Objective Physics for NEET

Abhay Kumar

Chapter 5

Work, Power, Energy and Circular Motion - all with Video Answers

Educators

Chapter Questions

Problem 1

A position dependent force $F=7-2 x+3 x^{2} \mathrm{~N}$ acts on a small body of mass $2 \mathrm{~kg}$ and displaces it from $x=0$ to $x=5 \mathrm{~m}$. The work done in joule is
(a) 70
(b) 270
(c) 35
(d) 135

Surendra Kumar

Surendra Kumar

Numerade Educator

Problem 2

A car of $1400 \mathrm{~kg}$ is moving on a circular path of radius 30 $\mathrm{m}$ with a speed of $40 \mathrm{~km} / \mathrm{h}$. When the driver applies the brakes and the car continues to move along the circular path, what is the maximum deceleration possible if the tyres are limited to a total horizontal friction of $10.6 \mathrm{kN}$ ?
(a) $10 \mathrm{~m} / \mathrm{s}^{2}$
(b) $6.36 \mathrm{~m} / \mathrm{s}^{2}$
(c) $4 \mathrm{~m} / \mathrm{s}^{2}$
(d) None of these.

Dheeraj Sharma

Numerade Educator

Problem 3

Three identical cars $A, B$ and $C$ are moving at the same speed on three bridges. The car $A$ goes on a plane bridge $B$ on a bridge convex upwards and $C$ goes on a bridge concave upwards. Let $F_{A}, F_{B}$ and $F_{C}$ be the normal forces exerted by the cars on the bridges when they are at the middle of the bridges. Then
(a) $F_{A}$ is maximum of the three forces
(b) $F_{B}$ is maximum of the three forces
(c) $F_{c}$ is maximum of the three forces
(d) $F_{A}=F_{B}=F_{C}$

Dheeraj Sharma

Numerade Educator

Problem 4

A string of length $L$ is fixed at one end and carries a mass $M$ at the other end. The string makes $2 / \pi$ revolutions per second around the vertical axis through the fixed end as shown in the figure, then tension in the string is
(a) $M L$
(b) $2 M L$
(c) $4 \overline{M L}$
(d) $16 \mathrm{ML}$

Dheeraj Sharma

Numerade Educator

Problem 5

Two spheres of equal masses are attached to a string of length $2 \mathrm{~m}$ as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant rate. What is the value of the ratio
$\left(\frac{\text { Tension in the string between } P \text { and } Q}{\text { Tension in the string between } P \text { and } O}\right)$
(a) $\frac{1}{2}$
(b) $\frac{2}{3}$
(c) $\frac{3}{2}$
(d) 2

Dheeraj Sharma

Numerade Educator

Problem 6

Two wires $A C$ and $B C$ are tied at $C$ to a small sphere of mass $5 \mathrm{~kg}$, which revolves at a constant speed $v$ in the horizontal circle of radius $1.6 \mathrm{~m}$. The minimum value of $\underline{y}$ is
(a) $3.01 \mathrm{~m} / \mathrm{s}$
(b) $4.01 \mathrm{~m} / \mathrm{s}$
(c) $8.2 \mathrm{~m} / \mathrm{s}$
(d) $3.96 \mathrm{~m} / \mathrm{s}$

Dheeraj Sharma

Numerade Educator

Problem 7

A stone of mass $1 \mathrm{~kg}$ tied to a light inextensible string of length $L=10 / 3 \mathrm{~m}$ is whirling in a circular path of radius $L$, in a vertical plane. If the ratio of the maximum tension in the string to the minimum tension is 4 and if $g$ is taken to be $10 \mathrm{~m} / \mathrm{s}^{2}$, the speed of the stone at the highest point of the circle is
(a) $20 \mathrm{~m} / \mathrm{s}$
(b) $10 \sqrt{3} \mathrm{~m} / \mathrm{s}$
(c) $5 \sqrt{2} \mathrm{~m} / \mathrm{s}$
(d) $10 \mathrm{~m} / \mathrm{s}$

Dheeraj Sharma

Numerade Educator

Problem 8

A heavy small-sized sphere is suspended by a string of length $l$. The sphere rotates uniformly in a horizontal circle with the string making an angle $\theta$ with the vertical. Then the time period of this conical pendulum is
(a) $t=2 \pi \sqrt{\frac{g}{l \sin \theta}}$
(b) $t=2 \pi \sqrt{\frac{l \sin \theta}{g}}$
(c) $t=2 \pi \sqrt{\frac{l \cos \theta}{g}}$
(d) $t=2 \pi \sqrt{\frac{g}{l \cos \theta}}$

