$
b. $\frac{\mu F}{M} \cos \phi$
c. $\frac{F}{M}(\cos \phi+\mu \sin \phi)-\mu g$
d. $\frac{F}{M} \sin \phi$

Ajay Singhal

Numerade Educator

Problem 98

A given object takes $n$ times more time to slide down $45^{\circ}$ rough inclined plane as it takes to slide down a perfectly smooth $45^{\circ}$ incline. The coefficient of kinetic friction between the object and the incline is
a. $\sqrt{\frac{1}{1-n^{2}}}$
b. $\sqrt{1-\frac{1}{n^{2}}}$
c. $1-\frac{1}{n^{2}}$
d. $\frac{1}{2-n^{2}}$

Ajay Singhal

Numerade Educator

Problem 99

A passenger is travelling in a.train which is moving at 40 $\mathrm{m} / \mathrm{s}$. His suitcase is kept on the berth. The driver of the train applies breaks such that the speed of the train decreases at a constant rate to $20 \mathrm{~m} / \mathrm{s}$ in $5 \mathrm{~s}$. What should be the minimum coefficient of friction between the suitcase and the berth if the suitcase is not to slide during retardation of the train?
a. $0.3$
b. $0.5$
c. $0.1$
d. $0.2$

Ajay Singhal

Numerade Educator

Problem 100

Starting from rest, a body slides down a $45^{\circ}$ inclined plane in twice the time it takes to slide the same distance in the absence of friction. What is the coefficient of friction between the body and the inclined plane?
a. $\sqrt{3} / 2$
b. $3 / 4$
c. $1 / 2$
d. $1 / 4$

Ajay Singhal

Numerade Educator

Problem 101

The tension in rope (rope is light)
a. $(M+m) g \sin \theta$
b. $(M+m) g \sin \theta-\mu m g \cos \theta$
c. zero
d. $(M+m) g \cos \theta$

Ajay Singhal

Numerade Educator

Problem 102

There is a chain of length $6 \mathrm{~m}$ and coefficient of friction $\frac{1}{2}$. What will be the maximum length of chain which can be held outside of table without sliding
a. $2 \mathrm{~m}$
b. $4 \mathrm{~m}$
c. $3 \mathrm{~m}$
d. $1 \mathrm{~m}$

Ajay Singhal

Numerade Educator

Problem 103

A given object takes $n$ times more time to slide down $45^{\circ}$ rough inclined plane as it takes to slide down a perfectly smooth $45^{\circ}$ incline. The coefficient of kinetic friction between the object and the incline is
a. $\frac{1}{2-n^{2}}$
b. $1-\frac{1}{n^{2}}$
c. $\sqrt{1-\frac{1}{n^{2}}}$
d. $\sqrt{\frac{1}{1-n^{2}}}$

Ajay Singhal

Numerade Educator

Problem 104

A block of mass $m$ is at rest with respect to a rough incline kept in clevator moving up with acceleration $a$ (Fig. 7.364). Which of following statement is correct?
a. The contact force between block and incline is parallel to the incline.
b. The contact force between block and incline is of magnitude $m(g+a)$
c. The contact force between block and incline is perpendicular to the incline.
d. The contact force is of magnitude $m g \cos \theta$.

Ajay Singhal

Numerade Educator

Problem 105

A block of mass $5 \mathrm{~kg}$ is at rest on a rough inclined plane as shown in the Fig. 7.365. The magnitude of net force exerted by the surface on the block will be
a. $25 \mathrm{~N}$
b. $50 \mathrm{~N}$
c. $10 \mathrm{~N}$
d. $30 \mathrm{~N}$

Ajay Singhal

Numerade Educator

Problem 106

A box of mass $8 \mathrm{~kg}$ is placed on a rough inclined plane of inclination $45^{\circ}$. Its downward motion can be prevented by applying an upward pull $F$ and it can be made to slide upwards by applying a force $2 F$. The coefficient of friction between the box and the inclined plane is
a. $\frac{1}{2}$
b. $\frac{1}{\sqrt{2}}$
c. $\frac{1}{2 \sqrt{2}}$
d. $\frac{1}{3}$

Ajay Singhal

Numerade Educator

Problem 107

A block of mass $m=3 \mathrm{~kg}$ is placed on the top of another block of mass $M=5 \mathrm{~kg}$ as shown in the Fig. $7.366$. The coefficient of friction between them is $\mu=0.4$. What is the maximum acceleration with which the block $M$ may move so that $m$ also moves along with it? $(M$ is on frictionless surface.)
a. $2 \mathrm{~m} / \mathrm{s}^{2}$
b. $1 \mathrm{~m} / \mathrm{s}^{2}$
c. $3 \mathrm{~m} / \mathrm{s}^{2}$
d. $4 \mathrm{~m} / \mathrm{s}^{2}$

Ajay Singhal

Numerade Educator

Problem 108

Two blocks $A(2 \mathrm{~kg})$ and $B(5 \mathrm{~kg})$ rest one over the other on a smooth horizontal plane (see Fig. 7.367). The coefficient of static and dynamic friction between $A$ and $B$ is the same and is equal to $0.80 .$ The maximum horizontal force that can be applied to $B$ in order that both $A$ and $B$ do not have relative motion is
a. $1.2 \mathrm{~N}$
b. $42 \mathrm{~N}$
c. $4.2 \mathrm{~N}$
d. $56 \mathrm{~N}$.

Ajay Singhal

Numerade Educator

Problem 109

The minimum acceleration that must be imparted to the cart in the Fig. $7.368$ so that the block $A$ will not fall (given $\mu=0.5$ is the coefficient of friction between the surfaces of block and cart) is given by
a. $2 \mathrm{~m} / \mathrm{s}^{2}$
b. $20 \mathrm{~m} / \mathrm{s}^{2}$
c. $5 \mathrm{~m} / \mathrm{s}^{2}$
d. $7.5 \mathrm{~m} / \mathrm{s}^{2}$

Ajay Singhal

Numerade Educator

Problem 110

A block of mass $m$, lying on a horizontal plane, is acted upon by a horizontal force $P$ and another force $Q$, inclined at an angle $\theta$ to the vertical. The block will remain in equilibrium, if the coefficient of friction between it and the surface is
a. $(P \sin \theta-Q) /(m g-\cos \theta)$
b. $(P-Q \sin \theta) /(m g+Q \sin \theta)$
c. $(P \cos \theta+Q) /(m g-Q \sin \theta)$
d. $(P+Q \sin \theta) /(m g+Q \cos \theta)$

Ajay Singhal

Numerade Educator

Problem 111

A horizontal force of $25 \mathrm{~N}$ is necessary to just hold a block stationary against a wall the coefficient of friction between the block and the wall is $0.4$. The weight of the block is
a. $2.5 \mathrm{~N}$
b. $20 \mathrm{~N}$
c. $10 \mathrm{~N}$
d. $5 \mathrm{~N}$

Ajay Singhal

Numerade Educator

Problem 112

A solid block of mass $2 \mathrm{~kg}$ is resting inside a cube as shown in Fig. 7.371. The cube is moving with a velocity $v=5 \hat{i}+2 \hat{j} \mathrm{~m} / \mathrm{s}$. If the coefficient of friction between the surface of cube and block is $0.2$. Then the force of friction between the block and cube is
a. $10 \mathrm{~N}$
b. $4 \mathrm{~N}$
c. $14 \mathrm{~N}$
d. 0

