IIT JEE Super Course in Physics: Mechanics II

Trishna Knowledge Systems

Chapter 1

Work, Power and Energy - all with Video Answers

Educators

Chapter Questions

Problem 1

A $20 \mathrm{~kg}$ box is pulled through a distance of $25 \mathrm{~m}$ along the floor, by a force, which makes angle $30^{\circ}$ with the horizontal. If the friction coefficient $\mu=0.3$, find the work done by the force for
(i) a constant speed $5 \mathrm{~m} \mathrm{~s}^{-1}$
(ii) variable speed $v=5+\frac{S}{2}$, where $S$ is the displacement. (Take $\sqrt{3}=1.7, g=10 \mathrm{~m} \mathrm{~s}^{-2}$ )

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 2

Man A pushes a block up a smooth incline upto certain height with uniform velocity $\mathrm{v}$, using power $\mathrm{P}_{1}$. Man B raises the same block vertically up with uniform velocity $2 \mathrm{v}$, through the same height, using power $3 \mathrm{P}_{\mathrm{1}}$. In which case is
the
(i) force applied less?
(ii) work done less?
(iii) time of travel less?
(iv) What is the ratio of times, $\frac{t_{A}}{t_{B}}$;

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 3

A $1000 \mathrm{~kg}$ aircraft accelerates uniformly on a $50 \mathrm{~m}$ runway to acquire a take off speed $80 \mathrm{~km} /$ hour. If the frictional force between the tyres and runway surface is $2000 \mathrm{~N}$, calculate:
(i) The force applied by the propeller on the plane.
(ii) Minimum engine power required for take off. $\left(\mathrm{g}=10 \mathrm{~m} \mathrm{~s}^{-2}\right)$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 4

Masses $\mathrm{m}_{1}$ and $\mathrm{m}_{2}$ are connected by a light inextensible string running over a light frictionless pulley attached to an elevator. The elevator starts to move up with an acceleration a. Find the work done by tension force on $\mathrm{m}_{1}$ and $\mathrm{m}_{2}$, in time $\mathrm{t}$ with respect to
(i) the elevator.
(ii) the ground.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 5

A point mass is released from rest from point $A$ of the track in the form of a trough as shown. $B C=3 \mathrm{~m}$. Height of A from $B C$ is $0.5 \mathrm{~m}$. The path $\mathrm{BC}$ is rough with $\mu=0.15$. Other parts of the path are smooth. Determine the point
on $\mathrm{BC}$ where the body comes to rest. $\left(\mathrm{g}=10 \mathrm{~m} \mathrm{~s}^{-2}\right)$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 6

A mass $\mathrm{m}$ is released on a stationary wedge of mass $\mathrm{M}$ Determine velocity $\mathrm{v}$ acquired by the mass $\mathrm{m}$ sliding through a distance $\mathrm{S}$ on the incline

Suman Saurav Thakur

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Numerade Educator

Problem 7

A ball of mass $9 \mathrm{~kg}$ is dropped on to a vertical light spring of force constant $\mathrm{k}=320 \mathrm{~N} \mathrm{~m}^{-1} .$ The block compresses the spring $1.7 \mathrm{~m}$ as shown in figure. Calculate

Suman Saurav Thakur

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Numerade Educator

Problem 8

A simple pendulum of length $\ell$ and mass $\mathrm{m}$ is suspended from ceiling of a trolley of mass $\mathrm{M}$. The system is released from rest when the string is horizontal. Find the velocity of trolley with respect to ground, when string is vertical

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 9

A ball is dropped from a height of $10 \mathrm{~m}$ above a point $\mathrm{A}$ on a fixed inclined plane inclined at an angle of $30^{\circ}$ upward with horizontal. If coefficient of restitution is $\frac{1}{\sqrt{3}}$, and if the ball hits the plane again at a point $B$, determine the distance $A B$ and time between collisions. $\left(g=10 \mathrm{~m} \mathrm{~s}^{-2}\right)$

Suman Saurav Thakur

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Numerade Educator

Problem 10

The figure shows $\mathrm{F}-\mathrm{x}$ graph for a body moving along a straight line. The work done by the force is
(a) $10 \mathrm{~J}$
(b) $6 \mathrm{~J}$
(c) $4 \mathrm{~J}$
(d) 2 J

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 11

The figure shows $\mathrm{F}-\mathrm{x}$ graph for a body moving along a straight line. The work done by the force is
(a) $10 \mathrm{~J}$
(b) $6 \mathrm{~J}$
(c) $4 \mathrm{~J}$
(d) 2 J

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 12

Two forces $\mathrm{F}$, and $\mathrm{F}_{2}$ act simultaneously on a body and it moves towards right with a velocity $2 \mathrm{~ms}^{-1} .$ If $\mathrm{F}_{1}=6 \mathrm{~N}$ and $F_{2}=3 \mathrm{~N}$, the net power is equal to
(a) $3 \mathrm{~W}$
(b) $6 \mathrm{~W}$
(c) $8 \mathrm{~W}$
(d) Zero

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 13

If the $\mathrm{KE}$ and momentum of a body are $100 \mathrm{~J}$ and $20 \mathrm{~kg} \mathrm{~m} \mathrm{~s}^{-1}$ respectively, the mass of the body is
(a) $1 \mathrm{~kg}$
(b) $2 \mathrm{~kg}$
(c) $3 \mathrm{~kg}$
(d) $4 \mathrm{~kg}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 14

A particle is released from the edge of a smooth semicircular track fixed in the vertical plane.
The force on the wire when the angle $\theta$ is $\frac{\pi}{4}$ is
(a) $\mathrm{mg}$
(b) $\frac{\mathrm{mg}}{\sqrt{2}}$
(c) $\frac{3 m g}{\sqrt{2}}$
(d) $\frac{\sqrt{5}}{2} \mathrm{mg}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 15

A particle slides down from rest from the top of a smooth inclined plane of length $\ell$ making angle $\theta$ with the horizontal. At the same instant, another particle is shot up the plane with velocity $v_{0}$. If the bodies came to rest with respect to each other after a perfectly inelastic collision, $\mathrm{v}_{0}$ is
(a) $\sqrt{2 \mathrm{~g} \ell}$
(b) $\sqrt{\mathrm{g} \ell \sin \theta}$
(c) $\sqrt{2 \mathrm{~g} \ell \sin \theta}$
(d) $\sqrt{g \ell \cos \theta}$

Suman Saurav Thakur

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Numerade Educator

Problem 16

Statement 1 When a ball is gently placed on the free end of a vertically fixed spring, its kinetic energy first increases and then decreases to zero.
and
Statement $\underline{2}$
Positive work done by gravity is proportional to displacement $\mathrm{x}$ and negative work done by spring is proportional to $\mathrm{x}^{2}$ and constants of proportionality differ.

Suman Saurav Thakur

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Problem 17

Statement 1
A spring vertically fixed at the bottom and a weight placed gently on top which is slowly lowered quasistatically till maximum compression, will not have mechanical energy conserved. and
Statement 2
Mechanical energy is conserved for conservative systems.

Suman Saurav Thakur

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Problem 18

Statement 1
A particle attached to the end of a string is in vertical circular motion with least possible kinetic energy at topmost point $\mathrm{P}$. If the string breaks, the particle cannot rise above the level of $\mathrm{P}$ irrespective of wherever the string breaks. and
Statement 2
Both vertical circular motion and projectile motion are processes conserving mechanical energy since tension does no work.

Suman Saurav Thakur

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Numerade Educator

Problem 19

Statement 1
If the exhaust gases out of a rocket are stationary, it means the rocket is in uniform motion. and
Statement 2
Relative speed of exhaust gases is one of the factors determining acceleration of the rocket.

Suman Saurav Thakur

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Problem 20

Statement 1 In a 1D collision between two bodies moving in same direction, the final kinetic energy of the trailing body can be more in inelastic collision, than what it would be if the collision were elastic.
and
Statement 2 Impulse J in inelastic collision is less than the impulse in elastic collision.

Suman Saurav Thakur

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Numerade Educator

Problem 21

If the collision of the body with the wall at $\mathrm{C}$ is elastic, the successive heights upto which the particle rises on $\mathrm{AB}$ form
(a) Arithmetic progression
(b) Geometric progression
(c) Harmonic progression
(d) None of the above

Suman Saurav Thakur

Suman Saurav Thakur

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Problem 22

If after several successive elastic collisions, the particle comes to rest just at the wall at $\mathrm{C}$, then $\mu$ can possibly be
(a) $0.1$
(b) $0.2$
(c) $0.5$
(d) $0.4$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 23

If the collision at wall $\mathrm{C}$ is inelastic with coefficient at restitution $\mathrm{e}$, velocity of the particle at $\mathrm{C}$ after the first collision is
(a) $\sqrt{\ell(1-\mathrm{e}) \mu}$
(b) $\sqrt{\ell(1-\mu \mathrm{e})}$
(c) $\sqrt{\ell(\mu-\mathrm{e})}$
(d) $e \sqrt{2 g \ell(1-\mu)}$

Suman Saurav Thakur

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Numerade Educator

Problem 24

Mass of $\mathrm{A}$ is
(a) $\frac{\mathrm{m}}{2}$
(b) $\frac{3}{4} \mathrm{~m}$
(c) $\mathrm{m}$
(d) $\frac{3}{2} \mathrm{~m}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 25

Mass of $\mathrm{C}$ is
(a) $\frac{\mathrm{m}}{2}$
(b) $\frac{3}{4} \mathrm{~m}$
(c) $\mathrm{m}$
(d) $\frac{3}{2} \mathrm{~m}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 26

When C's velocity is $\frac{\mathrm{u}}{2}$, the length of the spring differs from its natural length by
(a) $\frac{\mathrm{u}}{2} \sqrt{\frac{3 \mathrm{~m}}{2 \mathrm{k}}}$
(b) $\frac{\mathrm{u}}{2} \sqrt{\frac{3 \mathrm{~m}}{\mathrm{k}}}$
(c) $\mathrm{u} \sqrt{\frac{3 \mathrm{~m}}{\mathrm{k}}}$
(d) $\mathrm{u} \frac{3}{2} \sqrt{\frac{\mathrm{m}}{\mathrm{k}}}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 27

When C's velocity is $\frac{\mathrm{u}}{2}$, the length of the spring differs from its natural length by
(a) $\frac{\mathrm{u}}{2} \sqrt{\frac{3 \mathrm{~m}}{2 \mathrm{k}}}$
(b) $\frac{\mathrm{u}}{2} \sqrt{\frac{3 \mathrm{~m}}{\mathrm{k}}}$
(c) $\mathrm{u} \sqrt{\frac{3 \mathrm{~m}}{\mathrm{k}}}$
(d) $\mathrm{u} \frac{3}{2} \sqrt{\frac{\mathrm{m}}{\mathrm{k}}}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 28

A mass $\mathrm{m}$ is released on the top of the smooth surface of a wedge of mass $3 \mathrm{~m}$, kept on a smooth floor and having dimensions as shown.
(a) When the mass reaches the bottom position, the wedge would have moved $2 \mathrm{~m}$.
(b) When the mass reaches the bottom position its velocity is $\sqrt{\frac{720}{7}} \mathrm{~ms}^{-1}$
(c) The velocity of the mass in the reference frame of the wedge is less than its velocity as observed from ground
(d) If the mass hits the floor elastically and rebounds it has a projectile range of $\frac{72}{7} \mathrm{~m}$ on ground for its motion after impact.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 29