Dheeraj Sharma

Numerade Educator

Problem 9

A body of mass $M \mathrm{~kg}$ is on the top point of a smooth hemisphere of radius $5 \mathrm{~m}$. It is released to slide down the surface of the hemisphere. It leaves the surface when its velocity is $5 \mathrm{~m} / \mathrm{s}$. At this instant the angle made by the radius vector of the body with the vertical is (Acceleration due to gravity $=10 \mathrm{~m} / \mathrm{s}$ )
(a) $30^{\circ}$
(b) $45^{\circ}$
(c) $60^{\circ}$
(d) $90^{\circ}$

Dheeraj Sharma

Numerade Educator

Problem 10

A small block slides down from the top of a hemisphere of radius $r$. It is assumed that there is no friction between the block and the hemisphere. At what height, $h$ will the block lose contact with the surface of sphere?
(a) $\frac{r}{3}$
(b) $\frac{2 r}{3}$
(c) $\frac{r}{2}$
(d) $\frac{r}{4}$

Dheeraj Sharma

Numerade Educator

Problem 11

A hollow vertical cylinder of radius $R$ and height $h$ has smooth internal surface. A small particle is placed in contact with the inner side of the upper rim at a point $P$. It is given a horizontal speed $v_{0}$ tangential to rim. It leaves the lower rim at point $Q$, vertically below $P$. The number of revolutions made by the particle will be
(a) $\frac{h}{2 \pi R}$
(b) $\frac{v_{0}}{\sqrt{2 g h}}$
(c) $\frac{2 \pi R}{h}$
(d) $\frac{v_{0}}{2 \pi R}\left(\sqrt{\frac{2 h}{g}}\right)$

Dheeraj Sharma

Numerade Educator

Problem 12

A particle describes a horizontal circle in a conical funnel whose inner surface is smooth with speed of $0.5$ $\mathrm{m} / \mathrm{s}$. What is the height of the plane of circle from vertex of the funnel?
(a) $0.25 \mathrm{~cm}$
(b) $2 \mathrm{~cm}$
(c) $4 \mathrm{~cm}$
(d) $2.5 \mathrm{~cm}$

Dheeraj Sharma

Numerade Educator

Problem 13

A person with a mass of $M \mathrm{~kg}$ stands in contact against the wall of a cylindrical drum of radius $r$ rotating with an angular velocity $\omega$. If the coefficient of friction between the wall and the clothing is $\mu$, the minimum rotational speed of the cylinder which enables the person to remain stuck to the wall when the floor is suddenly removed, is
(a) $\omega_{\min }=\sqrt{\frac{g}{\mu r}}$
(b) $\omega_{\min }=\sqrt{\frac{\mu r}{g}}$
(c) $\omega_{\min }=\sqrt{\frac{2 g}{\mu r}}$
(d) $\omega_{\min }=\sqrt{\frac{r g}{\mu}}$

Dheeraj Sharma

Numerade Educator

Problem 14

A force $\vec{F}=(a \hat{i}+b \hat{j}) \mathrm{N}$ acts on a body and displace it by $\vec{s}=(c \hat{i}+d \hat{j}) \mathrm{m}$. The work done by the force is
(a) $a b+c d$
(b) $\frac{a b}{c d}$
(c) $a c+b d$
(d) $\frac{a c}{b d}$

Dheeraj Sharma

Numerade Educator

Problem 15

A body of mass $3.0 \mathrm{~kg}$ moves under the influence of some external force such that its position $s$ as a function of time $t$ is given by $s=6 t^{3}-t^{2}+1$ where $s$ is in metres and $t$ is in seconds. The work done by the force in first three seconds is
(a) $18 \mathrm{~J}$
(b) $1800 \mathrm{~J}$
(c) $3660 \mathrm{~J}$
(d) $36504 \mathrm{~J}$

Dheeraj Sharma

Numerade Educator

Problem 16

The work done by a force $F=k x^{2}$ acting on a particle at an angle $60^{\circ}$ with $x$ -axis to displace it from $x=2 \mathrm{~m}$ to $x$ $=3 \mathrm{~m}$ is
(a) $\frac{5}{6} k$
(b) $\frac{5}{2} k$
(c) $\frac{19}{2} k$
(d) $\frac{19}{6} k$