Ajay Singhal

Numerade Educator

Problem 113

A block of metal weighing $2 \mathrm{~kg}$ is resting on a frictionless plane. It is struck by a jet releasing water at a rate of $1 \mathrm{~kg} / \mathrm{s}$ and at a speed of $5 \mathrm{~m} / \mathrm{s}$. The initial acceleration of the block is
a. $\frac{5}{3} \mathrm{~m} / \mathrm{s}^{2}$
b. $\frac{25}{4} \mathrm{~m} / \mathrm{s}^{2}$
c. $\frac{25}{6} \mathrm{~m} / \mathrm{s}^{2}$
d. $\frac{5}{2} \mathrm{~m} / \mathrm{s}^{2}$

Ajay Singhal

Numerade Educator

Problem 114

Springs of spring cosnstant $K, 3 K, 9 K, 27 K, \cdots, \infty$ are connected in series. Equivalent spring constant of the combination is
a. $\frac{3 K}{2}$
b. $\frac{K}{2}$
c. $\frac{2 K}{3}$
d. $\infty$

Ajay Singhal

Numerade Educator

Problem 115

A block of mass $m$ is lying on a wedge having inclination angle $\alpha=\tan ^{-1}\left(\frac{1}{5}\right)$. Wedge is moving with a constant acceleration $a=2 \mathrm{~m} / \mathrm{s}^{2} .$ The minimum value of coefficient of friction $\mu$, so that $m$ remains stationary wr.t. to wedge is
a. $2 / 9$
b. $5 / 12$
c. $1 / 5$
d. $2 / 5$

Ajay Singhal

Numerade Educator

Problem 116

In the given Fig. $7.374$ the blocks are at rest and a force of $10 \mathrm{~N}$ acts on the block of $4 \mathrm{~kg}$ mass. The coefficient of static friction and the coefficient of kinetic friction are $\mu_{3}=0.2$ and $\mu_{k}=0.15$ for both the surfaces in contact. The magnitude of friction force acting between the surface of contact between the $2 \mathrm{~kg}$ and $4 \mathrm{~kg}$ block in this situation $\mathrm{s}$
a. $3 \mathrm{~N}$
b. $4 \mathrm{~N}$
c. $3.33 \mathrm{~N}$
d. 0

Ajay Singhal

Numerade Educator

Problem 117

An ideal liquid of density $\rho$ is pushed with velocity $v$ through the central limb of the tube shown in the Fig. $7.375$. What force does the liquid exert on the tube? The cross-sectional areas of the three limbs are equal to A each. Assume stream-line flow.
a. $\frac{9}{8} \rho A v^{2}$
b. $\frac{5}{4} \rho A v^{2}$
c. $\frac{3}{2} \rho A v^{2}$
d. $\rho A v^{2}$

Ajay Singhal

Numerade Educator

Problem 118

Two uniform solid cylinders $A$ and $B$ each of mass $1 \mathrm{~kg}$ are connected by a spring of constant $200 \mathrm{~N} / \mathrm{m}$ at their axles and are placed on a fixed wedge as shown in the Fig. 7.376. The coefficient of friction between the wedge and the cylinders is $0.2$. The angle made by the line $A B$ with the horizontal, in equilibrium, is
a. $0^{\circ}$
b. $15^{\circ}$
c. $30^{\circ}$
d. None of these

Prem Bijarniya

Numerade Educator

Problem 119

Velocity of point $A$ on the rod is $2 \mathrm{~m} / \mathrm{s}$ (leftward) at the instant shown in the Fig. 7.377. The velocity of the point $B$ on the rod at this instant is
a. $\frac{2}{\sqrt{3}} \mathrm{~m} / \mathrm{s}$
b. $1 \mathrm{~m} / \mathrm{s}$
c. $\frac{1}{2 \sqrt{3}} \mathrm{~m} / \mathrm{s}$
d. $\frac{\sqrt{3}}{2} \mathrm{~m} / \mathrm{s}$

Ajay Singhal

Numerade Educator

Problem 120

The masses of the blocks $A$ and $B$ are $m$ and $M .$ Between $A$ and $B$ there is a constant frictional force $F$, and $B$ can slide frictionlessly on horizontal surface (see Fig. $7.378$ ). $A$ is set in motion with velocity while $B$ is at rest. What is the distance moved by $A$ relative to $B$ before they move with the same velocity?
a. $\frac{m M v_{0}^{2}}{F(m-M)}$
b. $\frac{m M v_{0}^{2}}{2 F(m-M)}$
c. $\frac{m M v_{0}^{2}}{F(m+M)}$
d. $\frac{m M v_{0}^{2}}{2 F(m+M)}$

Prem Bijarniya

Numerade Educator

Problem 121

Two Small rings $O$ and $O^{\prime}$ are put on two vertical stationary rods $A B$ and $A^{\prime} B^{\prime}$, respectively (see Fig. 7.379). One end of an inextensible thread is tied at point $A^{\prime}$. The thread passes through ring $O^{\prime}$ and its other end is tied to ring $O$. Assuming that ring $O^{\prime}$ moves downwards at a constant velocity $v_{2}$ of the ring $O$, when $\angle A O O^{\prime}=\alpha$
a. $v_{1}\left[\frac{2 \sin ^{2} \alpha / 2}{\cos \alpha}\right]$
b. $v_{1}\left[\frac{2 \cos ^{2} \alpha / 2}{\sin \alpha}\right]$
c. $v_{1}\left[\frac{3 \cos ^{2} \alpha / 2}{\sin \alpha}\right]$
d. None of these

Narayan Hari

Numerade Educator

Problem 122

A fixed $U$ -shaped smooth wire has a semi-circular bending between $A$ and $B$ as shown in the Fig. $7.380 .$ A bead of mass $m$ moving with uniform speed $v$ through the wire enters the semicircular bend at $A$ and leaves at $B$. The average force exerted by the bead on the part $A B$ of the wire is
a. 0
b. $\frac{4 m v^{2}}{\pi d}$
c. $\frac{2 m v^{2}}{\pi d}$
d. None of these

Ajay Singhal

Numerade Educator

Problem 123

Find frictional force on block $30 \mathrm{~kg}$ (Fig. 7.381)
a. $20 \mathrm{~N}$
b. $30 \mathrm{~N}$
c. $40 \mathrm{~N}$
d. $50 \mathrm{~N}$

Ajay Singhal

Numerade Educator

Problem 124

Two identical particles $A$ and $B$, each of mass $m$, are interconnected by a spring of stiffness $k$. If the particle $B$ experiences a force $F$ and the elongation of the spring is $x$, the acceleration of particle $B$ relative to particle $A$ is equal to
a. $\frac{F}{2 m}$
b. $\frac{F-k x}{m}$
c. $\frac{F-2 k x}{m}$
d. $\frac{k x}{m}$

Ajay Singhal

Numerade Educator

Problem 125

The system shown in the Fig. $7.383$ is in equilibrium. Masses $m_{1}$ and $m_{2}$ are $2 \mathrm{~kg}$ and $8 \mathrm{~kg}$, respectively. Spring constants $k_{1}$ and $k_{2}$ are $50 \mathrm{~N} / \mathrm{m}$ and $70 \mathrm{~N} / \mathrm{m}$, respectively. If the compression in second spring is $0.5 \mathrm{~m}$. What is the compression in first spring? (Both springs have the same natural length.)
a. $1.3 \mathrm{~m}$
b. $-0.5 \mathrm{~m}$
c. $0.5 \mathrm{~m}$
d. $0.9 \mathrm{~m}$

Ajay Singhal

Numerade Educator

Problem 126

In Fig. $7.384$, the block of mass $M$ is at rest on the floor. The acceleration with which a boy of mass $m$ should climb along the rope of negligible mass so as to lift the block from the floor is
a. $\left(\frac{M}{m}-1\right) g$
b. $\left(\frac{M}{m}-1\right)$
c. $\frac{M}{m} g$
d. $>\frac{M}{m} g$