Two balls A and B having mass $m_{A}$ and $m_{B}$ are dropped on an inclined a plane of angle $\theta=45^{\circ}$, so that they fall on the same point $\mathrm{P}$ on the inclined plane with same K.E and rebound after perfectly elastic collision. B clears the inclined plane and reaches ground 2 s after collision, A falls just at the foot of the inclined plane.
(a) The ratio of their ranges on the ground level $\frac{\mathrm{R}_{\mathrm{A}}}{\mathrm{R}_{\mathrm{B}}}=\sqrt{\mathrm{m}_{\mathrm{B}} / \mathrm{m}_{\mathrm{A}}}$
(b) The ratio of their ranges on the ground level $=\mathrm{m}_{\mathrm{A}} / \mathrm{m}_{\mathrm{B}}$
(c) Height h from where $A$ is dropped is $25 \mathrm{~m}$ above ground level
(d) The velocity with which A reaches $\mathrm{P}$ is $\mathrm{v}=10 \mathrm{~m} \mathrm{~s}^{-1}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 30

$30 .$
$\begin{array}{ll}\text { Column I } & \text { Column II }\end{array}$
(a) Work done by a lifter in bringing down a weight from
(p) Work done is negative a height
(b) Work done by the external force $\overline{\mathrm{F}}$, A undergoing
(q) Work done is positive displacement $\overline{\mathrm{S}}$ while force $\overline{\mathrm{F}}$ is acting
(c) Work done by a compressed spring while it relaxes
(r) Mechanical energy is conserved and pushes a connected body
(d) Work done by a boy in catching a cricket ball
(s) Mechanical energy is not conserved

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 31

Which of the following is not a unit of energy
(a) joule
(b) $\mathrm{g} \mathrm{cm}^{2} \mathrm{~s}^{-2}$
(c) $\mathrm{kg}$
(d) erg

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 32

A force of $10 \mathrm{~N}$ acting on a body at an angle $30^{\circ}$ to the horizontal produces a displacement of $2 \mathrm{~m}$. The work done is
(a) $10 \sqrt{2} \mathrm{~J}$
(b) $10 \sqrt{3}$ J
(c) $\frac{10}{\sqrt{2}} \mathrm{~J}$
(d) $\frac{10}{\sqrt{3}}$ J

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 33

A man pushes a wall and fails to displace it. He does
(a) negative work
(b) positive but not maximum work
(c) no work at all
(d) maximum work

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 34

The incorrect relation is
(a) $1 \mathrm{~W}=1.34 \times 10^{-3} \mathrm{hp}$
(b) $1 \mathrm{kWh}=1000$ watt hour
(c) $1 \mathrm{kWh}=3.6 \times 10^{\xi} \mathrm{J}$
(d) $1 \mathrm{~W}=1 \mathrm{~J} / \mathrm{s}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 35

A force $\overline{\mathrm{F}}=(14 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}) \mathrm{N}$ is acting on a body and moves it through a distance $\mathrm{S}=(2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}) \mathrm{m}$. The work done
by the force is (in J)
(a) Zero
(b) 60
(c) 220
(d) 160

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 36

The amount of water a 1 hp electric motor can pump from a well $10 \mathrm{~m}$ deep is (in $\mathrm{kg} / \mathrm{s}$ )
(a) $1.5$
(b) $3.7$
(c) $5.8$
(d) $7.6$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 37

From a waterfall, water is pouring down at the rate of $100 \mathrm{~kg} / \mathrm{s}$ on the blades of a turbine. If the height of the fall is 100
$\mathrm{m}$, the power delivered to the turbine is approximately equal to
(a) $100 \mathrm{~kW}$
(b) $10 \mathrm{~kW}$
(c) $1 \mathrm{~kW}$
(d) $100 \mathrm{~W}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 38

When the speed of a body is doubled, its $\mathrm{KE}$ is
(a) Doubled
(b) One-fourth
(c) Halved
(d) Quadrupled

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 39

If the momentum of a body is numerically equal to four times its $\mathrm{KE}$, then the velocity of the body is
(a) 2
(b) 4
(c) $\frac{1}{2}$
(d) $\frac{1}{4}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 40

A body of mass $\mathrm{m}$ and momentum $\mathrm{p}$ has $\mathrm{KE}$
(a) $\frac{\mathrm{p}}{\mathrm{m}}$
(b) $\frac{\mathrm{p}^{2}}{\mathrm{~m}}$
(c) $\mathrm{p}^{2} \mathrm{~m}$
(d) $\frac{p^{2}}{2 m}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 41

Two masses $4 \mathrm{~m}$ and $9 \mathrm{~m}$ move with equal $\mathrm{KE}$. The ratio of the magnitude of their momenta is
(a) $1: 2$
(b) $2: 3$
(c) $1: 3$
(d) $3: 1$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 42

A body of mass $2 \mathrm{~kg}$ and momentum $4 \mathrm{~kg} \mathrm{~m} \mathrm{~s}^{-1}$ has $\mathrm{KE}$ (in J)
(a) 4
(b) 2
(c) 8
(d) 6

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 43

If the momentum of a body is increased 4 times, its KE will increase
(a) 4 times
(b) 8 times
(c) 12 times
(d) 16 times

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 44

A $300 \mathrm{~g}$ mass has a velocity of $(3 \mathrm{i}+4 \mathrm{j}) \mathrm{m} \mathrm{s}^{-1}$ at certain instant. Its kinetic energy is
(a) 9J
(b) $2.4 \mathrm{~J}$
(c) $3.75 \mathrm{~J}$
(d) $7.35 \mathrm{~J}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 45

If the linear momentum is increased by $50 \%$ the $K . E$ will be increased by
(a) $25 \%$
(b) $100 \%$
(c) $50 \%$
(d) $125 \%$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 46

A light and a heavy body have equal momenta. Which one has greater KE?
(a) The light body
(b) Both have equal KE
(c) The heavy body
(d) Data given is insufficient

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 47

If the $\mathrm{KE}$ of a body changes from $\mathrm{E}$ to $4 \mathrm{E}$ by the application of a force $4 \mathrm{~N}$, the displacement produced is (in the direction of the force)
(a) $\frac{2 \mathrm{E}}{3}$
(b) $\frac{\sqrt{3}}{2} \mathrm{E}$
(c) $\frac{15}{16} \mathrm{E}$
(d) $\frac{3 \mathrm{E}}{4}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 48

A body of mass $10 \mathrm{~kg}$ at rest is acted upon simultaneously by two forces $4 \mathrm{~N}$ and $3 \mathrm{~N}$ at right angles to each other. The $\mathrm{KE}$ at the end of 10 second is
(a) $100 \mathrm{~J}$
(b) $300 \mathrm{~J}$
(c) $125 \mathrm{~J}$
(d) $20 \mathrm{~J}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 49

A bullet of mass $8 \mathrm{~g}$ hits a fixed body at a speed of $500 \mathrm{~m} \mathrm{~s}^{-1}$ and pierces it through $20 \mathrm{~cm}$ before coming to rest. The average energy loss in the body is $(\mathrm{J} / \mathrm{m})$
(a) $8 \times 10^{2}$
(b) $50 . \times 10^{3}$
(c) $20 \times 10^{-3}$
(d) $5 \times 10^{3}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 50

A body of mass $2 \mathrm{~kg}$ is being rotated in a horizontal circle of radius $1 \mathrm{~m}$ with the help of a string. The energy spent by centripetal force is
(a) $20 \mathrm{~J}$
(b) $2 \mathrm{~J}$
(c) $5 \mathrm{~J}$
(d) Zero

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 51

A stationary body is slowly lowered on to a massive platform moving horizontally at a speed of $4 \mathrm{~m} \mathrm{~s}^{-1}$. The distance the body slides relative to the platform, if coefficient of friction is $0.2,\left(\mathrm{~g}=10 \mathrm{~m} \mathrm{~s}^{-2}\right)$ is
(a) $16 \mathrm{~m}$
(b) $8 \mathrm{~m}$
(c) $4 \mathrm{~m}$
(d) $2 \mathrm{~m}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 52

A constant retarding force is applied to stop a train. If the speed of the train is doubled and the same retarding force is applied, distance will be
(a) the same
(b) doubled
(c) three times
(d) four times

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 53

A body slides on a frictionless surface from $A$ to $B$. If the velocity of a ball at $A$ is $5 \mathrm{~m} \mathrm{~s}^{-1}$, its velocity at $\mathrm{B}\left(\mathrm{g}=10 \mathrm{~m} \mathrm{~s}^{-2}\right)$ is
(a) $\sqrt{25} \mathrm{~m} \mathrm{~s}^{-1}$
(b) $\sqrt{40} \mathrm{~ms}^{-1}$
(c) $\sqrt{50} \mathrm{~m} \mathrm{~s}^{-1}$
(d) $\sqrt{65} \mathrm{~m} \mathrm{~s}^{-1}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 54

A spring of original length $\mathrm{x}$ is compressed to half its length. If the force constant of the spring is $8 \mathrm{~N} / \mathrm{m}$, its potential energy is
(a) $8 x$
(b) $8 x^{2}$
(c) $\mathrm{x}^{2}$
(d) $4 x^{2}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 55

When a body is dropped from a certain height its $\mathrm{P.E}$ changes with distance as

Suman Saurav Thakur

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Problem 56

A body released from rest slides along a frictionless track AC. Its speed at $C$ is
(a) $\mathrm{mgh}$
(b) $\mathrm{gh}$
(c) $\sqrt{g h}$
(d) $\sqrt{2 \mathrm{gh}}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 57

If the energy stored in spring $A=10 \mathrm{~J}$, that stored in $\mathrm{B}$ is (under the same stretching force)
(a) $10 \mathrm{~J}$
(b) $20 \mathrm{~J}$
(c) $5 \mathrm{~J}$
(d) $15 \mathrm{~J}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 58

A ball is dropped along smooth inclines $\mathrm{AB}, \mathrm{AC}$ and $\mathrm{AD}$. The velocity acquired at $\mathrm{B}, \mathrm{C}, \mathrm{D}$ be $\mathrm{v}_{\mathrm{B}}, \mathrm{v}_{\mathrm{C}}$ and $\mathrm{v}_{\mathrm{D}}$ respectively. Then
(a) $v_{B}>v_{C}>v_{D}$
(b) $\mathrm{v}_{\mathrm{B}}<\mathrm{v}_{\mathrm{C}}<\mathrm{v}_{\mathrm{D}}$
(c) $\mathrm{v}_{\mathrm{B}}>\mathrm{v}_{\mathrm{C}}<\mathrm{v}_{\mathrm{D}}$
(d) $\mathrm{v}_{\mathrm{B}}=\mathrm{v}_{\mathrm{C}}=\mathrm{v}_{\mathrm{D}}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 59

A body of mass $500 \mathrm{~g}$, moving with uniform speed on the floor goes up along a smooth incline and stops at a height $10 \mathrm{~cm}$ with respect to the floor. Its speed on the floor was $\left(\mathrm{g}=10 \mathrm{~m} \mathrm{~s}^{-2}\right)$
(a) $1 \mathrm{~m} \mathrm{~s}^{-1}$
(b) $\sqrt{5} \mathrm{~m} \mathrm{~s}^{-1}$
(c) $\sqrt{2} \mathrm{~m} \mathrm{~s}^{-1}$
(d) $\frac{1}{\sqrt{50}} \mathrm{~ms}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 60