Dheeraj Sharma

Numerade Educator

Problem 17

A particle of mass $m$ is moving along $+y$ direction under the influence of force. The displacement of the particle is related with time as
$y=t^{2}-6 t+9$
The displacement (in metre) of the particle when it comes to rest, is
(a) zero
(b) 3
(c) 6
(d) 9

Dheeraj Sharma

Numerade Educator

Problem 18

In Q.No. 16 , the work done (in joule) by the particle in first six second is
(a) zero
(b) 3
(c) 6
(d) 9

Dheeraj Sharma

Numerade Educator

Problem 19

A particle moves along the $x$ -axis from $x=1$ to $x=3 \mathrm{~m}$ under the influence of a force $F=3 x^{2}-2 x+5$ newton. The work done in this process is
(a) $9 \mathrm{~J}$
(b) $28 \mathrm{~J}$
(c) $27 \mathrm{~J}$
(d) zero

Dheeraj Sharma

Numerade Educator

Problem 20

A force $\vec{F}=-k(y \hat{i}+x \hat{j})$, where $k$ is a constant, acts on a particle moving in the $x-y$ plane. Starting from the origin, the particle is taken from the origin $(0,0)$ to the point $(a, 0)$ and then from the point $(a, 0)$ to the point $(a,$,
a). The total work done by the forc $\vec{F}$ on the particle is
(a) $\mathrm{ka}^{2}$
(b) $-\mathrm{ka}^{2}$
(c) $2 \mathrm{ka}^{2}$
(d) $-2 \mathrm{ka}^{2}$

Dheeraj Sharma

Numerade Educator

Problem 21

A body of mass $6 \mathrm{~kg}$ is acted upon by a force which causes a displacement in it given by $x=\frac{t^{2}}{4} \mathrm{~m}$ where $t$
is the time in second. The work done by the force in 2 second is
(a) $12 \mathrm{~J}$
(b) $9 \mathrm{~J}$
(c) $6 \mathrm{~J}$
(d) $3 \mathrm{~J}$

Dheeraj Sharma

Numerade Educator

Problem 22

A body starts from rest and acquires a velocity $V$ in time $T$. The work done on the body in time $t$ will be proportional to
(a) $\frac{V}{T} t$
(b) $\frac{V^{2} t^{2}}{T}$
(c) $\frac{V^{2}}{T^{2}} t$
(d) $\frac{V^{2}}{T^{2}} t^{2}$

Dheeraj Sharma

Numerade Educator

Problem 23

Force-displacement curve for a body is shown in the figure.
Motion is in one dimensional. Work done by the force in displacing the body from displacement zero to $6 \mathrm{~m}$ is given by
(a) zero
(b) $10 \mathrm{~J}$
(c) $20 \mathrm{~J}$
(d) $60 \mathrm{~J}$

Dheeraj Sharma

Numerade Educator

Problem 24

A particle moves along a curve of unknown shape but magnitude of force $\dot{F}$ is constant and always acts along the tangent to the curve. Then,
(a) $\vec{F}$ must be conservative
(b) $\vec{F}$ may be conservative
(c) $\bar{F}$ must be non-conservative
(d) $\vec{F}$ may be non-conservative

Dheeraj Sharma

Numerade Educator

Problem 25

A rain drop of radius $r$ falls from a certain height $h$ above the ground. The work done by the gravitational force is proportional to
(a) $\frac{1}{r}$
(b) $r$
(c) $r^{2}$
(d) $r^{3}$

Dheeraj Sharma

Numerade Educator

Problem 26

If $\vec{F}=\hat{i} F_{x}+\hat{j} F_{y}+\hat{k} F_{t}$ is conservative, then
(a) $\frac{\partial F_{x}}{\partial y}=\frac{\partial F_{y}}{\partial x} ; \frac{\partial F_{y}}{\partial z}=\frac{\partial F_{z}}{\partial y} ; \frac{\partial F_{z}}{\partial x}=\frac{\partial F_{x}}{\partial z}$
(b) $\frac{\partial F_{x}}{\partial y} \neq \frac{\partial F_{y}}{\partial x} ; \frac{\partial F_{y}}{\partial z} \neq \frac{\partial F_{z}}{\partial y} ; \frac{\partial F_{z}}{\partial x} \neq \frac{\partial F_{x}}{\partial z}$
(c) $\frac{\partial F_{x}}{\partial x}=\frac{\partial F_{y}}{\partial y}=\frac{\partial F_{z}}{\partial z} \neq 0$
(d) $\frac{\partial F_{y}}{\partial y}=\frac{\partial F_{x}}{\partial x}+\frac{\partial F_{z}}{\partial z}$