Ajay Singhal

Numerade Educator

Problem 127

Three blocks $A, B$, and $C$ are of equal mass $m$ and are placed one over other on a frictionless surface (table) as shown in the Fig. $7.385$. Coefficient of friction between any blocks $A, B$ and $C$ is $\mu$. The maximum value of mass of block $M_{D}$ so that the block $A, B$, and $C$ move without slipping over each other is
a. $\frac{3 m \mu}{\mu+1}$
b. $\frac{3 m(1-\mu)}{\mu}$
c. $\frac{3 m(1+\mu)}{\mu}$
d. $\frac{3 m \mu}{(1-\mu)}$

Narayan Hari

Numerade Educator

Problem 128

Two blocks of masses $0.2 \mathrm{~kg}$ and $0.5 \mathrm{~kg}$, which are placed $22 \mathrm{~m}$ apart on a rough horizontal surface $(\mu=0.5)$, are acted upon by two forces of magnitude $3 \mathrm{~N}$ each as shown in Fig. $7.386$ at time $t=0$. Then, the time $t$ at which they collide each other is
a. $\mathrm{sec}$
b. $\sqrt{2} \mathrm{sec}$
c. 2 sec
d. None

Ajay Singhal

Numerade Educator

Problem 129

In an arrangement shown below in Fig. $7.387$, the acceleration of block $A$ and $B$ are given
a. $g / 3, g / 6$
b: $g / 6, g / 3$
c. $g / 2, g / 2$
d. 0,0

Ajay Singhal

Numerade Educator

Problem 130

A block of mass $1 \mathrm{~kg}$ lying on the floor is subjected to a horizontal force given by $f=2 \sin \omega t .$ The coefficient of friction between the block and the floor is $0.25$.
a. Acceleration of the block is positive and uniform.
b. Acceleration of the block depend on value of $W$.
c. The block always remains at rest.
d. Acceleration of the block is always zero.

Ajay Singhal

Numerade Educator

Problem 131

A solid block of mass $2 \mathrm{~kg}$ is resting inside a cube as shown in Fig. $7.388 .$ The cube is moving with a velocity $v=5 \hat{i}+2 \hat{j} \mathrm{~m} / \mathrm{s}$. If the coefficient of friction between the surface of cube and block is $0.2$. Then the force of friction between the block and the cube is
a. $10 \mathrm{~N}$
b. $4 \mathrm{~N}$
c. $14 \mathrm{~N}$
d. 0

Ajay Singhal

Numerade Educator

Problem 132

A block of metal weighing $2 \mathrm{~kg}$ is resting on a frictionless plane. It is struck by a jet releasing water at a rate of 1 $\mathrm{kg} / \mathrm{s}$ and at a speed of $5 \mathrm{~m} / \mathrm{s}$. The initial acceleration of the block is
a. $\frac{5}{3} \mathrm{~m} / \mathrm{s}^{2}$
b. $\frac{25}{4} \mathrm{~m} / \mathrm{s}^{2}$
c. $\frac{25}{8} \mathrm{~m} / \mathrm{s}^{2}$
d. $\frac{5}{2} \mathrm{~m} / \mathrm{s}^{2}$

Ajay Singhal

Numerade Educator

Problem 133

All surfaces are smooth and pulleys are ideal. The string is pulled with force $F$, mass of $A=$ mass of $B=-m$.
a. $a_{A}=a_{B}$
b. $a_{A}=0, a_{B} \neq 2 a_{B}$

Prem Bijarniya

Numerade Educator

Problem 134

A block of mass $m_{1}$ lies on top of fixed wedge as shown in Fig. $7.391(\mathrm{a})$ and another block of mass $m_{2}$ lies on top of wedge which is free to move as shown in Fig. $7.391(\mathrm{~b}) .$ At time $t=0$, both the blocks are released from rest from a vertical height $h$ above the respective horizontal surface on which the wedge is placed as shown. There is no friction between the block and the wedge in both the figures. Let $T_{1}$ and $T_{2}$ be the time taken by block in Fig. 7.391(a) and block in Fig. $7.391(b)$ respectively to just reach the horizontal surface, then
a. $T_{1}>T_{2}$
b. $T_{1}<T_{2}$
c. $T_{1}=T_{2}$
d. Data insufficient

Ajay Singhal

Numerade Educator

Problem 135

In the situation shown in Fig. $7.392$ all the string are light and inextensible and pullies are light. There is no friction at any surface and all block are of cuboidal shape. $\mathrm{A}$ horizontal force of magnitude $F$ is applied to right most free end of string in both the cases of Fig. $7.392(\mathrm{a})$ and Fig. $7.392(b)$ as shown. At the instant shown, the tension in all strings are non zero. Let the magnitude of the acceleration of large blocks (of mass $M$ ) in Fig. 7.392(a) and Fig. 7.392(b) are $a_{1}$ and $a_{2}$, respectively. Then
a. $a_{1}=a_{2} \neq 0$
b. $a_{1}=a_{2}=0$
c. $a_{1}>a_{2}$
d. $a_{1}<a_{2}$

Ajay Singhal

Numerade Educator

Problem 136

In Fig. $7.393$, a person wants to raise a block lying on the ground to a height $h$. In both the cases if time required is same then in which case he has to exert more force. Assume pulleys and stings lights.
a. (i)
b. (ii)
c. Same in both
d. Cannot be determined

Ajay Singhal

Numerade Educator

Problem 137

A chain of length $L$ is placed on a horizontal surface as shown in Fig. 7.394. At any instant $x$, the length of chain on rough surface and the remaining portion lies on smooth surface. Initially, $x=0$. A horizontal force $P$ is applied to the chain. In the duration $x$ changes from $x=0$ to $x=L$ for chain to move with a constant speed.
a. The magnitude of $P$ should increase with time.
b. The magnitude of $P$ should decrease with time.
c. The magnitude of $P$ should increase first and then decrease with time.
d. The magnitude of $P$ should decrease first and then increase with time.

Ajay Singhal

Numerade Educator

Problem 138

A $1.5 \mathrm{~kg}$ box is initially at rest on a horizontal surface when at $t=0$ a horizontal force $\vec{F}=(1.8 t) \hat{i} \mathrm{~N}$ (with $t$ in seconds), is applied to the box. The acceleration of the box as a function of time $t$ is given by $\vec{a}=0 \quad$ for $\quad 0 \leq t \leq 2.85$
$\vec{a}=(1.2 t-2.4) \hat{i} \mathrm{~m} / \mathrm{s}^{2} \quad$ for $\quad t>2.85$
The coefficient of kinetic friction between the box and the surface is
a. $0.12$
b. $0.24$
c. $0.36$
d. $0.48$

Ajay Singhal

Numerade Educator

Problem 139

A vehicle is moving with a velocity $v$ ou a curved road of width $b$ and radius of curvature $R$. For counteracting the centrifugal force on the vehicle, the difference in elevation required in between the outer and inner edges of the rod is
a. $v^{2} b / R g$
b. $v b / R g$
c. $v b^{2} / R g$
d. $v b / R^{2} g$

Ajay Singhal

Numerade Educator

Problem 140

A circular road of radius $1,000 \mathrm{~m}$ has a banking angle of $45^{\circ}$. What will be the maximum safe speed (in $\mathrm{m} / \mathrm{s}$ ) of a car whose mass is $2,000 \mathrm{~kg}$ and the coefficient of friction between the tyre and the road is $0.5$
a. 172
b. 124
c. 99
d. 86