A ball is being hit from the ground level at an angle of $45^{\circ}$ with the horizontal and has an initial kinetic energy E. The kinetic energy at the highest point of the path is
(a) 0
(b) $\frac{E}{\sqrt{2}}$
(c) $\mathrm{E}$
(d) $\frac{E}{2}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 61

A person holds a bucket of weight $50 \mathrm{~N}$. He walks $5 \mathrm{~m}$ along the horizontal and he climbs up a vertical distance of $5 \mathrm{~m}$. The work done by the person is
(a) $500 \mathrm{~J}$
(b) $250 \mathrm{~J}$
(c) Zero
(d) $5000 \mathrm{~J}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 62

In the arrangement shown both springs are in unstretched condition. If body $\mathrm{m}$ is released from rest, the maximum vertical distance that the body will come down is
(a) $\frac{2 m g \sin \theta}{k}$
(b) $\sqrt{\frac{4 m g \sin \theta}{k^{2}}}$
(c) $\sqrt{\frac{\mathrm{mg} \cos \theta}{2 \mathrm{k}}}$
(d) $\frac{m g \sin ^{2} \theta}{k}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 63

A spring of constant $80 \mathrm{~N} \mathrm{~m}^{-1}$ fixed at one end is having a mass of $0.05 \mathrm{~kg}$ at the other end. It is stretched by $0.1 \mathrm{~m}$ and released. The maximum velocity attained by the mass is
(a) $2 \mathrm{~m} \mathrm{~s}^{-1}$
(b) $2 \sqrt{2} \mathrm{~ms}^{-1}$
(c) $4 \mathrm{~m} \mathrm{~s}^{-1}$
(d) $1 \mathrm{~m} \mathrm{~s}^{-1}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 63

A spring of constant $80 \mathrm{~N} \mathrm{~m}^{-1}$ fixed at one end is having a mass of $0.05 \mathrm{~kg}$ at the other end. It is stretched by $0.1 \mathrm{~m}$ and released. The maximum velocity attained by the mass is
(a) $2 \mathrm{~m} \mathrm{~s}^{-1}$
(b) $2 \sqrt{2} \mathrm{~ms}^{-1}$
(c) $4 \mathrm{~m} \mathrm{~s}^{-1}$
(d) $1 \mathrm{~m} \mathrm{~s}^{-1}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 64

Potential energy of a particle as a function of its displacement from origin is given by $U=x^{4}-b x^{2}$. The body is in stable equilibrium at origin, if $\mathrm{b}$ is
(a) $-1$
(b) 2
(c) $-3$
(d) both a and $c$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 65

A pendulum of length $1 \mathrm{~m}$ and bob mass equal to $0.5 \mathrm{~kg}$ is projected from the bottom position with an initial horizontal velocity of $v<\sqrt{5 g \ell}$. If the bob loses the circular path when its velocity is $2 \mathrm{~m} \mathrm{~s}^{-1}$ (take this instant as $t=0$ ), then the time after this the bob will reach vertical position above the point of suspension is approximately.
(a) $11 / 8 \mathrm{~s}$
(b) $13 / 8 \mathrm{~s}$
(c) $1.5 \mathrm{~s}$
(d) $13 / 4 \mathrm{~s}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 66

In the above, its loss in potential energy at $t=0$ is
(a) $5 \mathrm{~J}$
(b) $6 \mathrm{~J}$
(c) 7
(d) 8 J

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 67

A rocket is held tight on a test pad and fuel is burnt at the rate of $10 \mathrm{~kg} \mathrm{~s}^{-1}$ and the exhaust speed is $1000 \mathrm{~m} \mathrm{~s}^{-1}$. The force required to hold he rocket stationary is
(a) $10,000 \mathrm{~N}$
(b) $12,000 \mathrm{~N}$
(c) $13,160 \mathrm{~N}$
(d) $14,820 \mathrm{~N}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 68

In the above case the power lost in the exhaust gas (exclude heat) is
(a) $2 \times 10^{6} \mathrm{~W}$
(b) $3 \times 10^{5} \mathrm{~W}$
(c) $4 \times 10^{6} \mathrm{~W}$
(d) $5 \times 10^{6} \mathrm{~W}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 69

A shell is fired from a cannon with a velocity $\mathrm{v}$ at an angle $\theta$ to the horizontal. At the highest point of its path, it explodes into two pieces of equal masses; one of the pieces retraces its path to the point of projection. The speed of the second piece immediately after the explosion is
(a) $3 v \cos \theta$
(b) $2 \mathrm{v} \cos \theta$
(c) $v \cos \theta$
(d) $\frac{2}{3} v \cos \theta$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 70

In elastic collisions
(a) Momentum is conserved
(b) KE is conserved
(c) Both momentum and $\mathrm{KE}$ are conserved
(d) Momentum not conserved but $\mathrm{KE}$ is conserved

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 71

A and $B$ are two identical balls attached to strings as shown. A is released from position OA. A collides with B elastically. Then,
(a) $\mathrm{A}$ and $\mathrm{B}$ move with different non-zero velocities
(b) $\mathrm{B}$ moves forward and $\mathrm{A}$ rebounds with the same velocity
(c) A and B stick together and move with same velocity
(d) A comes to rest and $B$ moves with the velocity of $A$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 72

An inelastic ball falls from a height of $100 \mathrm{~m}$. It loses $20 \%$ energy due to impact. The ball will again rise to a height of
(a) $80 \mathrm{~m}$
(b) $98 \mathrm{~m}$
(c) $60 \mathrm{~m}$
(d) $40 \mathrm{~m}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 73

A ball hits the floor and rebounds after inelastic collision. In this case
(a) the momentum of the ball just after the collision is the same as that just before the collision
(b) the mechanical energy of the ball remains the same in the collision
(c) the total momentum of the ball and the earth is conserved
(d) the total energy of the ball and the earth is conserved

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 74

Which one of the following Statements is true?
(a) Momentum is conserved in elastic collisions but not in inelastic collisions
(b) Total KE is conserved in elastic collisions but momentum is not conserved in elastic collisions
(c) Total KE is not conserved but momentum is conserved in inelastic collisions
(d) $\mathrm{KE}$ and momentum both are conserved in all types of collisions

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 75

A particle A suffers an oblique elastic collision with a particle B that is at rest. If their masses are the same, then, after the collision
(a) They will move in opposite directions
(b) A continues to move in the original direction while $\mathrm{B}$ remains at rest
(c) They will move in mutually perpendicular directions
(d) A comes to rest and $\mathrm{B}$ starts moving in the direction of the original motion of $\mathrm{A}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 76

Which one of the following Statements does not hold good when two balls of masses $\mathrm{m}_{1}$ and $\mathrm{m}_{2}$ undergo elastic collision?
(a) When $\mathrm{m}_{1}<\mathrm{m}_{2}$ and $\mathrm{m}_{2}$ at rest, there will be maximum transfer of momentum
(b) When $\mathrm{m}_{1}>\mathrm{m}_{2}$ and $\mathrm{m}_{2}$ at rest, after collision the ball of mass $\mathrm{m}_{2}$ can move with two times the velocity of $\mathrm{m}$.
(c) When $m_{1}=m_{2}$ and $m_{2}$ at rest, there will be maximum transfer of $K E$
(d) When collision is oblique and $\mathrm{m}$, at rest with $\mathrm{m}_{1}=\mathrm{m}_{2}$, after collisions the ball moves in opposite directions

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 77

A body is dropped from height h while another identical body is thrown up with velocity $\sqrt{2 g h}$. If they have a completely inelastic collision, they will reach ground in
(a) $\sqrt{\frac{\mathrm{h}}{2 \mathrm{~g}}}$
(b) $\sqrt{\frac{2 \mathrm{~h}}{\mathrm{~g}}}$
(c) $\sqrt{\frac{3 \mathrm{~h}}{2 \mathrm{~g}}}$
(d) $\sqrt{\frac{\mathrm{h}}{\mathrm{g}}}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 78

A stationary bomb explodes into two parts, $4 \mathrm{~kg}$ and $8 \mathrm{~kg}$. The velocity of the $8 \mathrm{~kg}$ mass is $6 \mathrm{~m} \mathrm{~s}^{-1}$. The $\mathrm{KE}$ of the other body is
(a) $48 \mathrm{~J}$
(b) $24 \mathrm{~J}$
(c) $288 \mathrm{~J}$
(d) $16 \mathrm{~J}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 79

A stationary body of mass $m$ explodes into 3 parts with mass ratio of $1: 3: 3$. The two fragments with equal mass move at right angles to each other with velocity of $15 \mathrm{~m} \mathrm{~s}^{-1}$. The velocity of the third fragment is $\left(\mathrm{m} \mathrm{s}^{-1}\right)$
(a) $\sqrt{2}$
(b) 5
(c) $20 \sqrt{2}$
(d) $45 \sqrt{2}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 80

A space craft of mass $\mathrm{M}$ is moving with velocity $\mathrm{v}$ in free space when it explodes and breaks in two. After the explosion, a mass $\mathrm{m}$ of the spacecraft is left stationary. The velocity of other part is
(a) $\frac{M v}{(M-m)}$
(b) $\frac{M v}{(M+m)}$
(c) $\left(\frac{\mathrm{m}}{\mathrm{M}+\mathrm{m}}\right) \times \frac{1}{\mathrm{v}}$
(d) $\frac{(M+m) v}{M}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 81

A particle of mass $\mathrm{m}$ moves in the positive direction with speed $\mathrm{v}_{0}$ at the origin. If a force $\mathrm{F}=-\mathrm{kx}^{3}$ acts on the particle, the distance from the origin where the particle stops is
(a) $\left(\frac{m v_{0}^{2}}{k}\right)^{1 / 4}$
(b) $\left(\frac{2 m v_{0}^{2}}{k}\right)^{1 / 4}$
(c) $\left(\frac{\mathrm{mv}_{0}^{2}}{\mathrm{k}}\right)^{1 / 3}$
(d) $\left(\frac{m v_{0}^{2}}{2 k}\right)^{1 / 4}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 82

A body moves a distance of $10 \mathrm{~m}$ along a straight line under the action of a force of $5 \mathrm{~N}$. If the work done is $25 \mathrm{~J}$, the angle, which the force makes with the direction of motion of the body, is
(a) $0^{\circ}$
(b) $30^{\circ}$
(c) $60^{\circ}$
(d) $45^{\circ}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 83

A car climbs up a gradient of 1 in 20 at a speed $5 \mathrm{~m} \mathrm{~s}^{-1}$. The car weighs $6000 \mathrm{~kg}$ and the coefficient of friction is $0.01$. The power required (in $\mathrm{kW}$ ) is $\left(\mathrm{g}=10 \mathrm{~m} \mathrm{~s}^{-2}\right)$
(a) 10
(b) 18
(c) 24
(d) 32

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 84

A $25000 \mathrm{~kg}$ airplane takes off from rest on the run away, and reaches an altitude of $5000 \mathrm{~m}$ and cruises at $900 \mathrm{~km} /$ hour in 8 minute. Assuming no viscous losses, the average power generated by the engines during this period (in $M W$ ) is
(a) $10.5$
(b) $12.8$
(c) $3.4$
(d) $4.2$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 85