Dheeraj Sharma

Numerade Educator

Problem 27

The closed line integral of force $\vec{F}$ taken along the closed curve is given by
$$
\oint \vec{F} \cdot d \hat{l}
$$
If the force $\vec{F}$ is conservative, then the value of this integral is
(a) zero
(b) not zero
(c) greater than zero
(d) smaller than zero

Dheeraj Sharma

Numerade Educator

Problem 28

A block of mass $m$ is pulled along a horizontal surface by applying force at an angle $\theta$ with the horizontal. If the block moves with uniform velocity and has a displacement $x$ while the coefficient of friction between the block and the surface is $\mu$ the work done by the applied force is
(a) $\frac{\mu m g x \cos \theta}{\mu \cos \theta+\sin \theta}$
(b) $\frac{m g x \cos \theta}{\cos \theta+\mu \sin \theta}$
(c) $\frac{\mu m g x}{\mu \cos \theta+\sin \theta}$
(d) $\frac{\mu m g x \cos \theta}{\cos \theta+\mu \sin \theta}$

Dheeraj Sharma

Numerade Educator

Problem 29

During the swinging of simple pendulum,
(a) the work done by tension force is always zero
(b) the work done by the gravitational force is zero
(c) the mechanical energy of the bob remains constant in the presence of air resistance
(d) the mechanical energy of the bob does not remain constant in the absence of air

Dheeraj Sharma

Numerade Educator

Problem 30

It is easier to draw up a wooden block along an inclined plane then to haul it up vertically, principally because
(a) the friction is reduced
(b) the mass becomes smaller
(c) only a part of the weight has to be overcome
(d) $G$ becomes smaller.

Dheeraj Sharma

Numerade Educator

Problem 31

The figure shows the vertical section of a frictionless surface. A block of mass $2.0 \mathrm{~kg}$ is released from position $A$. Its kinetic energies (in $\mathrm{J}$ ) as it reaches positions $B, C$ and $D$ (Given gravitational field $9.8 \mathrm{~J} / \mathrm{m} / \mathrm{kg}$ ) are
(a) $176.4 ; 274.4 ; 137.2$
(b) $137.2 ; 274.4 ; 176.4$
(c) $176.4 ; 137.2 ; 274.4$
(d) $274.4 ; 137.2 ; 176.4$

Dheeraj Sharma

Numerade Educator

Problem 32

A boy whose mass is $51 \mathrm{~kg}$ climbs, with constant speed, a vertical rope, $6 \mathrm{~m}$ long in 10 seconds. How much work does the boy perform and what will be the power output of the boy during the climb? (Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$ )
(a) $30 \mathrm{~J}, 3.6 \mathrm{~W}$
(b) $306 \mathrm{~J}, 306 \mathrm{~W}$
(c) $3060 \mathrm{~J}, 306 \mathrm{~W}$
(d) $36000 \mathrm{~J}, 3060 \mathrm{~W}$

Dheeraj Sharma

Numerade Educator

Problem 33

$A B C D E$ is a channel in the vertical plane, part $B C D E$ being circular with radius $r$. A block is released from $A$ and slides without friction and without rolling. The block will complete the loop if $h$ is
(a) $h \leq \frac{3}{2} r$
(b) $h \geq \frac{5}{2} r$
(c) $h \geq \frac{3}{2} r$
(d) $h \leq \frac{5}{2} r$

Dheeraj Sharma

Numerade Educator

Problem 34

A ball is thrown up from the earth by a person and then caught by him on its return. When the ball falls towards the earth, the ratio of kinetic energies gained by the ball $\left(K_{R}\right)$ and the earth $\left(K_{k}\right)$ is
(a) $\frac{K_{E}}{K_{B}}=$ zero
(b) $\frac{K_{E}}{K_{B}}=\frac{m_{E}}{m_{\mathbb{B}}}$
(c) $\frac{K_{E}}{K_{B}}=\frac{m_{D}}{m_{k}}$
(d) $\frac{K_{E}}{K_{B}}=$ Infinite