Ajay Singhal

Numerade Educator

Problem 141

A circular table of radius $0.5 \mathrm{~m}$ has a smooth diametrical groove. A ball of mass $90 \mathrm{~g}$ is placed inside the groove along with a spring of spring constant $10^{2} \mathrm{~N} / \mathrm{cm} .$ One end of the spring is tied to the edge of the table and the other end to the ball. The ball is at a distance of $0.1 \mathrm{~m}$ from the centre when the table is at rest. On rotating the table with a constant angular frequency of $10^{2} \mathrm{rad}-\mathrm{s}^{-1}$, the ball moves away from the centre by a distance nearly equal to
a. $10^{-1} \mathrm{~m}$
b. $10^{-2} \mathrm{~m}$
c. $10^{-3} \mathrm{~m}$
d. $2 \times 10^{-1} \mathrm{~m}$

Ajay Singhal

Numerade Educator

Problem 142

A coin is placed at the edge of a horizontal disc rotating about a vertical axis through its axis with a uniform angular speed $2 \mathrm{rad} / \mathrm{s}$. The radius of the disc is $50 \mathrm{~cm}$. Find the minimum coefficient of friction between disc and coin so that the coin ?oes not slip $\left(g=10 \mathrm{~ms}^{-2}\right)$.
a. $0.1$
b. $0.2$
c. $0.3$
d. $0.4$

Ajay Singhal

Numerade Educator

Problem 143

Mark out the incorrect statement
a. Second law of motion is a local relation, i.e., if $\vec{a}$ at a point charges at any time $t$, then $\vec{F}$ has to change at the same point at same time $t$.
b. In Newton's third law, action and reaction start acting at the same instant.
c. If pseudo force acting on an object in a non-inertial frame is $\vec{F}$, then its reaction on inertial frame is $-\vec{F}$.

Ajay Singhal

Numerade Educator

Problem 144

Friction force can be reduced to a great extent by
a. Lubricating the two moving parts.
b. Using ball bearings between two moving parts.
c. Introducing a thin cushion of air maintained between two relatively moving surfaces.
d. All of the above.

Ajay Singhal

Numerade Educator

Problem 145

An intersteller spacecraft far away from the influence of any star or planet is moving at high speed under the influence of fusion rockets (due to thrust exerted by fusion rockets, the spacecraft is accelerating). Suddenly the engine malfunctions and stops. The spacecraft will
a. immediately stops, throwing all of the occupants to the front
b. begins slowing down and eventually comes to rest
c. keep moving at constant speed for a while, and then. begins to slow down
d. keeps moving forever with constant speed

Ajay Singhal

Numerade Educator

Problem 146

Three arrangement of a light spring balance are shown in the Fig. $7.396,7.397$, and $7.398$ below. The readings of the spring scales in three arrangements are, respectively
a. $20 \mathrm{~g}, 20 \mathrm{~g}, 10 \mathrm{~g}$
b. $20 \mathrm{~g}, 20 \mathrm{~g}, \frac{40 \mathrm{~g}}{3}$
c. zero, $20 \mathrm{~g}, 10 \mathrm{~g}$
d. zero, $20 \mathrm{~g}, \frac{40 \mathrm{~g}}{3}$

Ajay Singhal

Numerade Educator

Problem 147

Mark out the most appropriate statement.
a. The normal force is the same thing as the weight.
b. The normal force is different from the weight, but always has the same magnitude.
c. The normal force is different from the weight, but the two form an action-reaction pair according to the Newton's third law.
d. The normal force is different from the weight, but the two may have same magnitude in certain cases.

Ajay Singhal

Numerade Educator

Problem 148

A system of two blocks, a light string and a light and frictionless pulley is arranged as shown in Fig. $7.399$. The coefficient of friction between fixed incline and $10 \mathrm{~kg}$ block is given by $\mu_{s}=0.25$ and $\mu_{k}=0.20 .$ If the system is released from rest, then find the acceleration of $10 \mathrm{~kg}$ block?
a. 0 .
b. $0.114 \mathrm{~m} / \mathrm{s}^{2}$
c. $0.228 \mathrm{~m} / \mathrm{s}^{2}$
d. $2.97 \mathrm{~m} / \mathrm{s}^{2}$

Prem Bijarniya

Numerade Educator

Problem 149

A wooden box is placed on a table. The normal force on the box from the table is $N_{1}$. Now another identical box is kept on first box and the normal force on lower block due to upper block is $N_{2}$ and normal force on lower block by the table is $N_{3}$. For this situation mark out the correct statement(s)
a. $N_{1}=N_{2}=N_{3}$
b. $N_{1}<N_{2}=N_{3}$
c. $N_{1}=N_{2}<N_{3}$
d. $N_{1}=N_{2}>N_{3}$

Ajay Singhal

Numerade Educator

Problem 150

Consider a 14-tyre truck, whose only rear 8 wheels are power driven (means only these 8 wheels can produce acceleration). These 8 wheels are supporting approximately half of the entire load. If coefficient of friction between rod and each of the tyres is $0.6$, then what could be the maximum attainable acceleration by this truck?
a. $6 \mathrm{~m} / \mathrm{s}^{2}$
b. $24 \mathrm{~m} / \mathrm{s}^{2}$
c. $3 \mathrm{~m} / \mathrm{s}^{2}$
d. $10 \mathrm{~m} / \mathrm{s}^{2}$

Ajay Singhal

Numerade Educator

Problem 151

A house is built on the top of a hill with $45^{\circ}$ slope. Due to sliding of the material and the sand from the top to the bottom of the hill the slope angle has been reduced. If the coefficient of static friction between the sand particles is $0.75$, what is the final angle attained by the hili? $\left[\tan ^{-1}\left(0.75 \simeq 37^{\circ}\right)\right]$
a. $8^{\circ}$
b. $45^{\circ}$
c. $37^{\circ}$
d. $30^{\circ}$

Ajay Singhal

Numerade Educator

Problem 152

A block of mass $4 \mathrm{~kg}$ is pressed against the wall by a force of $80 \mathrm{~N}$ as shown in the Fig. $7.401$. Determine the value of friction force and block's acceleration, (Take $\mu_{s}=0.2$, $\mu_{k}=0.15$ ).
a. $8 \mathrm{~N}, 0 \mathrm{~m} / \mathrm{s}^{2}$
b. $32 \mathrm{~N}, 6 \mathrm{~m} / \mathrm{s}^{2}$
c. $8 \mathrm{~N}, 6 \mathrm{~m} / \mathrm{s}^{2}$
d. $32 \mathrm{~N}, 2 \mathrm{~m} / \mathrm{s}^{2}$

Ajay Singhal

Numerade Educator

Problem 153

For the situation shown in the Fig. $7.402$, the block is stationary w.r.t. incline fixed in an elevator. The elevator has an acceleration of $\sqrt{5} a_{0}$ whose components are shown in the figure. The surface is rough and coefficient of static friction between the incline and block is $\mu_{s} .$ Determine the magnitude of force exerted by incline on the block. $\left[\right.$ Take $a_{0}=\frac{g}{2}$ and $\left.\theta=37^{\circ}, \mu_{s}=0.6\right]$
a. $\frac{m g}{10}$
b. $\frac{9 m g}{25}$
c. $\frac{3 m g}{25} \times \sqrt{41}$
d. $\frac{\sqrt{13} m g}{2}$

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 154

A block of mass $m$ is placed on a rough table, which is kept in a gravity free hall. Coefficient of friction between block and table is $\mu$ and a horizontal force $F$ is applied to the block. The reaction force exerted by the table on the block is
a. mg
b. $m g \sqrt{1+\mu^{2}}$
c. $F$
d. $\left(\sqrt{1+\mu^{2}}\right) F$