A particle of mass $\mathrm{m}$ released from top position on one side of a smooth semicircular surface of radius $\mathrm{r}$ in the vertical plane. Maximum power generated by gravity is when $\theta$ is:
(a) 0
(b) $\frac{\pi}{4}$
(c) $\frac{\pi}{3}$
(d) $\sin ^{-1} \frac{1}{\sqrt{3}}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 86

If the displacement of a body of mass $m$ is given by $x=a \sin \omega t$, then its kinetic energy is
(a) $\frac{1}{2} \operatorname{ma}^{2} \omega^{2}[1+\cos \omega \mathrm{t}]$
(b) $\frac{1}{4} \operatorname{ma}^{2} \omega^{2}[1+\sin 2 \omega \mathrm{t}]$
(c) $\frac{1}{4} \operatorname{ma}^{2} \omega^{2}[1+\cos 2 \omega \mathrm{t}]$
(d) $\frac{1}{4} \operatorname{ma}^{2} \omega^{2} \sin ^{2} \omega \mathrm{t}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 87

A rocket with mass $\mathrm{m}$ and flow rate $\mathrm{m}$ at velocity u travels on a horizontal circular wire of radius $\mathrm{r}$ and friction coef-
ficient $\mu$. The constant velocity with which it travels is (neglect gravity)
(a) $\frac{\mathrm{m}}{\mathrm{m}} \frac{\mathrm{u}}{\mu}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 88

A body of mass $1 \mathrm{~kg}$ has velocity $1 \mathrm{~m} \mathrm{~s}^{-1}$, up an inclined plane of angle of $30^{\circ}$ to the horizontal. The friction coefficient is $\frac{1}{\sqrt{3}} .$ The distance the body travels before stopping is $\left(g=10 \mathrm{~m} \mathrm{~s}^{2}\right)$
(a) $5 \mathrm{~cm}$
(b) $7.5 \mathrm{~cm}$
(c) $10 \mathrm{~cm}$
(d) $6.7 \mathrm{~cm}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 89

A spring is held between two supports and it is in its normal length. If the center is displaced by $\delta \mathrm{x}$ normal to its length, the energy stored is the spring in proportional to $(\delta \mathrm{x})^{\mathrm{n}}$ where $\mathrm{n}$ is
(a) 1
(b) 2
(c) 3
(d) 4

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 90

Water flows from the top of an inclined plane of height $\mathrm{H}$ and is leaves at the end of a tap arranged for maximum range. The maximum value of $\mathrm{R}$ is obtained for $\mathrm{h}$ of $\left(\mathrm{R}=\frac{\mathrm{u} \sqrt{\mathrm{u}^{2}+2 \mathrm{gh}}}{\mathrm{g}}\right)$
(a) $\mathrm{H}$
(b) $\frac{\mathrm{H}}{2}$
(c) $\frac{\mathrm{H}}{3}$
(d) 0

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 91

A small body of mass $\mathrm{m}$ slides down from the top of a hemisphere of radius $\mathrm{r}$. The surface of block and hemisphere are frictionless. The height at which the body loses contact with the surface of the sphere is
(a) $\frac{3}{2} \mathrm{r}$
(b) $\frac{2}{3} \mathrm{r}$
(c) $\frac{1}{2} \mathrm{gt}^{2}$
(d) $\frac{1}{2} \mathrm{r}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 92

A person standing on a vehicle moving with speed $v$ is thrown against a wall when it comes to rest suddenly. Assuming the mass of the person is $\mathrm{m}$ and the wall acts as a spring of constant $\mathrm{k}$, the maximum force experienced is
(a) $\sqrt{\mathrm{mk}} \mathrm{v}$
(b) $\sqrt{2 \mathrm{mk}} \mathrm{v}$
(c) $\sqrt{\frac{\mathrm{mk}}{2}} \mathrm{v}$
(d) $\sqrt{3 \mathrm{mk}} \mathrm{v}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 93

A body of mass $1 \mathrm{~kg}$ is whirled in a vertical circle of radius $0.5 \mathrm{~m}$. What is the velocity of the body when the string makes an angle of $37^{\circ}$ with the vertical, if the tension in the string in this position is $10 \mathrm{~N} ?\left(\tan 37^{\circ}=\frac{3}{4}, \mathrm{~g}=10 \mathrm{~m} \mathrm{~s}^{-2}\right)$
(a) $1 \mathrm{~m} \mathrm{~s}^{-1}$
(b) $2 \mathrm{~m} \mathrm{~s}^{-1}$
(c) $\sqrt{6} \mathrm{~m} \mathrm{~s}^{-1}$
(d) $\sqrt{3} \mathrm{~m} \mathrm{~s}^{-1}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 94

A trough shaped body $M$ of mass $2 \mathrm{~kg}$ and having smooth surface is kept on a smooth floor. Sections $\mathrm{AB}$ and $\mathrm{EF}$ of the inner surface are perfectly vertical while section CD is horizontal and the inside curvature is smooth. A body $\mathrm{m}=1 \mathrm{~kg}$ is released at the top most point A inside M. Neglect friction everywhere.
Kinetic energy of $\mathrm{m}$ when it is at mid position of $\mathrm{CD}$ is
(a) 12 J
(b) $10 \mathrm{~J}$
(c) $8 \mathrm{~J}$
(d) $7.1 \mathrm{~J}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 95

A mass $\mathrm{m}$ resting on a smooth horizontal slot is connected to another equal mass over a smooth pulley. The system is at rest when $\theta=\theta_{0}\left(\Delta \mathrm{y}<<\ell=\ell^{\prime}+\Delta \mathrm{y}\right)$. The velocity of the mass on the floor at $\theta=90^{\circ}$ is
(a) $\sqrt{2 \mathrm{gh}}$
(b) $\sqrt{2 \operatorname{gh}\left(1-\cos \theta_{0}\right)}$
(c) $\sqrt{2 \operatorname{gh}\left(1-\sin \theta_{0}\right)}$
(d) $\sqrt{2 \mathrm{gh}\left(\operatorname{cosec} \theta_{0}-1\right)}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 96

A block of mass $\mathrm{m}$ at the end of a string is whirled round in a vertical circle of radius $\mathrm{R}$. The critical speed of the block at the top of its string below which the string would slacken before the block reaches the top is
(a) $\mathrm{Rg}$
(b) $\frac{1}{\sqrt{\mathrm{Rg}}}$
(c) $\frac{\mathrm{R}}{\mathrm{g}}$
(d) $\sqrt{\mathrm{Rg}}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 97

A body of mass $\mathrm{m}_{2}$ has one of its surfaces in the form of a quarter circle of radius $\mathrm{R}$. A mass $\mathrm{m}_{1}$ is placed on top of a plunger assembly. Assume all surfaces are frictionless. The final velocity of $m_{2}$, if plunger starts to descend from initial position $B$ as shown, is
(a) $\sqrt{2 \mathrm{gR} \cdot \frac{\mathrm{m}_{1}}{\mathrm{~m}_{2}}}$
(b) $\sqrt{2 \mathrm{gR}(1-\sin \theta) \frac{\mathrm{m}_{1}}{\mathrm{~m}_{2}}}$
(c) $\sqrt{2 g R(1-\cos \theta) \frac{m_{1}}{m_{2}}}$
(d) $\sqrt{2 g R \cdot \frac{m_{2}}{m_{1}}}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 98

Potential energy of a particle free to move along $\mathrm{x}$ -axis is given by $\left(\mathrm{x}^{3}-\frac{\mathrm{x}^{2}}{2}\right) \mathrm{J}$, where $\mathrm{x}$ is in metre. The particle is initially kept at $x=0$ and then given a slight displacement $\Delta x$ in $+x$ direction. The forces acting on the particle will displace it to:
(a) $x=0$
(b) $x=+\infty$
(c) $x=-\infty$
(d) $x=+\frac{1}{3} m$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 99

Two springs $\mathrm{P}$ and $\mathrm{Q}$ are identical except that $\mathrm{P}$ is stiffer than $\mathrm{Q}$. If they are stretched by the same amount and $\mathrm{W}_{\mathrm{p}}$ and
$\mathrm{W}_{\mathrm{Q}}$ represent the work expended on the springs then
(a) $\mathrm{W}_{\mathrm{p}}>\mathrm{W}_{\mathrm{Q}}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 100

Two equal masses are sent down on two inclined planes as shown. Both surfaces have same coefficient of friction $\mu$. The loss in kinetic energy is
(a) more in (1)
(b) more in (2)
(c) same in both cases
(d) data insufficient

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 101

Two masses m each are connected by a spring. An identical system of masses moving with a speed $v_{0}$ collides with this system along the axis. The ratio of maximum compression of spring in moving system to that in rest system is(Assume both springs are initially in their normal lengths)
(a) $\frac{1}{\sqrt{2}}$
(b) $\frac{1}{2}$
(c) $\frac{1}{\sqrt{3}}$
(d) 1

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 102

$\mathrm{A}$ and $\mathrm{B}$ are two points on the edge of the floor of a circular room of radius $\mathrm{R}$. If the collisions with the walls are elastic, and if the particle has exactly two collisions with the wall before reaching B from $A, A B$ is a side of a square inscribed with the circumference of the room
(a) $\frac{\mathrm{R}}{2}$
(b) $\frac{\sqrt{3} \mathrm{R}}{2}$
(c) $\sqrt{2} \mathrm{R}$
(d) $\mathrm{R}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 103

Three identical balls of mass $m$ are placed on a straight line on a smooth table separated by a distance between each pair. The critical condition that a striker ball of mass $\mathrm{m}^{\prime}(=\mathrm{k}$ times $\mathrm{m}$ where $\mathrm{k}$ is a constant), knocks all three balls off the table is
(a) $k>3$
(b) $k>6$
(c) $k>1$
(d) $\mathrm{k}>0.5$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 104

A particle moving with a velocity $\mathrm{v}$ in $+\mathrm{x}$ direction has an inelastic collision with a wall moving in opposite direction with a speed $\mathrm{V}$ and coefficient of restitution is e. If the speed of the ball remains same after collision then $\mathrm{V}$ is (modulus value)
(a) $\frac{1-\mathrm{e}}{1+\mathrm{e}} \mathrm{v}$
(b) $\frac{1-\mathrm{e}}{1+2 \mathrm{e}} \mathrm{v}$
(c) $\frac{1-e}{1+e} v^{2}$
(d) $\frac{1-2 e}{1+e} v$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 105

A block $m$, rests on a smooth floor and is connected to a spring of constant $\mathrm{k}$. A mass $\mathrm{m}_{2}$ is placed on top of $\mathrm{m}$, and coefficient of friction between them is $\mu$. The minimum velocity of a body of mass $\mathrm{m}$, which strikes $\mathrm{m}_{1}$ and sticks to it so that $\mathrm{m}_{2}$ slips is
(a) $\mu \mathrm{g} \sqrt{\frac{\mathrm{m}_{1}+\mathrm{m}_{2}+\mathrm{m}_{3}}{\mathrm{k}}}$
(b) $\mu \mathrm{g} \sqrt{\frac{\left(\mathrm{m}_{1}+\mathrm{m}_{2}\right) \mathrm{m}_{3}}{\mathrm{~m}_{1} \mathrm{k}}}$
(c) $\mu \mathrm{g} \sqrt{\frac{\left(\mathrm{m}_{1}+\mathrm{m}_{3}\right) \mathrm{m}_{1}}{\mathrm{~m}_{3} \mathrm{k}}}$
(d) slips for all values