Dheeraj Sharma

Numerade Educator

Problem 35

A rubber ball is dropped from a height of $5 \mathrm{~m}$ on the surface of a planet where the acceleration due to gravity is not known. On bouncing, it rises to $1.8 \mathrm{~m}$. The ball loses its velocity on bouncing by a factor of
(a) $\frac{16}{25}$
(b) $\frac{2}{5}$
(c) $\frac{3}{5}$
(d) $\frac{9}{2}$

Dheeraj Sharma

Numerade Educator

Problem 36

10 litres of water per second is lifted from a well through $20 \mathrm{~m}$ and delivered with a velocity of $10 \mathrm{~m} / \mathrm{s}$, then the power (in $\mathrm{kW}$ ) of the motor is (Take $\left.g=10 \mathrm{~m} / \mathrm{s}^{2}\right)$
(a) $0.5$
(b) $1.25$
(c) $2.5$
(d) $5.0$

Dheeraj Sharma

Numerade Educator

Problem 37

Two springs $P$ and $Q$ are stretched by applying forces of equal magnitudes at the four ends. If the spring constant of $P$ is 2 times greater than that of $Q$ and the energy stored in $P$ is $E$, then the energy stored in $Q$ is
(a) $\frac{E}{4}$
(b) $\frac{E}{2}$
(c) $E$
(d) $2 E$

Dheeraj Sharma

Numerade Educator

Problem 38

The potential energy of a particle of mass $5 \mathrm{~kg}$ moving in the $x-y$ plane is given by $U=-7 x+24 y$ joule, $x$ and $y$ being in metre. Initially at $t=0$ the particle is at the origin and moving with velocity $(2 \hat{i}+3 \hat{j}) \mathrm{m} / \mathrm{s}$. The magnitude of the force on the particle and the acceleration of the particle are
(a) $35 \mathrm{~N}, 7 \mathrm{~m} / \mathrm{s}^{2}$
(b) $30 \mathrm{~N}, 6 \mathrm{~m} / \mathrm{s}^{2}$
(c) $25 \mathrm{~N}, 5 \mathrm{~m} / \mathrm{s}^{2}$
(d) None of these

Dheeraj Sharma

Numerade Educator

Problem 39

The potential energy of a body is given by
$$
U=40+6 x^{2}-7 x y+8 y^{2}+32 z
$$
where $U$ is in joule and $x, y, z$ in metre. Deduce the $x, y$ and $z$ components of the force (in newton) on the body when it is in position $(-2,0,+5)$
(a) $F_{x}=-32 ; F_{y}=-14 ; F_{z}=24$
(b) $F_{x}=-14 ; F_{y}=-32 ; F_{z}=24$
(c) $F_{x}=24 ; F_{y}=-32 ; F_{z}=-14$
(d) $F_{x}=24 ; F_{y}=-14 ; F_{\tau}=-32$

Dheeraj Sharma

Numerade Educator

Problem 40

An elastic string of unstretched length $I$ and force constant $k$ is stretched by a small length $x .$ It is further stretched by another small length $y$. The work done in the second stretching is
(a) $\frac{1}{2} k x^{2}$
(b) $\frac{1}{2} k\left(x^{2}+y^{2}\right)$
(c) $\frac{1}{2} k y(2 x+y)$.
(d) $\frac{1}{2} k x(x+2 y)$

Dheeraj Sharma

Numerade Educator

Problem 41

In the equilibrium position, a body has
(a) maximum potential energy
(b) minimum potential energy
(c) minimum kinetic energy
(d) neither maximum nor minimum potential energy

Dheeraj Sharma

Numerade Educator

Problem 42

The force $F$ acting on a body moving along $x$ -axis varies with the position $x$ of the particle as shown in the graph. The body is in stable equilibrium at
(a) $x=x_{1}$
(b) $x=x_{2}$
(c) both $x_{1}$ and $x_{2}$
(d) neither $x_{1}$ nor $x_{2}$

Dheeraj Sharma

Numerade Educator

Problem 43

Power is
(a) the distance derivative of force
(b) the distance derivative of work
(c) the time derivative of force
(d) the time derivative of kinetic energy