Ajay Singhal

Numerade Educator

Problem 155

If coefficient of friction between all surfaces (seee Fig. $7.403$ ) is $0.4$, then find the minimum force $F$ to have equilibrium of the system.
a. $62.5 \mathrm{~N}$
b. $150 \mathrm{~N}$
c. $135 \mathrm{~N}$
d. $50 \mathrm{~N}$

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 156

In the Fig. $7.404$ shown below, if acceleration of $B$ is $\vec{a}$, then find the acceleration of $A$.
a. $a \sin \alpha$
b. $a \cot \theta$
c. $a \tan \theta$
d. $a \sin \alpha \cot \theta$

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 157

If the acceleration of wedge in the Fig. $7.405$ shown is $a \mathrm{~m} / \mathrm{s}^{2}$ towards left, then at this instant acceleration of the block (magnitude only) would be
a. $4 a \mathrm{~m} / \mathrm{s}^{2}$
b. $a \sqrt{17-8 \cos \alpha} \mathrm{m} / \mathrm{s}^{2}$
c. $\sqrt{17 a} \mathrm{~m} / \mathrm{s}^{2}$
d. $\sqrt{17} \cos \frac{\alpha}{2} \times a \mathrm{~m} / \mathrm{s}^{2}$

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 158

In the arrangement shown in the Fig. $7.406$ below at a particular instant the roller is coming down with a speed of $12 \mathrm{~m} / \mathrm{s}^{2}$ and $C$ is moving up with $4 \mathrm{~m} / \mathrm{s}$. At the same instant it is also known that w.r.t. pulley $P$, block $A$ is moving down with speed $3 \mathrm{~m} / \mathrm{s}$. Determine the motion of block $B$ (velocity) w.r.t. ground.
a. $4 \mathrm{~m} / \mathrm{s}$ in downward direction
b. $3 \mathrm{~m} / \mathrm{s}$ in upward direction
c. $7 \mathrm{~m} / \mathrm{s}$ in downward direction
d. $7 \mathrm{~mm} / \mathrm{s}$ in upward direction

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 159

A professor holds an eraser against a vertical chalkboard by pushing horizontally on it. He pushes with a force that is much greater than it required to hold the eraser. The force of friction exerted by the board on the eraser increases if he
a. pushes eraser with slightly greater force.
b. pushes eraser with slightly less force.
c. raises his elbow so that the force he exerts is slightly downward but has same magnitude.
d. lowers his elbow so that the force he exerts is slightly upward but has the same magnitude.

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 160

A particle of mass $2 \mathrm{~kg}$ moves with an intial velocity of $(4 \hat{i}+2 \hat{j}) \mathrm{m} / \mathrm{s}$ on the $x-y$ plane. A force $\vec{F}=(2 \hat{i}-8 \hat{j})$
$\mathrm{N}$ acts on the particle. The initial position of the particle is $(2 \mathrm{~m}, 3 \mathrm{~m})$. Then for $y=3 \mathrm{~m}$
a. The possible value of $x$ is only $x=2 \mathrm{~m}$.
b. The possible value of $x$ is not only $x=2 \mathrm{~m}$, but there exists some other value of $x$ also.
c. Time taken is $2 \mathrm{~s}$.
d. All of the abe"e.

Ajay Singhal

Numerade Educator

Problem 161

Find the least horizontal force $P$ to start motion of any part of the system of the three blocks resting upon one another as shown in Fig. $7.407$. The weights of blocks are $A=300 \mathrm{~N}, B=100 \mathrm{~N}$ and $C=200 \mathrm{~N}$. Between Aand $B$ coefficient of friction is $0.3$, between $B$ and $C$ is $0.2$ and between $C$ and the ground is $0.1$.
a. $60 \mathrm{~N}$
b. $90 \mathrm{~N}$
c. $80 \mathrm{~N}$
d. $70 \mathrm{~N}$

Ajay Singhal

Numerade Educator

Problem 162

A person is drawing himself up and a trolley on which he stands with some acceleration. Mass of the person is more than the mass of the trolley. As the person increases his force on the string, the normal reaction between person and the trolley will
a. increase
b. decrease
c. remain same
d. cannot be predicted as data is insufficient

Ajay Singhal

Numerade Educator

Problem 163

A block of mass $m$ is attached with a mass less spring of torce constant $k .$ The block is placed over a rough inclined surface for which the coefficient of friction is $0.5 . M$ is released from rest when the spring was unstretched (see Fig. 7.409). The minimum value of $M$ required to move the block $m$ up the plane is (neglect mass of string and pulley and friction in pulley)
a. $m / 2$
b. $m / 3$
c. $m / 4$
d. None of these

Ajay Singhal

Numerade Educator

Problem 164

Figure $7.410$ shows the variation of force acting on a body, with time. Assuming the body to start from rest, the variation of its momentum with time is best represented by which plot?
$\mathbf{a}$.
b.
c.
d.

Ajay Singhal

Numerade Educator

Problem 165

The system shown in Fig. $7.411$ is released from rest when the spring was in its normal length. On releasing, the spring starts elongating
a. If $M>m$
b. If $M>2 m$
c. If $M<\frac{m}{2}$
d. For any value of $M$

Ajay Singhal

Numerade Educator

Problem 166

In Fig. 7.412 string does not slip on pulley $P$, but pulley $P$ is free to rotate about its own axis. Block $A$ is displaced towards left, then pulley $P$
a. rotates clockwise and translates
b. rotates anticlockwise and translates
c. only translates
d. only rotates (clockwise or anticlockwise)

Ajay Singhal

Numerade Educator

Problem 167

In the system in Fig. $7.413$, the friction coefficient between and bigger block $\mu .$ There is no friction between both the blocks. The string connecting both the blocks is light; all three pulleys are light and frictionless. Then the minimum limiting value of $\mu$ so that the system remains in equilibrium is
a. $\frac{1}{2}$
b. $\frac{3}{2}$
c. $\frac{2}{3}$
d. $\frac{3}{2}$

Ajay Singhal

Numerade Educator

Problem 168

Two blocks $A$ of $6 \mathrm{~kg}$ and $B$ of $4 \mathrm{~kg}$ are placed in contact with each other as shown in Fig. $7.414$. There is no friction between $A$ and ground and between both the blocks. Coefficient of friction between $B$ and ground is $0.5 . A$ horizontal force $F$ is applied on $\mathrm{A}$. Find the minimum and maximum value of $F$ which can be applied so that both blocks can move combinely without any relative motion between them.
a. $10 \mathrm{~N}, 50 \mathrm{~N}$
b. $12 \mathrm{~N}, 50 \mathrm{~N}$
c. $12 \mathrm{~N}, 75 \mathrm{~N}$
d. None of these

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 169

Two blocks are resting on ground with masses $5 \mathrm{~kg}$ and $7 \mathrm{~kg}$. A string connects them which goes over a massless pulley $A$. There is no friction between pulley and string. A force $F=124 \mathrm{~N}$ is applied on pulley $A$. The acceleration of centre of mass of $7 \mathrm{~kg}$ block and $5 \mathrm{~kg}$ block in vertical direction is
a. 0
b. $5 \mathrm{~ms}^{-2}$
c. $2.5 \mathrm{~ms}^{-2}$
d. $1 \mathrm{~ms}^{-2}$

Ajay Singhal

Numerade Educator

Problem 170

Two masses $\sqrt{3} \mathrm{~m}$ and $\sqrt{2} \mathrm{~m}$ tied by a light string are placed on a wedge of mass $4 \mathrm{~m}$. The wedge is placed on a smooth horizontal surface (see Fig. 7.416). Find out the value of $\theta$ so that the wedge does not move after the system is set free from the state of rest.
a. $30^{\circ}$
b. $45^{\circ}$
c. $60^{\circ}$
d. None of these