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 106

A large wedge of base angle $\alpha$ and mass $\mathrm{M}$, moving with a velocity $\mathrm{v}$ on a smooth floor has an elastic collision with the bob of mass $\mathrm{m}$, of a pendulum hung from the ceiling and $\mathrm{m}<<\mathrm{M}$. The velocity of the bob after collision is:
(a) $\operatorname{vsin}^{2} \frac{\alpha}{2}$
(b) $\mathrm{vcos}^{2} \frac{\alpha}{2}$
(c) $2 \mathrm{vcos} \alpha$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 107

A body $\mathrm{m}$, rests on the smooth floor while another body $\mathrm{m}_{2}$ is placed on top. All surfaces are smooth. If a velocity $\mathrm{v}_{0}$ is given to $\mathrm{m}_{1}$ towards the right, and the collision of $\mathrm{m}_{2}$ with the side of $\mathrm{m}_{1}$ is elastic the time taken for $\mathrm{m}_{2}$ to slide off is $\left(\frac{\mathrm{m}_{2}}{\mathrm{~m}_{1}}=\mathrm{k}\right)$
(a) $\left(\frac{2 \mathrm{k}+3}{\mathrm{k}+1}\right)\left(\frac{\mathrm{d}}{\mathrm{v}_{0}}\right)$
(b) $\frac{2 \mathrm{~d}}{\mathrm{v}_{0}}, \frac{1}{\mathrm{k}+1}$
(c) $\frac{3 \mathrm{~d}}{\mathrm{v}_{0}}$
(d) $\frac{1-k}{1+k} \cdot \frac{d}{v_{0}}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 108

The system shown (the spring is without tension) travels towards the wall and has an elastic collision. The maximum compression of the spring is
(a) $v_{0} \sqrt{\frac{m}{k}}$
(b) $2 v_{0} \sqrt{\frac{m}{k}}$
(c) $2 v_{0} \sqrt{\frac{2 m}{3 k}}$
(d) $2 v_{0} \sqrt{\frac{m}{3 k}}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 109

A mass $m$ is released from a height $h$ on a block of mass $m$, which rests on a smooth floor. After elastic collision with the surface the mass will rise to a height of
(a) $\mathrm{h}$
(b) $\frac{2 \mathrm{~h}}{3}$
(c) $\frac{4 \mathrm{~h}}{9}$
(d) $\frac{\mathrm{h}}{2}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 110

A particle strikes a quarter circular disc and rebounds elastically with a velocity as shown. The angle by which its velocity vector is rotated is
(a) $\bar{\theta}$
(b) $2 \theta$
(c) $\pi-2 \theta$
(d) $\frac{\pi}{2}+\theta$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 111

Statement 1 If the distance s moved by the body is not in the direction of force but makes an angle $\theta$ with it, then the work done is given by $\mathrm{W}=\mathrm{F}(\mathrm{s} \cos \theta)=\overline{\mathrm{F}} \cdot \overline{\mathrm{s}}$
and
Statement 2 If the force and displacement are at right angles to each other the work done is zero.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 112

Statement 1
The amount of work that we must do in order to bring a moving body to rest is equal to the negative value of kinetic energy of the body. and
Statement 2 The kinetic energy of a moving body is equal to the total work that is done on the body starting from rest.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 113

Statement 1 A negative mechanical energy implies that its potential energy is negative. and
Statement 2
The potential energy value for the reference state is arbitrary.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 114

How much does the body move on the inclined plane before it comes to a halt?
(a) $1 \mathrm{~m}$
(b) $2 \mathrm{~m}$
(c) $3 \mathrm{~m}$
(d) $3.2 \mathrm{~m}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 115

The acceleration of the body during the motion is
(a) $0.5 \mathrm{~m} \mathrm{~s}^{-2}$
(b) $\frac{2}{3} \mathrm{~m} \mathrm{~s}^{-2}$
(c) $2 \mathrm{~m} \mathrm{~s}^{-2}$
(d) $2.3 \mathrm{~m} \mathrm{~s}^{-2}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 116

The time it takes for the body to stop is
(a) $1 \mathrm{~s}$
(b) $2 \mathrm{~s}$
(c) $3 \mathrm{~s}$
(d) $4 \mathrm{~s}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 117

In which of the following is work done by the external agent described?
(a) The motor of a lift going up with the same acceleration as the acceleration due to gravity
(b) A man carrying a bucket of water on level ground.
(c) A crane lowering a load vertically with a constant velocity
(d) A man carrying a bucket of water on a stair case

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 118

A ball of mass $2 \mathrm{~kg}$ is dropped from a height of $0.4 \mathrm{~m}$ above the free end of a vertically fixed spring of constant $\mathrm{k}=200 \mathrm{~N} \mathrm{~m}^{-1}$
(a) The maximum $\mathrm{K} . \mathrm{E}$ attained by the ball is $9 \mathrm{~J}$
(b) The maximum K.E is attained $0.5 \mathrm{~m}$ below original position $\mathrm{P}$.
(c) The maximum potential energy attained by the spring is $16 \mathrm{~J}$
(d) The maximum potential energy attained by the spring is 9 J

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 119

A ball falls on an inclined plane of angle of inclination $37^{\circ}$, from a height of $\frac{20}{9} \mathrm{~m}$ above the point of impact. The coefficient of restitution of the impact is, $\mathrm{e}=\frac{9}{16}$. Then,
(a) The maximum vertical height reached by the ball above the point of impact is $0.8 \mathrm{~m}$
(b) The maximum vertical height reached by the ball above the point of impact is $0.92 \mathrm{~m}$
(c) The velocity of the ball after the impact is $5 \mathrm{~m} \mathrm{~s}^{-1}$
(d) The ball will fall back on the inclined plane after $\frac{3}{4} s$ after the impact

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 120

Match the physical quantities, which are unchanged in the process in column II Column I Column II
(a) Momentum
(p) elastic collision between two bodies
(b) Total energy
(q) inelastic collision between two bodies
(c) Kinetic energy
(r) explosion
(d) Relative velocity
(s) recoil

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 121

The track in the vertical plane has a curved portion of total height $4 \mathrm{R}$, a straight level portion of length $10.5 \mathrm{R}$ and a circular portion of radius $\mathrm{R}$. The straight portion alone is rough. An object starts at $\mathrm{A}$ from rest. When it is at $\mathrm{B}$, the resultant force acting on it makes an angle $\tan ^{-1} \frac{3}{4}$ with the vertical. Determine

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 122

A block of mass $\mathrm{m}=5 \mathrm{~kg}$ slides from the top of a smooth fixed inclined plane of altitude $\mathrm{h}=9 \mathrm{~m}$ and angle of inclination $\tan ^{-1} 0.75$ with the horizontal. Calculate

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 123

If a mass of $10 \mathrm{~kg}$ is kept on the inclined plane at the end of a spring and slowly allowed to come to rest, the spring has a maximum compression of $25 \mathrm{~cm} .$ The friction coefficient between plane and mass is $\mu=\frac{1}{2 \sqrt{3}}$.
(i) What is the spring constant $\mathrm{k}$ ?
(ii) Now the mass is kept at a farther point $\mathrm{A}$ on the plane and released from rest. The spring has a maximum compression of $2 \mathrm{~m}$. What is the height descended by the mass? $\left(\mathrm{g}=10 \mathrm{~m} \mathrm{~s}^{-2}\right)$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 124

(i) A block of mass $\mathrm{m}$ moving at a speed $\mathrm{v}_{0}$ compresses a spring through a distance $\mathrm{x}_{0}$ before its speed is halved. Find the spring constant.
(ii) Suppose this solid block is pushed against a spring of same spring constant as above. The natural length of the spring is $\mathrm{L}$ and it is now held compressed to half its natural length and then the block is released. Find the velocity of the block when the spring relaxes to its original length for the first time.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 125

A block $m$, rests on a smooth floor and is connected to a spring of constant $\mathrm{k}$. A block of mass $\mathrm{m}_{2}$ rests on top. The coefficient of friction between the masses is $\mu$. The mass $\mathrm{m}_{2}$ is suddenly given a velocity $\mathrm{v}_{0}$. If it stops slipping when the compression of spring is maximum, find the maximum compression of the spring

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 126

A $10 \mathrm{H} . \mathrm{P}$ pump, working at $80 \%$ efficiency is used for drawing water from a well. The water level in the well is $6 \mathrm{~m}$ below the location of the pump. The water outlet from pump is horizontal and the water flow rate through the pump is 50 litre per second.
(i) What is the energy spent by the pump in 10 hour?
(ii) What is the speed with which water will come out of the pump?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 127

For a projectile, launched from the ground at an angle ' $\theta$ ' with the horizontal, the ratio of its kinetic energy to its potential energy at the maximum height is $3: 7$. What is the corresponding ratio when the projectile is at.
(i) $\frac{3}{4}$ of its maximum height?
(ii) $\frac{1}{2}$ of its maximum height?
(iii) $\frac{1}{4}$ of its maximum height?
(iv) At what height, expressed as fraction of maximum height, will its $\mathrm{KE}$ and $\mathrm{PE}$ be equal?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 128

A block of mass $\mathrm{m}=12.25 \mathrm{~kg}$ is placed on a table of mass $\mathrm{M}=37.5 \mathrm{~kg}$ which can move without friction on a level floor. A particle of mass $m_{0}=0.25 \mathrm{~kg}$ moving horizontally strikes the block totally inelastically with velocity $300 \mathrm{~m} \mathrm{~s}^{-1}(\mu$ between block and table $=0.25$ ). Calculate
(i) the final velocity of the combined mass.
(ii) the kinetic energy acquired by the combined mass.
(iii) the relative retardation of the block.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 129

A stone is attached to a light inextensible string of length $2 \mathrm{~m}$ and rotated in a vertical circle with the axis of rotation passing through the other end of string.
(i) If the speed of the stone at the top most position is $10 \mathrm{~m} \mathrm{~s}^{-1}$, what is the speed of the stone at the lower most position? Take $g=10 \mathrm{~m} \mathrm{~s}^{-2}$
(ii) If the speed of the stone at the lowest position is $8 \mathrm{~m} \mathrm{~s}^{-1}$, will it be possible to complete the vertical circular motion?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 130

Two particles of mass $\mathrm{m}$ and $\mathrm{km}$ respectively are connected to the ends of a light, inextensible string of length $2 \mathrm{r}$ and tied to a nail at A. Initially the strings are held horizontal as shown. When released, the particles collide in an inelastic collision $(e=0.5)$
(i) What is the speed of the heavy and light particles immediately after collision?
(ii) What is the value of $\mathrm{k}$ for which the lighter particle reaches its initial position after collision?
(iii) For the value of $\mathrm{k}$ in (ii) above, what is the speed of the heavy particle after collision?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 131

A travelling bullet is brought to rest by a wooden block. Let F' be the time average of the resistance force and $\mathrm{F}^{\prime \prime}$ be the distance average of the resistance force, Then
(a) $\mathrm{F}^{\prime}=\mathrm{F}^{n}$
(b) $\mathrm{F}^{\prime}>\mathrm{F}^{\prime}$
(c) $\mathrm{F}^{\prime}<\mathrm{F}^{\prime}$
(d) cannot be concluded with given data