Dheeraj Sharma

Numerade Educator

Problem 44

An ideal spring with spring constant $k$ is hung from the ceiling and a block of mass $m$ is attached to its lower end. The mass is released with the spring initially unstreched. The maximum extension in the spring is
(a) $\frac{4 m g}{k}$
(b) $\frac{2 m g}{k}$
(c) $\frac{m g}{k}$
(d) $\frac{m g}{2 k}$

Dheeraj Sharma

Numerade Educator

Problem 45

The potential energy of a $1 \mathrm{~kg}$ particle free to move along the $x$ -axis is given by $V=\left(\frac{x^{4}}{4}-\frac{x^{2}}{2}\right) \mathrm{J}$. The total mechanical energy of the particle is $2 \mathrm{~J}$. Then the maximum speed is
(a) $\frac{3}{\sqrt{2}} \mathrm{~m} / \mathrm{s}$
(b) $\sqrt{2} \mathrm{~m} / \mathrm{s}$
(c) $\frac{1}{\sqrt{2}} \mathrm{~m} / \mathrm{s}$
(d) $2 \mathrm{~m} / \mathrm{s}$

Dheeraj Sharma

Numerade Educator

Problem 46

A body of mass $m$ accelerates uniformly from rest to $v_{1}$ in time $t_{1}$. The instantaneous power delivered to the body as a function of time $t$ is
(a) $\frac{m v_{1} t}{t_{1}}$
(b) $\frac{m v_{1}^{2} l}{t_{1}^{2}}$
(c) $\frac{m v_{1} t^{2}}{t_{1}}$
(d) $\frac{m v_{1}^{2} t}{t_{1}}$

Dheeraj Sharma

Numerade Educator

Problem 47

One man takes 1 minute to raise a box to a height of $1 \mathrm{~m}$ and another man takes $1 / 2$ minute to do so. The energy of the two is
(a) different
(b) same
(c) energy of the first is more
(d) energy of the second is more

Dheeraj Sharma

Numerade Educator

Problem 48

If the potential energy of a gas molecule is
$$
U=\frac{M}{r^{6}}-\frac{N}{r^{12}}
$$
$M$ and $N$ being positive constants, then the potential energy at equilibrium must be
(a) zero
(b) $M^{2} / 4 N$
(c) $N^{2} / 4 M$
(d) $M N^{2} / 4$

Dheeraj Sharma

Numerade Educator

Problem 49

The block of mass $M$ moving on the frictionless horizontal surface collides with a spring of spring constant $k$ and compresses it by length $L$. The maximum momentum of the block after collision is
(a) zero
(b) $\frac{M L^{2}}{k}$
(c) $\sqrt{M k} L$
(d) $\frac{k L^{2}}{2 M}$

Dheeraj Sharma

Numerade Educator

Problem 50

A body of mass $4 \mathrm{~kg}$ is moving with momentum of 8 $\mathrm{kg}-\mathrm{m} / \mathrm{s}$. A force of $0.2 \mathrm{~N}$ acts on it in the direction of motion of the body for $10 \mathrm{~s}$. The increase in $\mathrm{KE}$ in joule is
(a) 10
(b) $8.5$
(c) $4.5$
(d) 4

Dheeraj Sharma

Numerade Educator

Problem 51

A running man has the same kinetic energy as that of a boy of half his mass. The man speeds up by $2 \mathrm{~m} / \mathrm{s}$ and the boy changes his speed by $x \mathrm{~m} / \mathrm{s}$ so that the kinetic energies of the boy and the man are again equal. Then $x$ in $\mathrm{m} / \mathrm{s}$ is
(a) $-2 \sqrt{2}$
(b) $2 \sqrt{2}$
(c) $\sqrt{2}$
(d) 2

Dheeraj Sharma

Numerade Educator

Problem 52

A body is moving along a straight line by a machine delivering a constant power. The distance moved by the body in time $t$ is proportional to
(a) $t^{1 / 2}$
(b) $t$
(c) $t^{1 / 2}$
(d) $t^{2}$

Dheeraj Sharma

Numerade Educator

Problem 53

In Q. No. 52, the power, delivered by the machine is
(a) proportional to $t^{1 / 2}$
(b) proportional to $l^{2}$
(c) inversely proportional to $\frac{1}{t^{2}}$
(d) independent of time