Ajay Singhal

Numerade Educator

Problem 171

Two blocks $A$ and $B$ each of mass $m$ are placed one over another on an incline as shown in Fig. $7.417 .$ When the system is released from rest the block slides down with constant velocity while block $B$ rests on top of $A$. If the coefficient of friction between $A$ and $B$ and between $B$ and incline are same, then value of coefficient of friction would be
a. $\frac{1}{4}$
b. $\frac{3}{5}$
c. $\frac{4}{5}$
d. Information insufficient

Dheeraj Sharma

Numerade Educator

Problem 172

Two blocks of masses $3 \mathrm{~kg}$ and $2 \mathrm{~kg}$ are placed side by side on an incline as shown in the Fig. $7.418$. A force, $F=20 \mathrm{~N}$ is acting on $2 \mathrm{~kg}$ block along the incline. The coefficient of friction between the block and the incline is same and equal to $0.1$. find the normal contact force exerted by $2 \mathrm{~kg}$ block on $3 \mathrm{~kg}$ block.
a. $18 \mathrm{~N}$
b. $30 \mathrm{~N}$
c. $12 \mathrm{~N}$
d. $27.6 \mathrm{~N}$

Ajay Singhal

Numerade Educator

Problem 173

Determine the time in which the smaller block reaches other end of bigger block in the Fig. $7.419$
a. $4 \mathrm{~s}$
b. 8
c. $2.19 \mathrm{~s}$
d. $2.13 \mathrm{~s}$

Ajay Singhal

Numerade Educator

Problem 174

A body of mass $m$ is held at rest at a height $h$ on two smooth wedges of mass $M$ each which are themselves at rest on a horizontal frictionless surface (see Fig. 7.420). When the mass $m$ is released, it moves down, pushing aside the wedges. The velocity with which the wedges recede from each other, when $m$ reaches the ground is
a. $\sqrt{\frac{8 m g h}{m+2 M}}$
b. $\sqrt{\frac{40 m g h \times 4}{5 m+6 M}}$
c. $\sqrt{\frac{32 m g h \times 4}{32 M+9 m}}$
d. None of these

Ajay Singhal

Numerade Educator

Problem 175

For the system shown in Fig. $7.421, m_{1}>m_{2}>m_{3}>$ $m_{4}$. Initially, the system is at rest in equilibrium condition. If the string joining $m_{4}$ and ground is cut, then just atter the string is cut:
Statement I: $m_{1}, m_{2}, m_{3}$ remain stationary. Statement II: the value of acceleration of all the 4 blocks can be determined. Statement III: Only $m_{4}$ remains stationary. Statement IV: Only $m_{4}$ accelerates. Statement $\mathrm{V}:$ All the four blocks remain stationary. Now, choose the correct options.
a. All the statement are correct.
b. Only I, II and IV are correct.
c. Only II and $\mathrm{V}$ are correct.
d. Only II and IV are correct.

Ajay Singhal

Numerade Educator

Problem 176

A particles is moving in $x-y$ plane. At certain instant.of time, the components of its velocity and acceleration are as follows: $v_{x}=3 \mathrm{~ms}^{-1}, v_{y}=4 \mathrm{~ms}^{-1}, a_{x}=2 \mathrm{~ms}^{-2}$, and
$a_{y}=1 \mathrm{~ms}^{-2}$. The rateof change of speed at this moment is
a. $\sqrt{10} \mathrm{~ms}^{-2}$
b. $4 \mathrm{~ms}^{-2}$
c. $\sqrt{5} \mathrm{~ms}^{-2}$
d. $2 \mathrm{~ms}^{-2}$

Ajay Singhal

Numerade Educator

Problem 177

If block $A$ is moving horizontally with velocity $v_{A}$ then find the velocity of block $B$ at the instant as shown in the Fig. $7.422 .$
a. $\frac{h v_{A}}{2 \sqrt{x^{2}+h^{2}}}$
b. $\frac{x v_{A}}{\sqrt{x^{2}+h^{2}}}$
c. $\frac{x v_{A}}{2 \sqrt{x^{2}+h^{2}}}$
d. $\frac{h v_{A}}{\sqrt{x^{2}+h^{2}}}$

Ajay Singhal

Numerade Educator

Problem 178

Figure $7.423$ shows two blocks, each of mass $m$. The system is released form rest. If accelerations of blocks $A$ and $B$ at any instant (not initially) are $a_{1}$ and $a_{2}$, respectively, then
a. $a_{1}=a_{2} \cos \theta$
b. $a_{2}=a_{1} \cos \theta$
c. $a_{1}=a_{2}$
d. None of these

Ajay Singhal

Numerade Educator

Problem 179

A small block of mass $m$ rests on a smooth wedge of angle
$\theta$. With what horizontal acceleration $a$ should the wedge be pulley, as shown in the Fig. $7.424$, so that the block falls freely.
a. $g \cos \theta$
b. $g \sin \theta$
c. $g \cot \theta$
d. $g \tan \theta$

Ajay Singhal

Numerade Educator

Problem 180

In the given Fig. $7.425$, man $A$ is standing on a movable plank while man $B$ is standing on a stationary platform. Both are pulling the string down such that the plank moves slowly up. As a result of this, the string slips through the hands of the men. Find the ratio of length of the string that slips through the hands of $A$ to $B$.
a. $3 / 2$
b. $3 / 4$
c. $4 / 3$
d. $2 / 3$

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 181

A uniform chain is placed at rest on a rough surface of base length $l$ and height $h$ on an irregular surface as shown in Fig. $7.426 .$ Then, the minimum coefficient of friction between the chain and the surface must be equal to
a. $\mu=\frac{h}{2 l}$
b. $\mu=\frac{h}{l}$
c. $\mu=\frac{3 h}{2 l}$
d. $\mu=\frac{2 h}{3 l}$

Ajay Singhal

Numerade Educator

Problem 182

In the Fig. $7.427$, the block of mass $m$ is at rest relative to the wedge of mass $M$ and the wedge is at rest with respect to ground. This implies that
a. Net force applied by $m$ on $M$ is $\mathrm{mg}$.
b. Normal force applied by $m$ on $M$ is mg.
c. Force of friction applied by $m$ on $M$ is $\mathrm{mg}$.
d. None
-sts

Ajay Singhal

Numerade Educator

Problem 183

A triangular prism of mass $M$ with a block of mass $m$ placed on it is released from rest on a smooth inclined olane of inclination $\theta$. The block does not slip on the prism. Then
a. The acceleration of the prism is $g \cos \theta$.
b. The acceleration of the prism is $g \tan \theta$.
c. The minimum coefficient of friction between the block and the prism is $\mu_{\operatorname{mia}}=\cot \theta$.
d. The minimum coefficient of friction between the block and the prism is $\mu_{\min }=\tan \theta$.