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 132

Work done in time $t\left(\leq t_{0}\right)$ on a body of mass $m$, which is accelerated from rest to speed $u$ in time $t_{0}$, as a function of time $\mathrm{t}$ is given by
(a) $\frac{1}{2} \mathrm{~m} \frac{\mathrm{u}^{2}}{\mathrm{t}_{0}^{2}} \mathrm{t}^{2}$
(b) $\frac{1}{2} \mathrm{~m} \frac{\mathrm{u}^{2}}{\mathrm{t}_{0}} \mathrm{t}$
(c) $\mathrm{m} \frac{\mathrm{u}}{\mathrm{t}_{0}} \mathrm{t}^{2}$
(d) $\frac{1}{2} \mathrm{~m} \frac{\mathrm{u}}{\mathrm{t}_{0}} \mathrm{t}^{2}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 133

A particle is in motion along a straight line under the action of a force $\mathrm{F}$ which varies with velocity $\mathrm{v}$ as per law $\mathrm{F}=$ $\frac{\mathrm{A}}{\mathrm{v}}$ where $\mathrm{A}$ is a constant. The work done by the force in time $\mathrm{t}$ is
(a) At
(b) $\mathrm{At}^{2}$
(c) $\frac{\mathrm{A}}{\mathrm{t}}$
(d) $\frac{\mathrm{A}}{\mathrm{t}^{2}}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 134

A force $(3 \hat{i}-2 \hat{j}) \mathrm{N}$ acting on a particle, does zero work when the particle is displaced from point $(1,-1)$ to a point $(2, a)$. Then a is (position co-ordinates are in metre)
(a) $\frac{1}{2} \mathrm{~m}$
(b) $1 \mathrm{~m}$
(c) $\frac{3}{2} \mathrm{~m}$
(d) $-\frac{3}{2}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 135

A loco-engine of mass 40 ton moves on a straight track having $\mu=0.01$. When its speed is $72 \mathrm{~km} \mathrm{~h}^{-1}$, power developed by engine is $880 \mathrm{~kW}$. Its acceleration at that instant is (in $\left.\mathrm{m} \mathrm{s}^{-2}\right)\left(\mathrm{g}=10 \mathrm{~m} \mathrm{~s}^{-2}\right)$
(a) $0.1$
(b) 1
(c) $0.2$
(d) 2

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 136

A body of mass $4 \mathrm{~kg}$ is projected at $20 \mathrm{~m} \mathrm{~s}^{-1}$ at an angle $57^{\circ}$ to horizontal. Power of the gravitational force on the block at its highest point is
(a) $480 \mathrm{~W}$
(b) $240 \mathrm{~W}$
(c) $640 \mathrm{~W}$
(d) zero

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 137

Power of frictional force on a body of mass $\mathrm{m}$ as a function of time $\mathrm{t}$, if the body is released at $\mathrm{t}=0$ on a rough inclined plane of angle $\theta$ and coefficient of friction $\mu(<\tan \theta)$ is
(a) $\mu \mathrm{mg}^{2} \mathrm{t} \cos \theta$
(b) $\mu \mathrm{mg}^{2} \mathrm{t} \sin \theta$
(c) $\mu \mathrm{mg}^{2} \mathrm{t} \sin \theta(\sin \theta-\mu \cos \theta)$
(d) $\mu m g^{2} \operatorname{tcos} \theta(\sin \theta-\mu \cos \theta)$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 138

If a body moves from rest along a straight line under constant power, its displacement is proportional to time raised to power
(a) $\frac{1}{2}$
(b) 1
(c) $\frac{3}{2}$
(d) 2

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 139

A particle of mass $\mathrm{m}$ is subjected to constant power $\mathrm{P}$. Its displacement when velocity increases from $\mathrm{u}$ to $\mathrm{v}$ is
(a) $\frac{\left(\mathrm{v}^{2}-\mathrm{u}^{2}\right) \mathrm{m}}{2 \mathrm{P}}$
(b) $\frac{\left(\mathrm{v}^{3}-\mathrm{u}^{3}\right) \mathrm{m}}{3 \mathrm{P}}$
(c) $\frac{2 \mathrm{~m}\left(\mathrm{v}^{2}-\mathrm{u}^{2}\right)}{\mathrm{P}}$
(d) $\frac{3 m\left(v^{3}-u^{3}\right)}{P}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 140

Power of a force acting on a particle of mass $2 \mathrm{~kg}$ varies with time as per $\mathrm{P}=\frac{2 \mathrm{t}^{2}}{3} .$ where $\mathrm{P}$ is in watt and $\mathrm{t}$ is in second. At $t=0$, the particle is at rest. Its velocity at $t=3 s$ is $\left(i n m s^{-1}\right)$
(a) 2
(b) $\sqrt{6}$
(c) $2 \sqrt{2}$
(d) 4

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 141

A block is moved from rest by constant power $\mathrm{P}$ along a rough horizontal plane (coefficient of friction $\mu$ ). Then the maximum velocity attained by the block is
(a) $\frac{\mathrm{P}}{\mu \mathrm{mg}}$
(b) $\frac{\mu \mathrm{mg}}{\mathrm{P}}$
(c) $\mu \mathrm{Pmg}$
(d) $\sqrt{\frac{\mathrm{P}}{\mu \mathrm{mg}}}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 142

If the kinetic energy of a body increases by $800 \%$, its momentum increases by
(a) $400 \%$
(b) $200 \%$
(c) $141 \%$
(d) $100 \%$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 143

Ratio of a particle's momentum to kinetic energy is inversely proportional to time. Then the particle executes
(a) uniform motion
(b) uniformly accelerated motion
(c) simple harmonic motion
(d) none of the above

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 145

A $1 \mathrm{~kg}$ stone is dropped down from a height of $2 \mathrm{~m}$ on a vertically fixed spring of spring constant $\mathrm{k}=200 \mathrm{~N} \mathrm{~m}^{-1} .$ The maximum energy stored in the spring subsequently is. (in joule). Take $\mathrm{g}=10 \mathrm{~m} \mathrm{~s}^{-2}$.
(a) $20.05$
(b) 25
(c) $26.2$
(d) 28

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 146

A body of mass $\mathrm{m}$ is projected along a rough inclined plane (having an angle of inclination with horizontal $\theta$, equal to angle of repose) with a velocity $\mathrm{v}$. It travels up a maximum distance s before it comes to a halt. Then $\mathrm{v}$ is
(a) $\sqrt{g s \cos \theta}$
(b) $2 \sqrt{g \sin \theta}$
(c) $2 \sqrt{g \operatorname{stan} \theta}$
(d) $\sqrt{g \tan ^{2} \theta}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 147

In the set up shown the mass $\mathrm{m}=1 \mathrm{~kg}$, kept on a rough floor is projected towards the spring with an initial velocity of $\mathrm{v}=10 \mathrm{~m} \mathrm{~s}^{-1}$. The final energy stored in the spring is $0.04 \mathrm{~J} .$ Then the $\mu$ of the floor can be

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 148

A mass of $1 \mathrm{~kg}$ is sliding down on a rough inclined plane inclined up at $37^{\circ}$ to horizontal and $\mu=0.2 .$ When the velocity of the mass is $12 \mathrm{~m} \mathrm{~s}^{-1} \mathrm{a}$ horizontal force $\mathrm{F}=10 \mathrm{~N}$ starts acting on the body and brings it to halt. The work done by the force is
(a) +120 J
(b) $-10 \mathrm{~J}$
(c) $-80 \mathrm{~J}$
(d) $-120 \mathrm{~J}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 149

In the set up shown, the floor is smooth, $m_{1}=1 \mathrm{~kg}, \mathrm{~m}_{2}=10 \mathrm{~kg}$ and the coefficient of friction between their surfaces is $\mu=0.6$. Assume the surface of $\mathrm{m}$, is large enough so that when $\mathrm{m}_{2}$ is given a velocity $\mathrm{v}$ as shown, it slips on $\mathrm{m}_{1}$ till
the spring is compressed to maximum i.e., till $m_{1}$ stops. The spring constant $\mathrm{k}=200 \mathrm{~N} \mathrm{~m}^{-1}$. The maximum velocity achieved by $\mathrm{m}_{1}$ is
(a) $3 \sqrt{2} \mathrm{~m} \mathrm{~s}^{-1}$
(b) $5 \sqrt{2} \mathrm{~m} \mathrm{~s}^{-1}$
(c) $3 \sqrt{3} \mathrm{~m} \mathrm{~s}^{-1}$
(d) $\sqrt{7} \mathrm{~m} \mathrm{~s}^{-1}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 150

Block A (mass $=10 \mathrm{~kg}$ ) and block $\mathrm{B}$ (mass $=20 \mathrm{~kg}$ ) are connected as shown. The pulleys are light and smooth. The strings are light and inextensible. Initially the system is at rest. When released, the work done by the tension in string on block A, in moving it trough $3 \mathrm{~m}$ upwards along the smooth inclined plane (angle of inclination with horizontal $\left.=30^{\circ}\right)$ is $\left(\mathrm{g}=10 \mathrm{~m} \mathrm{~s}^{-2}\right)$
(a) $450 \mathrm{~J}$
(b) $500 \mathrm{~J}$
(c) $300 \mathrm{~J}$
(d) $250 \mathrm{~J}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 151

A simple pendulum oscillates so that $\theta$ is the maximum angle with vertical. Its maximum speed is $\sqrt{\frac{7}{5}}$ times its speed at $\frac{\theta}{2}$. Then, $\theta$ is
(a) $\cos ^{-1} \frac{1}{2}$
(b) $\cos ^{-1} \frac{1}{4}$
(c) $\cos ^{-1} \frac{1}{6}$
(d) $\cos ^{-1} \frac{1}{8}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 152

A simple pendulum oscillates so that $\theta$ is the maximum angle with vertical. Its maximum speed is $\sqrt{\frac{7}{5}}$ times its speed at $\frac{\theta}{2} .$ Then, $\theta$ is
(a) $\cos ^{-1} \frac{1}{2}$
(b) $\cos ^{-1} \frac{1}{4}$
(c) $\cos ^{-1} \frac{1}{6}$
(d) $\cos ^{-1} \frac{1}{8}$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 153

A block dropped from a height h over a vertically held spring, compresses the spring to maximum of $\mathrm{x}$ at which instant its acceleration is $3 \mathrm{~g}$. Then, $\frac{\mathrm{h}}{\mathrm{x}}$ is
(a) 1
(b) 2
(c) 3
(d) 4

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 154

A block released from top of an inclined plane, of angle $\theta$ with the horizontal, with an initial velocity u down the plane has a speed $v$ after travelling a distance s down the plane. Coefficient of friction is $\mu$. Then which of the following will be constant?
(a) $\mathrm{v}^{2}-2 \mathrm{~g} \sin \theta-2 \mu \mathrm{gscos} \theta$
(b) $\mathrm{v}^{2}-2 \mathrm{~g} \operatorname{ssin} \theta+2 \mu \mathrm{gscos} \theta$
(c) $\mathrm{v}^{2}+2 \mathrm{~g} \sin \theta-2 \mu \mathrm{gs} \cos \theta$
(d) $\mathrm{v}^{2}+2 \mathrm{~g} \sin \theta+2 \mu \mathrm{gscos} \theta$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 155