Dheeraj Sharma

Numerade Educator

Problem 54

The kinetic energy $K$ of a particle moving along a circular path of radius $r$ depends upon the distance $s$ as $K=A s^{2}$
The force acting on the particle is
(a) 2 As
(b) $2 A \sqrt{r^{2}+s^{2}}$
(c) $2 \mathrm{As} \sqrt{1+\frac{s^{2}}{r^{2}}}$
(d) $2 A r$

Dheeraj Sharma

Numerade Educator

Problem 55

The kinetic energy $K$ of a particle of mass $m$ moving along a straight line depends upon the displacement $s$ as
$$
K=A s^{2}
$$
The force acting on the particle is
(a) $2 \mathrm{As}$
(b) $A s$
(c) $2 \mathrm{mAs}$
(d) $m A s$

Dheeraj Sharma

Numerade Educator

Problem 56

A particle of mass $m$ is moving in a circular path of constant radius $r$ such that its centripetal acceleration $a_{c}$ is varying with time $t$ as $a_{c}=k^{2} r t^{2}$, where $k$ is a constant. The power delivered to the particle by the forces acting on it is
(a) zero
(b) $m k^{2} r^{2} t^{2}$
(c) $m k^{2} r^{2} t$
(d) $m k^{2} r t$

Dheeraj Sharma

Numerade Educator

Problem 57

A spherical ball of mass $20 \mathrm{~kg}$ is stationary at the top of a hill of height $100 \mathrm{~m}$. It rolls down a smooth surface to the ground, then climbs up another hill of height $30 \mathrm{~m}$ and finally rolls down to a horizontal base at a height of $20 \mathrm{~m}$ above the ground. The velocity attained by the ball is
(a) $10 \mathrm{~m} / \mathrm{s}$
(b) $30 \mathrm{~m} / \mathrm{s}$
(c) $40 \mathrm{~m} / \mathrm{s}$
(d) $20 \mathrm{~m} / \mathrm{s}$

Dheeraj Sharma

Numerade Educator

Problem 58

A particle of mass $1 \mathrm{~g}$ executes an oscillatory motion on a concave surface of spherical dish of radius $2 \mathrm{~m}$ placed on a horizontal plane. If the motion of the particle begins from a point on the dish at the height $1 \mathrm{~cm}$ from the horizontal plane and coefficient of friction is $0.01$, find the total distance covered by the particle before it comes to rest (Assume the radius of curvature of the concave surface to be very large).
(a) $10 \mathrm{~cm}$
(b) $28 \mathrm{~cm}$
(c) $58 \mathrm{~cm}$
(d) $100 \mathrm{~cm}$

Dheeraj Sharma

Numerade Educator

Problem 59

If $W_{1}, W_{2}$ and $W_{3}$ represent the work done in moving a particle from $A$ to $B$ along three different paths, 1,2 and 3 (as shown in the figure) in the gravitational field of a point mass $m$, find the correct relation between $W_{1}, W_{2}$ and $W_{3}$.
(a) $W_{1}>W_{3}>W_{2}$
(b) $W_{1}=W_{3}=W_{2}$
(c) $W_{1}<W_{3}^{3}<W_{2}^{2}$
(d) $W_{1}<W_{2}<W_{3}$

Dheeraj Sharma

Numerade Educator

Problem 60

A particle in a certain conservative force field has a potential energy given by $U=\frac{20 x y}{z} .$ The force exerted on it is
(a) $\left(\frac{20 y}{z}\right) \hat{i}+\left(\frac{20 x}{z}\right) \hat{j}+\left(\frac{20 x y}{z^{2}}\right) \hat{k}$
(b) $-\left(\frac{20 y}{z}\right) \hat{i}-\left(\frac{20 x}{z}\right) \hat{j}+\left(\frac{20 x y}{z^{2}}\right) \hat{k}$
(c) $-\left(\frac{20 y}{z}\right) \hat{i}-\left(\frac{20 x}{z}\right) \hat{j}-\left(\frac{20 x y}{z^{2}}\right) \hat{k}$
(d) $\left(\frac{20 y}{z}\right) \hat{i}+\left(\frac{20 x}{z}\right) \hat{j}-\left(\frac{20 x y}{z^{2}}\right) \hat{k}$

Dheeraj Sharma

Numerade Educator