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 184

Two trolleys 1 and 2 are moving with accelerations $a_{1}$ and $a_{2}$, respectively, in the same direct. A bock of mass $m$ on trolley 1 is in equilibrium from the frame of observer stationary with respect to trolley 2 . The magnitude of friction force on block due to trolley is (assume that no horizontal force other than friction force is acting on block)
a. $m\left(a_{1}-a_{2}\right)$
b. $m a_{2}$
c. $m a_{1}$
d. Data insufficient

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 185

In Fig. $7.430$ all the surfaces are frictionless while pulley and string are massless. Mass of block $A$ is $2 m$ and that of block $B$ is $m .$ Acceleration of block $B$ immediately after system is released from rest is
a. $g / 2$
b. $g$
c. $g / 3$
d. None of these

Narayan Hari

Numerade Educator

Problem 186

In two pulley-particle Figs. $7.431$ (i) and (ii), the acceleration and force imparted by the string on the pulley and tension in the strings are, $\left(a_{1}, a_{2}\right),\left(N_{1}, N_{2}\right) ;$ and $\left(T_{1}, T_{2}\right)$ respectively. Ignoring friction in all contacting surfaces Study the following statements
a. $\frac{a_{1}}{a_{2}}=1$.
b. $\frac{T_{1}}{T_{2}}<1$
c. $\frac{N_{1}}{N_{2}}>1$
d. $\frac{a_{1}}{a_{2}}<1$
Now mark correct answer
(i) Relation (ii) and (iii) always follows.
(ii) Relation (ii) and (iv) always follows.
(iii) Relation (i) only always follows.
(iv) Relation (iv) always follows.

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 187

A block of mass $m$ is pressed against a vertical wall with a horizontal force $F=m g$ (see Fig. 7.432), Another force $F^{\prime}=\frac{m g}{2}$ is acting vertically upon the block. If the $\mathrm{co-}$ efficient of friction between the block and wall is $\frac{1}{2}$, the friction between them is
a. $\frac{m g}{2}$ up
b. $\frac{m g}{2}$ down
c. $m g$ up
d. 0

Narayan Hari

Numerade Educator

Problem 188

A particle has initial velocity, $\vec{v}=3 \hat{i}+4 \hat{j}$ and a constant force $\vec{F}=4 \hat{i}-3 \hat{j}$ acts on the particle. The path of the particle is
a. straight line
b. parabolic
c. circular
d. elliptical

Narayan Hari

Numerade Educator

Problem 189

In the Fig. $7.433$ shown the acceleration of $A$ is, $\vec{a}_{A}=15 \hat{i}+15 \hat{j}$ then the acceleration of $B$ is $(A$ remains in contact with $B$ )
a. $6 \hat{i}$
b. $-15 \hat{i}$
c. $-10 \hat{i}$
d. $-5 \hat{i}$

Narayan Hari

Numerade Educator

Problem 190

Two blocks $A$ and $B$ of masses $m$ and $2 m$ (Fig. 7.434), respectively, are held at rest such that the spring is in natural length. Find out the accelerations of both the blocks just after release.
a. $g \downarrow, g \downarrow$
b. $\frac{g}{3} \downarrow, \frac{g}{3} \uparrow$
c. 0,0
d. $g+c$

Dheeraj Sharma

Numerade Educator

Problem 191

A bob is hanging over a pulley inside a car through a string. The second end of the string is in the hand of a person standing in the car. The car is moving with constant acceleration $a$ directed horizontally as shown in Fig. $7.435$. Other end of the string is pulled with constant acceleration $a$ vertically. The tension in the string is equal to
a. $m \sqrt{g^{2}+a^{2}}$
b. $m \sqrt{g^{2}+a^{2}}-m a$
c. $m \sqrt{g^{2}+a^{2}}+m a$
d. $m(g+a)$

Dheeraj Sharma

Numerade Educator

Problem 192

Inside a horizontally moving box, an experimenter finds that when an object is placed on a smooth horizontal table and is released, it moves with an acceleration of $10 \mathrm{~m} / \mathrm{s}^{2}$. In this box of $1 \mathrm{~kg}$ body is suspended with a light string, the tension in the string in equilibrium position, (w.r.t. experimenter) will be (take $g=10 \mathrm{~m} / \mathrm{s}^{2}$ )
a. $10 \mathrm{~m} / \mathrm{s}^{2}$
b. $10 \sqrt{2} \mathrm{~m} / \mathrm{s}^{2}$
c. $20 \mathrm{~m} / \mathrm{s}^{2}$
d. zero

Narayan Hari

Numerade Educator

Problem 193

Two blocks $A$ and $B$ cach of mass $m$ are placed on a smooth horizontal surface. Two horizontal force $F$ and $2 F$ are applied on both the blocks $A$ and $B$, respectively. as shown in Fig. $7.436 .$ The block $A$ does not slide on block $B$. Then the normal reaction acting between the two blocks is
a. $F$
b. $F / 2$
c. $\frac{F}{\sqrt{3}}$
d. $3 F$

Dheeraj Sharma

Numerade Educator

Problem 194

In the given Fig. $7.437$, by what acceleration the boy must go up so that $100 \mathrm{~kg}$ block remains stationary on the wedge. The wedge is fixed and friction is absent everywhere (take $g=10 \mathrm{~m} / \mathrm{s}^{2}$ )
a. $2 \mathrm{~m} / \mathrm{s}^{2}$
b. $4 \mathrm{~m} / \mathrm{s}^{2}$
c. $6 \mathrm{~m} / \mathrm{s}^{2}$
d. $8 \mathrm{~m} / \mathrm{s}^{2}$

Narayan Hari

Numerade Educator

Problem 195

A system is shown in the Fig. $7.438$. A man standing on the block is pulling the rope. Velocity of the point of string in contact with the hand of the man is $2 \mathrm{~m} / \mathrm{s}$ downwards. The velocity of the block will be [assume that the block does not rotate]
a. $3 \mathrm{~m} / \mathrm{s}$
b. $2 \mathrm{~m} / \mathrm{s}$
c. $1 / 2 \mathrm{~m} / \mathrm{s}$
d. $1 \mathrm{~m} / \mathrm{s}$

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 196

In the Fig. $7.439$ shown the velocity of lift is $2 \mathrm{~m} / \mathrm{s}$ while string is winding on the motor shaft with velocity $2 \mathrm{~m} / \mathrm{s}$ and block $A$ is moving downwards with velocity of $2 \mathrm{~m} / \mathrm{s}$, then find out the velocity of block $B$.
a. $2 \mathrm{~m} / \mathrm{s} \uparrow$
b. $2 \mathrm{~m} / \mathrm{s} \downarrow$
c. $4 \mathrm{~m} / \mathrm{s} \uparrow$
d. None of th

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 197

A system is shown in the Fig. $7.440$. Assume that cylinder remains in contact with the two wedges hence the velocity of cylinder is
a. $\sqrt{19-4 \sqrt{3}} \frac{\mathrm{u}}{2} \mathrm{~m} / \mathrm{s}$
b. $\frac{\sqrt{13} u}{2} \mathrm{~m} / \mathrm{s}$
c. $\sqrt{3} u \mathrm{~m} / \mathrm{s}$
d. $\sqrt{7} u \mathrm{~m} / \mathrm{s}$

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 198

Two beads $A$ and $B$ move along a semicircular wire frame as shown in Fig. $7.441 .$ The beads are connected by an inelastic string with always remains tight. At as instant the speed of $A$ is $u, \angle B A C=45^{\circ}$ and $\angle B O C=75^{\circ}$, where $\mathrm{O}$ is the centre of the semicircular arc. The speed of bead $B$ at that instant is
a. $\sqrt{2} u$
b. $u$
c. $\frac{u}{2 \sqrt{2}}$
d. $\sqrt{\frac{2}{3}} u$

Dheeraj Sharma

Numerade Educator

Problem 199

A plank is held at an angle $\alpha$ to the horizontal (Fig. 7.442) on two fixed supports $A$ and $B$. The plank can slide against the supports (without friction) because of the weight $M g$. Acieration and direction in which a man of mass $m$ should move so that the plank does not move.
a. $g \sin \alpha\left(1+\frac{m}{M}\right)$ down the incline
b. $g \sin \alpha\left(1+\frac{M}{m}\right)$ down the incline
c. $g \sin \alpha\left(1+\frac{m}{M}\right)$ up the incline
d. $g \sin \alpha\left(1+\frac{M}{m}\right)$ up the incline

Dheeraj Sharma

Numerade Educator

Problem 200

Object $A$ and $B$ each of mass $m$ are connected by light inextensible cord. They are constrained to move on a frictionless ring to a vertical plane as shown in Fig. $7.443 .$ The objects are released form rest at the positions shown. The tension in the cord just after release will be
a. $m g \sqrt{2}$
b. $\frac{m g}{\sqrt{2}}$
c. $\frac{m g}{2}$
d. $\frac{m g}{4}$

Dheeraj Sharma

Numerade Educator

Problem 201

A pendulum of mass $m$ hangs from a support fixed to a trolley. The direction of the string when the trolley rolls up a plane of inclination $\alpha$ with acceleration $a_{0}$ is
a. $\theta=\tan ^{-1} \alpha$
b. $\theta=\tan ^{-1}\left(\frac{a_{0}}{g}\right)$
c. $\theta=\tan ^{-1}\left(\frac{g}{a_{0}}\right)$
d. $\theta=\tan ^{-1}\left(\frac{a_{0}+g \sin \alpha}{g \cos \alpha}\right)$

Dheeraj Sharma

Numerade Educator

Problem 202

Find the acceleration of the block: $B$ in the Fig. $7.445$, assuming that the surfaces and the pulleys $P_{1}$ and $P_{2}$ are all smooth.
a. $\frac{F}{4 m}$
b. $\frac{F}{6 m}$
c. $\frac{F}{2 m}$
d. $\frac{3 F}{17 m}$

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 203

In the Fig. $7.446$ shown all blocks are of equal mass $m .$ All surfaces are smooth. The acceleration of the block $A$ with respect to the ground is
a. $\frac{4 g \sin \theta}{1+3 \sin ^{2} \theta}$
b. $\frac{4 g \sin ^{2} \theta}{1+3 \sin ^{2} \theta}$
c. $\frac{4 g \sin ^{2} \theta}{\sqrt{1+3 \sin ^{2} \theta}}$
d. None of these

Prem Bijarniya

Numerade Educator

Problem 204

In the question 203 the acceleration $B$ w.r.t. ground is
a. $\frac{2 g \sin \theta}{1+3 \sin ^{2} \theta}$
b. $\frac{4 g \sin \theta}{1+3 \sin ^{2} \theta}$
c. $\frac{2 g \sin \theta}{\sqrt{1+3 \sin ^{2} \theta}}$
d. $\frac{4 g \sin \theta}{\sqrt{1+3 \sin ^{2} \theta}}$

Gregory Higby

Numerade Educator

Problem 205

In the question 203 the acceleration $C$ w.r.t. ground is
a. $\frac{2 g \sin \theta \cos \theta}{1+3 \sin ^{2} \theta}$
b. $\frac{g \sin \theta \cos \theta}{1+3 \sin ^{2} \theta}$
c. $\frac{g \sin 2 \theta}{\sqrt{1+3 \sin ^{2} \theta}}$
d. $\frac{g \sin \theta \cos \theta}{\sqrt{1+3 \sin ^{2} \theta}}$

Gregory Higby

Numerade Educator

Problem 206

In the arrangement shown in the Fig. $7.447$, the block of mass $m=2 \mathrm{~kg}$ lies on the wedge of mass $M=8 \mathrm{~kg}$. The initial acceleration of the wedge if the surfaces are smooth
a. $\frac{\sqrt{3} g}{23} \mathrm{~m} / \mathrm{s}^{2}$
b. $\frac{3 \sqrt{3} g}{23} \mathrm{~m} / \mathrm{s}^{2}$
c. $\frac{3 g}{23} \mathrm{~m} / \mathrm{s}^{2}$
d. $\frac{g}{23} \mathrm{~m} / \mathrm{s}^{2}$

Dheeraj Sharma

Numerade Educator

Problem 207

An object moving with a constant acceleration in a noninertial frame
a. must have non-zero net force acting on it.
b. may have zero net force acting on it.
c. may have no force acting on it.
d. this situation is practically impossible. (The pseudo force acting on the object has also to be considered)

Vysakh M

Numerade Educator

Problem 208

An object moving with constant velocity in a non-inertial frame of reference
a. must have non-zero net force acting on it.
b. may have zero net force acting on it.
c. must have zero net force acting on it.
d. may have non-zero net force acting on it (Consider only the real forces).

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 209

In the Fig. $7.448$ if $f_{1}, f_{2}$, and $T$ be the frictional forces on $2 \mathrm{~kg}$ block, $3 \mathrm{~kg}$ block $\%$ tension in string, respectively, then their values are
a. $2 \mathrm{~N}, 6 \mathrm{~N}, 3.2 \mathrm{~N}$
$\mathbf{b}_{+} 2 \mathrm{~N}, 6 \mathrm{~N}, 0 \mathrm{~N}$
c. $1 \mathrm{~N}, 6 \mathrm{~N}, 2 \mathrm{~N}$
d. Data insufficient to calculate the required value

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 210

A rope of length $4 \mathrm{~m}$ having mass $1.5 \mathrm{~kg} / \mathrm{m}$ lying on a horizontal frictionless surface is pulled at one end by a force of $12 \mathrm{~N}$. What is the tension in the rope at a point $1.6 \mathrm{~m}$ from the other end?
a. $5 \mathrm{~N}$
b. $4.8 \mathrm{~N}$
c. $7.2 \mathrm{~N}$
d. $6 \mathrm{~N}$

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 211

A block of mass $5.0 \mathrm{~kg}$ slides down from the top of an inclined plane of length $3 \mathrm{~m}$. The first $1 \mathrm{~m}$ of the plane is smooth and the next $2 \mathrm{~m}$ is rough. The block is released from rest and again comes to rest at the bottom of the plane. If the plane is inclined at $30^{\circ}$ with the horizontal (Fig. $7.449$ ), find the coefficient of friction on the rough portion.
a. $\frac{2}{\sqrt{3}}$
b. $\frac{\sqrt{3}}{2}$
c. $\frac{\sqrt{3}}{4}$
d. $\frac{\sqrt{3}}{5}$

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 212

A body of mass $m$ starting from rest slides down a frictionless inclined surface of gradient $\tan \alpha$ fixed on the floor of a lift accelerating upward with acceleration $a$. Taking width of inclined plane as $W$, the time taken by body to slide from top to bottom of the plane is
a. $\left(\frac{2 W}{(g+a) \sin \alpha}\right)^{\frac{1}{2}}$
b. $\left(\frac{4 W}{(g-a) \sin \alpha}\right)^{\frac{1}{2}}$
c. $\left(\frac{4 W}{(g+a) \sin 2 \alpha}\right)^{\frac{1}{2}}$
d. $\left(\frac{W}{(g+a) \sin 2 \alpha}\right)^{\frac{1}{2}}$

Prem Bijarniya

Numerade Educator

Problem 213

A rope is stretched between two boats at rest. A sailor in the first boat pulls the rope with a constant force of $100 \mathrm{~N}$. First boat with the sailor has a mass of $250 \mathrm{~kg}$ whereas the mass of second boat is double of this mass (see Fig. $7.451$ ). If the initial distance between the boats was $100 \mathrm{~m}$, the time taken for two boats to meet each other is (neglect water resistance between boats and water)
a. $13.8 \mathrm{~s}$
b. $18.3 \mathrm{~s}$
c. $3.18 \mathrm{~s}$
d. $31.8 \mathrm{~s}$

RZ

Rubeena Zulfiqar

Numerade Educator