The potential energy of a particle as a function of its position is given by $U=x^{3}-6 x^{2}$ (in SI unit). Then
(a) at $\mathrm{x}=3 \mathrm{~m}$ it is at stable equilibrium
(b) at $x=4 \mathrm{~m}$ it is unstable equilibrium
(c) at $\mathrm{x}=4 \mathrm{~m}$ it is at stable equilibrium
(d) at $\mathrm{x}=1 \mathrm{~m}$ the force on it is $+8 \mathrm{~N}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 156

The track in the vertical plane has a curved portion of total height $4 \mathrm{R}$, a straight portion of length $10.5 \mathrm{R}$ and a semi circular portion of radius $\mathrm{R}$. The straight line portion is rough $\mu=\frac{1}{4}$, curved portions are smooth Mass $\mathrm{m}=1 \mathrm{~kg}$ released at $\mathrm{A}$, when it reaches $\mathrm{B}$; the normal reaction on it is:

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 157

A pendulum consists of a bob of mass 'm' suspended by a light, inextensible string of length $' \ell^{\circ}$ fixed at $\mathrm{O}$. What is the horizontal velocity ' $\mathrm{V}^{3}$ to be given to the bob at position $\mathrm{A}$, so that it passes position B but does not reach position $\mathrm{C} ?$
(a) $\sqrt{2 \mathrm{~g} \ell}<\mathrm{V}<2 \sqrt{\mathrm{g} \ell}$
(b) $\sqrt{\mathrm{g} \ell}<\mathrm{V}<\sqrt{2 \mathrm{~g} \ell}$
(c) $\sqrt{\mathrm{g} \ell}<\mathrm{V}<\sqrt{5 \mathrm{~g} \ell}$
(d) $\sqrt{2 \mathrm{~g} \ell}<\mathrm{V}<\sqrt{5 \mathrm{~g} \ell}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 158

A $1 \mathrm{~kg}$ mass is released from the top surface of a smooth surface of wedge kept on another smooth surface, as shown. The kinetic energy of the wedge when the mass reaches bottom is approximately
(a) $8.6 \mathrm{~J}$
(b) $10.1 \mathrm{~J}$
(c) $12 \mathrm{~J}$
(d) $15 \mathrm{~J}$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 159

Two masses are attached, each to either end of a spring. The spring is compressed till its potential energy is $\mathrm{U}$ and then released. Their velocity of separation at the instant the heavier mass has a momentum $\mathrm{p}$ is less than or equal to
(a) $\frac{U}{2 p}$
(b) $\frac{\mathrm{U}}{\mathrm{p}}$
(c) $\frac{2 \mathrm{U}}{\mathrm{p}}$
(d) $\frac{4 \mathrm{U}}{\mathrm{p}}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 160

From a hopper, cement is falling vertically on to a horizontal conveyor belt at a constant rate of $10 \mathrm{~kg} / \mathrm{s}$. If the belt is moving at a constant speed of $15 \mathrm{~m} \mathrm{~s}^{-1}$, what is the minimum power required to keep the belt moving?
(a) zero
(b) $1.125 \mathrm{~kW}$
(c) $0.15 \mathrm{~kW}$
(d) $2.25 \mathrm{~kW}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 162

A body $A$ is dropped from a height h. At that instant, another body $B$ is projected up from ground below with a velocity of $+25 \mathrm{~m} \mathrm{~s}^{-1}$. [Take upward direction as +ve] The masses of the bodies are equal and the bodies have an elastic collision in mid-air. After collision the velocity of $\mathrm{A}$ is equal to $-7 \mathrm{~m} \mathrm{~s}^{-1} .$ Then velocity of $\mathrm{A}$ before collision is (in $\left.\mathrm{m} \mathrm{s}^{-1}\right)$
(a) $-25$
(b) $-28$
(c) $-32$
(d) $-35$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 163

A body of mass M moving with kinetic energy K collides elastically with a mass $\mathrm{m}$ at rest and kinetic energy of $\mathrm{M}$ after collision is $\frac{\mathrm{K}}{4}$ and it moves in the same direction. Then the ratio $\frac{\mathrm{M}}{\mathrm{m}}$ :
(a) 2
(b) 3
(c) $3.5$
(d) $4.2$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 164

The bob of a pendulum, $3.2 \mathrm{~m}$ long and fixed at $\mathrm{O}$ as shown is released from a horizontal position as shown. The coefficient of restitution for collision of the bob with the wall is $0.5$. After the fourth collision, the bob will rise to a height of $\left(\mathrm{g}=10 \mathrm{~m} \mathrm{~s}^{-2}\right)$
(a) $0.125 \mathrm{~m}$
(b) $0.5 \mathrm{~m}$
(c) $0.75 \mathrm{~m}$
(d) $1.25 \mathrm{~cm}$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 165

Two bodies are projected vertically upward from the same point at the same instant with velocities $u$ and $1.3 \mathrm{u}$ in $\mathrm{m}$ $s^{-1}$. The coefficient of restitution of the floor is $e=0.5$. The bodies will meet each other after time tequal to (in second) $\left(g=10 \mathrm{~m} \mathrm{~s}^{-2}\right)$
(a) $1.2 \mathrm{u}$
(b) $0.8 \mathrm{u}$
(c) $0.73 \mathrm{u}$
(d) $0.25 \mathrm{u}$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 166

A block of mass $\mathrm{M}$ is suspended by a string. Bullets, each of mass $\mathrm{m}$, are fired vertically upward towards the centre of the block at speed u. If the coefficient of restitution is e, how many bullets per second are to be fired so that the string is just taut?
(a) $\frac{\mathrm{Mu}(1+\mathrm{e})}{\mathrm{mg}}$
(b) $\frac{\mathrm{Mg}(1+\mathrm{e})}{\mathrm{mu}}$
(c) $\frac{\mathrm{Mg}}{\mathrm{mu}(1+\mathrm{e})}$
(d) $\frac{\mathrm{Mg}}{\text { mue }}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 167

A ball falling from height $\mathrm{H}$ rebounds from floor to height $\frac{\mathrm{H}}{2}$. The time of contact is proportional to the time of fall. The average force exerted by the floor is proportional to $\mathrm{H}^{n}$ where $n$ is
(a) 0
(b) $\frac{1}{2}$
(c) 1
(d) none of these

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 168

After an elastic collision between a particle of speed $v$ and a particle at rest, the lighter particle moves at $\frac{v}{2}$. The ratio of their masses (heavier : lighter) is
(a) $\frac{3}{2}$
(b) 2
(c) 3
(d) 4

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 169

A heavy body and a light body travel toward each other along a straight path, with speed $5 \mathrm{~m} \mathrm{~s}^{-1}$ and $10 \mathrm{~m} \mathrm{~s}^{-1}$ respectively. If the coefficient of restitution is $2 / 3$, then the velocity of the light body after collision is
(a) $5 \mathrm{~m} \mathrm{~s}^{-1}$
(b) $10 \mathrm{~ms}^{-1}$
(c) $15 \mathrm{~m} \mathrm{~s}^{-1}$
(d) $20 \mathrm{~ms}^{-1}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 170

A bomb explodes into two parts, after which one part is at rest. The kinetic energy of the other part of mass $2 \mathrm{~kg}$ forms ratio $3: 4$ or $4: 3$ with the (chemical) energy of explosion. The mass of the bomb is (in $\mathrm{kg}$ ) [Assume that the chemical energy is fully released as mechanical energy only]
(a) 5
(b) 8
(c) 10
(d) 6

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 171

Statement 1 A force $\bar{F}$ acting on a particle is given by the expression $\overline{\mathrm{F}}=\mathrm{q}(\overline{\mathrm{v}} \times \overline{\mathrm{B}})$, where $\overline{\mathrm{v}}$ is the velocity of the particle and $\mathrm{q}$ and $\overline{\mathrm{B}}$ are constants. The work done by the force on the particle is zero. and
Statement 2 The force $\overline{\mathrm{F}}$ is perpendicular to $\overline{\mathrm{v}}$.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 172

Statement 1 If the speed of a particle is $2 \mathrm{~m} \mathrm{~s}^{-1}$, the magnitude of its linear momentum (in $\left.\mathrm{kg} \mathrm{m} \mathrm{s}^{-1}\right)$ is numerically equal to its kinetic energy (in joule), irrespective of the mass of the particle.
and
Statement 2
Mass of a particle is equivalent to energy (in joule).

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 173

Statement 1 A block released from top of a rough inclined plane will reach bottom with the same kinetic energy whatever be the angle of inclination and height as long as the horizontal displacement remains same and
Statement 2
In an inclined plane situation, for a given horizontal dimension both the magnitudes of friction force and displacement change as angle of inclination changes.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 174

Statement 1 A body is projected upward along an inclined plane from its bottom with a speed $\mathrm{v}$, when another identical body is projected down along on the same inclined plane with same speed, v. When they travel the same distance on the inclined plane, the loss in $\mathrm{KE}$ is same for both. and
Statement 2 Same magnitude of frictional force in the opposite direction of motion develop in each case and does same negative work.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 175

Statement 1
If there are conservative and non-conservative forces acting, the mechanical energy, is conserved. and
Statement 2
Energy may be transformed from one kind to another but it cannot be created or destroyed.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 176

Statement 1 The conservative force is the negative gradient of potential energy $\left(\mathrm{F}=-\frac{\mathrm{dU}}{\mathrm{dx}}\right)$ and
Statement 2 Potential energy can be defined only for conservative forces.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 177

Statement 1 If the work done by a force on a particle does not depend on path followed by the particle, then the force is called conservative. and

Statement 2 The work done by conservative forces acting on a particle depend only on the initial and final positions of the particle.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 178

Statement 1
A block $A$ is released from rest on top of a movable wedge $B$ as shown. No friction anywhere. At the end of the journey, the block $A$ cannot be found at a point such as $P$ shown, whatever may be the masses, angle, height etc. and
Statement 2
Momentum is conserved in the horizontal direction due to absence of external forces.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 179

Statement 1
Linear momentum of a body at rest in a moving train is zero relative to a man sitting in the train. and
Statement 2
Linear momentum of a body at rest on a moving train is not zero for a man standing on the ground.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 180

Statement 1 A ball dropped on an inclined plane $\left(\theta=30^{\circ}\right.$ with horizontal) from a height $\mathrm{h}$, rebounds and reaches ground. Smaller the value of co-efficient of restitution e, upto the limit that the body clears the inclined plane and reaches ground, slower it reaches ground after rebounding. and
Statement 2 Smaller the value of $\mathrm{e}$, smaller the velocity after impact.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 181

The mechanical energy of the system is
(a) $20 \mathrm{~J}$
(b) $29 \mathrm{~J}$
(c) $40 \mathrm{~J}$
(d) $49 \mathrm{~J}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 182

The value of $\mathrm{x}$ for which the kinetic energy is maximum is
(a) 0
(b) $2 \mathrm{~m}$
(c) $1 \mathrm{~m}$
(d) $4 \mathrm{~m}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 183

The maximum value of kinetic energy is
(a) $20 \mathrm{~J}$
(b) $49 \mathrm{~J}$
(c) $29 \mathrm{~J}$
(d) $40 \mathrm{~J}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 184

If $\mathrm{u}$ is the initial velocity of moving hydrogen atom, after collision, the combined velocity is
(a) $\underline{u}$
(b) $\frac{\mathrm{u}}{\mathrm{s}}$
(c) $\underline{\mathrm{u}}$
(d) 0

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 185

Kinetic energy lost after collision is
(a) $\mathrm{m}_{\mathrm{H}} \frac{\mathrm{u}^{2}}{2}$
(b) $\mathrm{m}_{\mathrm{H}} \frac{\mathrm{u}^{2}}{4}$
(c) $\mathrm{m}_{\mathrm{H}} \frac{\mathrm{u}^{2}}{8}$
(d) $2 \mathrm{~m}_{\mathrm{H}} \mathrm{u}^{2}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 186

The value of initial velocity of the hydrogen atom is
(a) $6.25 \times 10^{4} \mathrm{~m} \mathrm{~s}^{-1}$
(b) $5.6 \times 10^{4} \mathrm{~m} \mathrm{~s}^{-1}$
(c) $6.8 \times 10^{5} \mathrm{~m} \mathrm{~s}^{-1}$
(d) $7.2 \times 10^{4} \mathrm{~m} \mathrm{~s}^{-1}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 187

The modulus value of relative velocity of the ball with respect to the wall after collision is
(a) $6 \mathrm{~m} \mathrm{~s}^{-1}$
(b) $8 \mathrm{~m} \mathrm{~s}^{-1}$
(c) $9 \mathrm{~m} \mathrm{~s}^{-1}$
(d) $10 \mathrm{~m} \mathrm{~s}^{-1}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 188

The actual velocity of the ball after collision is (take $+\mathrm{X}$ as positive direction
(a) $-8 \mathrm{~m} \mathrm{~s}^{-1}$
(b) $-18 \mathrm{~m} \mathrm{~s}^{-1}$
(c) $-2 \mathrm{~m} \mathrm{~s}^{-1}$
(d) $2 \mathrm{~m} \mathrm{~s}^{-1}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 189

Loss in kinetic energy of the ball in percentage is:
(a) 90
(b) 85
(c) 96
(d) 99

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 190

A force $\bar{F}=k\left(y^{2} \hat{i}+x \hat{j}\right)$, where $k$ is a positive constant, acts on a particle when it is at position $(x, y)$. First, the particle is taken through path $\mathrm{OAB}$ and the work done by the force is $\mathrm{W}_{\mathrm{OAB}^{*}}$. Then the particle is taken on through path $\mathrm{OCB}$ and the work done by the force is $\mathrm{W}_{\mathrm{ocB}^{\prime}}$. Then
(a) $\mathrm{W}_{\text {OAB }}=\mathrm{ka}^{2}$
(b) $\mathrm{W}_{\mathrm{OCB}}=\mathrm{ka}^{2}$
(c) the force is conservative
(d) the force is non-conservative

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 191

A force $F=(4 i+5 j) \mathrm{N}$ acts on a particle located at the origin $\mathrm{O}$
(a) The work done in taking the particle parallel to the axes to a point $\mathrm{B}(2 \mathrm{~m}, 2.4 \mathrm{~m})$ is $15 \mathrm{~J}$
(b) The work done in taking the particle from $\mathrm{O}$ along the $\mathrm{X}$ -axis to a point $\mathrm{A}(3 \mathrm{~m}, 0)$ is $15 \mathrm{~J}$
(c) The work done in taking the particle from $\mathrm{O}$ to $(3,0)$ and then to $(2,2.4)$ is $20 \mathrm{~J}$
(d) The work done in taking the particle from $\mathrm{O}$ directly to $(2,2.4)$ is $20 \mathrm{~J}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 192

Figure shows the force F (in newton) acting on a body as a function of $x$. The work done in moving the body.
(a) from $x=0$ to $x=2 \mathrm{~m}$ is $8 \mathrm{~J}$.
(b) from $x=2 \mathrm{~m}$ to $\mathrm{x}=4 \mathrm{~m}$ is $16 \mathrm{~J}$
(c) from $x=0$ to $x=6 \mathrm{~m}$ is $30 \mathrm{~J}$
(d) from $x=0$ to $x=6 \mathrm{~m}$ is $32 \mathrm{~N}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 193

A block of mass $5 \mathrm{~kg}$, initially at rest on a horizontal floor, moves under the action of a horizontal force of $20 \mathrm{~N}$. The coefficient of friction between the block and the floor is $0.2$. If $g=10 \mathrm{~m} \mathrm{~s}^{-2}$
(a) The work done by the applied force in $8 \mathrm{~s}$ is $1280 \mathrm{~J} .$
(b) The work done by the frictional force in $8 \mathrm{~s}$ in $640 \mathrm{~J}$
(c) The work done by the net force in $8 \mathrm{~s}$ in $720 \mathrm{~J}$.
(d) The change in $\mathrm{K} . \mathrm{E}$ of the block in $10 \mathrm{~s}$ is $1000 \mathrm{~J} .$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 194

A block of mass $4 \mathrm{~kg}$ is hanging over a smooth and light pulley through a light and inextensible string. The other end of the string is pulled by a constant force $F=60 \mathrm{~N}$. The kinetic energy of the particle increases by $60 \mathrm{~J}$ in a given interval of time. Then $\left(g=10 \mathrm{~m} \mathrm{~s}^{-2}\right)$.
(a) The tension in the string is $60 \mathrm{~N}$
(b) The displacement of the block in the given interval of time is 3 metre.
(c) Work done by gravity is $120 \mathrm{~J}$
(d) Work done by the force is $180 \mathrm{~J}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 195

A body of mass $1 \mathrm{~kg}$ moving along a straight line with a velocity of $4 \mathrm{~m} \mathrm{~s}^{-1}$, collides head on with a body of mass $2 \mathrm{~kg}$ moving along the same line with a velocity of $3 \mathrm{~m} \mathrm{~s}^{-1}$. After collision the two bodies stick together and move with a common velocity of magnitude $\left(\mathrm{m} \mathrm{s}^{-1}\right)$
(a) $\frac{3}{2}$
(b) $\frac{10}{3}$
(c) $\frac{2}{3}$
(d) $\frac{3}{4}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 196

A body of mass $5 \mathrm{~kg}$ moving a velocity of $4 \mathrm{~m} \mathrm{~s}^{-1}$ along a straight line, collides with a body of $2 \mathrm{~kg}$ moving along the same line with a velocity of $8 \mathrm{~m} \mathrm{~s}^{-1}$. If the collision is perfectly inelastic, the magnitude of the velocity of composite mass after the collision is
(a) $\frac{30}{7} \mathrm{~m} \mathrm{~s}^{-1}$
(b) $\frac{36}{7} \mathrm{~m} \mathrm{~s}^{-1}$
(c) $\frac{7}{4} \mathrm{~m} \mathrm{~s}^{-1}$
(d) $\frac{4}{7} \mathrm{~ms}^{-1}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 197

A $0.8 \mathrm{~kg}$ ball is attached to a light, inextensible string of length $\ell$ and the other end of the string is fixed at $\mathrm{P}$ at a height $\ell$ above a smooth floor. $\mathrm{B}$ is a ball of $\mathrm{kg}$ mass kept on the smooth floor vertically below P. A spring of spring constant $1000 \mathrm{~N} \mathrm{~m}^{-1}$ is kept as shown with one end fixed. $\mathrm{A}$ is held in a horizontal position as shown and released, and has a collision with $B$, with co-efficient of restitution, $\mathrm{e}=0.8 .$ A comes to a stop and $\mathrm{B}$ moves ahead and compresses spring and the maximum acceleration the spring produces on B is $80 \mathrm{~m} \mathrm{~s}^{-2}$. Then
(a) length $\ell=\frac{1}{2} \mathrm{n}$
(b) $\ell=1 \mathrm{~m}$
(c) Maximum P.E attained by the spring is $3.2 \mathrm{~J}$.
(d) After B returns and hits $\mathrm{A}, \mathrm{B}$ will come to a halt.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 198

A particle of mass $\mathrm{m}$ is projected vertically upwards from point $\mathrm{D}$ on ground with a kinetic energy K. $\mathrm{PE}_{\mathrm{A}}, \mathrm{PE}_{\mathrm{B}}, \mathrm{PE}_{\mathrm{C}}$ and $\mathrm{PE}_{\mathrm{D}}$ are the potential energies at positions $A, B, C$ and $D$ respectively. Given $P E_{D}=0$ and maximum height is at $A$, $g=10 \mathrm{~m} \mathrm{~s}^{-2} .$ Match the columns $\left(\mathrm{KE}_{A}, \mathrm{KE}_{\mathrm{E}}, \mathrm{KE}_{\mathrm{C}}=\right.$ kinetic energies at $\mathrm{A}, \mathrm{B}$ and $\left.\mathrm{C}\right)$
Column I Column II
(a) $\mathrm{KE}_{\mathrm{B}}$
(p) Half of kinetic energy at C
(b) $\mathrm{PE}_{\mathrm{c}}$
(q) One third of total mechanical energy at $\mathrm{B}$
(c) $\frac{\mathrm{PE}_{\mathrm{B}}}{\mathrm{KE}_{\mathrm{B}}}$
(r) Greater than 1
(d) $\frac{\mathrm{KE}_{\mathrm{C}}}{\mathrm{KE}_{\mathrm{B}}}$
(s) $\frac{\mathrm{KE}_{\mathrm{C}}}{\mathrm{PE}_{\mathrm{C}}}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 199

Two identical spherical balls, each of mass $\mathrm{m}$, travelling along the same line with speeds $\mathrm{u}_{1}$ and $\mathrm{u}_{2}$ undergo one dimensional elastic collision. The resulting speeds are $\mathrm{v}_{1}$ and $\mathrm{v}_{2}$ respectively Column I Column II
(a) $\overrightarrow{\mathrm{u}_{2}}=0$
(p) speeds exchanged after collision.
(b) $\overrightarrow{\mathrm{u}}_{2}=-\overrightarrow{\mathrm{u}}_{1}$
(q) motion of balls after collision is as if no collision took place.
(c) $\overrightarrow{\mathrm{u}_{1}}$ and $\overrightarrow{\mathrm{u}}_{2}$ are in opposite directions and $\mathrm{u}_{1}>\mathrm{u}_{2}$
(r) $\vec{v}_{2}=\vec{u}_{1}$
(d) $\overrightarrow{\mathrm{u}}_{1}$ and $\overrightarrow{\mathrm{u}}_{2}$ are in the same direction and $\mathrm{u}_{1}>\mathrm{u}_{2}$
(s) both balls turn back with exchanged speeds.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 200

A body of mass $m_{1}$ collides one dimensionally with another stationary body of mass $m_{2}$. The initial and final velocities of mass $\mathrm{m}_{1}$ are $\mathrm{u}_{1}$ and $\mathrm{v}_{1}$ and the final velocity of mass $\mathrm{m}_{2}$ is $\mathrm{v}_{2}$. The coefficient of restitution is e. After the collision Column I Column II
(a) Velocity of the second body is maximum when
(p) $e=1$
(b) Momentum of the second body is maximum when
(q) $\mathrm{m}_{1}>>\mathrm{m}_{2}$
(c) Kinetic energy of the second body is maximum when
(r) $e=0$
(d) Velocity of the two bodies are equal when
(s) $\mathrm{m}_{1}<<\mathrm{m}_{2}$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator