Chapter 7

A stone tied to a string of length \(L\) is whirled in a vertical circle with the other end of the string at the centre. At a certain instant of time, the stone is at its lowest position and has a speed \(u\). The magnitude of the change in its velocity as it reaches a position where the string is horizontal is (a) \(\sqrt{u^{2}-2 g l}\) (b) \(\sqrt{2 g l}\) (c) \(\sqrt{u^{2}-g l}\) (d) \(\sqrt{2\left(u^{2}-g L\right)}\)

The potential energy function for a particle executing linear SHM is given by \(V(x)=\frac{1}{2} k x^{2}\) where \(k\) is the force constant of the oscillator. For \(k=0.5 \mathrm{~N} / \mathrm{m}\), the graph of \(V(x)\) versus \(x\) is shown in the figure. A particle of total energy \(E\) turns back when it reaches \(x=\pm x_{m}\). If \(V\) and \(K\) indicate the \(\mathrm{PE}\) and \(\mathrm{KE}\) respectively of the particle at \(x=\pm x_{m}\) then which of the following is correct? \(\quad\) [NCERT Exemplar] (a) \(V=0, \bar{K}=\mathrm{E}\) (b) \(V=\mathrm{E}, \bar{K}=0\) (c) \(V

An elastic string of unstretched length \(L\) 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) \(\frac{1}{2} k y^{2}\) (b) \(\frac{1}{2} k\left(x^{2}+y^{2}\right)\) (c) \(\frac{1}{2} k(x+y)^{2}\) (d) \(\frac{1}{2} k y(2 x+y)\)

A ball is projected vertically upwards with a certain initial speed. Another ball of the same mass is projected at an angle of \(60^{\circ}\) with the vertical with the same initial speed. At highest point of their journey, the ratio of their potential energies will be (a) \(1: 1\) (b) \(2: 1\) (c) \(3: 2\) (d) \(4: 1\)

The kinetic energy \(K\) of a particle moving in straight line depends upon the distance \(s\) as $$ K=a s^{2} $$ The force acting on the particle is (a) 2 as (b) 2 mas (c) \(2 a\) (d) \(\sqrt{a s^{2}}\)

A body of mass \(0.5 \mathrm{~kg}\) travels in a straight line with velocity \(v=a x^{3 / 2}\) where \(a=5 \mathrm{~m}^{-1 / 2} \mathrm{~s}^{-1}\). The work done by the net force during its displacement from \(x=0\) to \(x=2 \mathrm{~m}\) is \(\quad\) [NCERT Exemplar] (a) \(1.5 \mathrm{~J}\) (b) \(50 \mathrm{~J}\) (c) \(10 \mathrm{j}\) (d) \(100 \mathrm{~J}\)

When a man increases his speed by \(2 \mathrm{~ms}^{-1}\), he finds that his kinetic energy is doubled, the original speed of the man is (a) \(2(\sqrt{2}-1) \mathrm{ms}^{-1}\) (b) \(2(\sqrt{2}+1) \mathrm{ms}^{-1}\) (c) \(4.5 \mathrm{~ms}^{-1}\) (d) None of these

A stone of mass \(2 \mathrm{~kg}\) is projected upward with kinetic energy of \(98 \mathrm{~J}\). The height at which the kinetic energy of the body becomes half its original value, is given by (Take \(g=10 \mathrm{~ms}^{-2}\) ) (a) \(5 \mathrm{~m}\) (b) \(2.5 \mathrm{~m}\) (c) \(1.5 \mathrm{~m}\) (d) \(0.5 \mathrm{~m}\)

Assertion The change in kinetic energy of a particle is equal to the work done on it by the net force. Reason Change in kinetic energy of particle is equal to the work done only in case of a system of one particle.

A ball whose kinetic energy is \(E\), is projected at an angle \(45^{\circ}\) to the horizontal. The kinetic energy of the ball at the highest point of its flight will be (a) \(E\) (b) \(\frac{E}{\sqrt{2}}\) (c) \(\frac{E}{2}\) (d) zero

Assertion Power developed in circular motion is always zero. Reason Work done in case of circular motion is zero.

A body is falling freely under the action of gravity alone in vacuum. Which of the following quantities remain constant during the fall? [NCERT Exemplar] (a) Kinetic energy (b) Potential energy (c) Total mechanical energy (d) Total linear energy

Assertion Stopping distance \(=\frac{\text { Kinetic energy }}{\text { Stopping force }}\) Reason Work done in stopping a body is equal to change in kinetic energy of the body.

The potential energy as a function of the force between two atoms in a diatomic molecules is given by \(U(x)=\frac{a}{x^{12}}-\frac{b}{x^{6}}\), where \(a\) and \(b\) are positive constants and \(x\) is the distance between the atoms. The position of stable equilibrium for the system of the two atoms is given (a) \(x=\frac{a}{b}\) (b) \(x=\sqrt{\frac{a}{b}}\) (c) \(x=\frac{\sqrt{3 a}}{b}\) (d) \(x=\sqrt[6]{\left(\frac{2 a}{b}\right)}\)

Assertion Two springs of force constants \(k_{1}\) and \(k_{2}\) are stretched by the same force. If \(k_{1}>k_{2}\), then work done in stretching the first \(\left(W_{1}\right)\) is less than work done in stretching the second \(\left(W_{2}\right.\) ). Reason \(F=k_{1} x_{1}=k_{2} x_{2}\) $$ \begin{aligned} \frac{x_{1}}{x_{2}} &=\frac{k_{2}}{k_{1}} \\ \frac{W_{1}}{W_{2}} &=\frac{\frac{1}{2} k_{1} x_{1}^{2}}{\frac{1}{2} k_{2} x_{2}^{2}}=\frac{k_{1}}{k_{2}}\left(\frac{k_{2}}{k_{1}}\right)^{2}=\frac{k_{2}}{k_{1}} \end{aligned} $$ As \(k_{1}>k_{2}, W_{1}

The potential energy of a particle of mass \(5 \mathrm{~kg}\) moving in the \(x y\)-plane is given by \(U=(-7 x+24 y) \mathrm{J}, x\) and \(y\) being in metre. If the particle starts from rest from origin, then speed of particle at \(t=2 \mathrm{~s}\) is (a) \(5 \mathrm{~ms}^{-1}\) (b) \(01 \mathrm{~ms}^{-1}\) (c) \(17.5 \mathrm{~ms}^{-1}\) (d) \(10 \mathrm{~ms}^{-1}\)

Assertion Mass and energy are not conserved separately, but are conserved as a single entity called 'mass-energy'. Reason This is because one can be obtained at the cost of the other as per Einstein equation. $$ E=m c^{2} $$

A running man has half the kinetic energy of that of a boy of half of his mass. The man speeds up by \(1 \mathrm{~m} / \mathrm{s}\), so as to have same kinetic energy as that of the boy. The original speed of the man will be (a) \(\sqrt{2} \mathrm{~m} / \mathrm{s}\) (b) \(\sqrt{2-1} \mathrm{~m} / \mathrm{s}\) (c) \(\frac{1}{\sqrt{2}-1} \mathrm{~m} / \mathrm{s}\) (d) \(\frac{1}{\sqrt{2}} \mathrm{~m} / \mathrm{s}\)

If a body looses half of its velocity on penetrating \(3 \mathrm{~cm}\) in a wooden block, then how much will it penetrate more before coming to rest? (a) \(1 \mathrm{~cm}\) (b) \(2 \mathrm{~cm}\) (c) \(3 \mathrm{~cm}\) (d) \(4 \mathrm{~cm}\)

This question has statement I and statement II. Of the four choices given after the statements, choose the one that best describes the two statements. Statement I A point particle of mass \(m\) moving with speed \(v\) collides with stationary point particle of mass \(M\). If the maximum energy loss possible is given as \(f\left(\frac{1}{2} m v^{2}\right)\), then \(f=\left(\frac{m}{M+m}\right)\) Statement II Maximum energy loss occurs when the particles get stuck together as a result of the collision. [UEE Main 2013] (a) Statement 1 is true, Statement \(\|\) is true, and Statement II is the correct explanation of Statement 1 (b) Statement 1 is true, Statement \(\|\) is true, but Statement II is not the correct explanation of Statement 1 (c) Statement \(I\) is true, Statement \(\|\) is false (d) Statement 1 is false, Statement Il is true

Two springs have force constants \(k_{1}\) and \(k_{2}\). These are extended through the same distance \(x\). If their elastic energies are \(E_{1}\) and \(E_{2}\), then \(\frac{E_{1}}{E_{2}}\) is equal to (a) \(k_{1}: k_{2}\) (b) \(k_{2}: k_{1}\) (c) \(\sqrt{k_{1}}: \sqrt{k_{2}}\) (d) \(k_{1}^{2}: k_{2}^{2}\)

A force \((4 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}) \mathrm{N}\) acting on a body maintains its velocity at \((2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}) \mathrm{ms}^{-1}\). The power exerted is [Kerala CET 2010] (a) \(4 \mathrm{~W}\) (b) \(5 \mathrm{~W}\) (c) \(2 \mathrm{~W}\) (d) \(8 \mathrm{~W}\)

A body of mass \(M\) is moving with a uniform speed of \(10 \mathrm{~m} / \mathrm{s}\) on frictionless surface under the influence of two forces \(F_{1}\) and \(F_{2}\). The net power of the system is [MP PET 2010] (a) \(10 F F_{12} M\) (b) \(10\left(F_{i}+F_{2}\right) M\) (c) \(\left(F_{1}+F_{2}\right) M\) (d) zero

A car is moving with a speed of \(100 \mathrm{kmh}^{-1}\). If the mass of the car is \(950 \mathrm{~kg}\), then its kinetic energy is (a) \(0.367 \mathrm{M}\rfloor\) (b) \(3.67 \mathrm{~J}\) (c) \(3.67 \mathrm{M} \mathrm{J}\) (d) \(367 \mathrm{~J}\)

An engine pumps water through a hose pipe. Water passes through the pipe and leaves to with a velocity of \(2 \mathrm{~m} / \mathrm{s}\). The mass per unit length of water in the pipe is \(100 \mathrm{~kg} / \mathrm{m}\). What is the power of the engine? [CBSE PMT 2010] (a) \(800 \mathrm{~W}\) (b) \(400 \mathrm{~W}\) (c) \(200 \mathrm{~W}\) (d) \(100 \mathrm{~W}\)

Two masses of \(1 \mathrm{~g}\) and \(4 \mathrm{~g}\) are moving with equal kinetic energies. The ratio of the magnitudes of their linear momenta is (a) \(4: 1\) (b) \(\sqrt{2}: 1\) (c) \(1: 2\) (d) \(1: 16\)

A variable force given by the two-dimensional vector \(\mathbf{F}=\left(3 x^{2} \hat{\mathbf{i}}+4 \hat{\mathbf{j}}\right)\) acts on a particle. The force is in newton and \(X\) is in metre. What is the change in the kinetic energy of the particle as it moves from the point with coordinates \((2,3)\) to \((3,0)\) (The coordinates are in metres)? [AMU (Med) 2010] (a) \(-7]\) (b) zero (c) +7 J (d) \(+19 \mathrm{~J}\)

A mass of \(5 \mathrm{~kg}\) is moving along a circular path of radius \(1 \mathrm{~m}\). If the mass moves with 300 revolutions per minute, its kinetic energy would be [NCERT Exemplar] (a) \(250 \pi^{2}\) (b) \(100 \pi^{2}\) (c) \(5 \pi^{2}\) (d) 0

Water falls from a height of \(60 \mathrm{~m}\) at the rate of \(15 \mathrm{~kg} / \mathrm{s}\) to operate a turbine. The losses due to frictional force are \(10 \%\) of energy. How much power is generated by the turbine \(\left(g=10 \mathrm{~m} / \mathrm{s}^{2}\right)\) ? [CBSE PMT 2008] (a) \(12.3 \mathrm{~kW}\) (b) \(7.0 \mathrm{~kW}\) (c) \(8.1 \mathrm{~kW}\) (d) \(10.2 \mathrm{~kW}\)

A body of mass \(2 \mathrm{~kg}\) is thrown up vertically with kinetic energy of \(490 \mathrm{~J}\). The height at which the kinetic energy of the body becomes half of its original value is (a) \(50 \mathrm{~m}\) (b) \(12.25 \mathrm{~m}\) (c) \(25 \mathrm{~m}\) (d) \(10 \mathrm{~m}\)

A body of mass \(m\) is accelerated uniformly for rest to a speed \(v\) in a time \(T\). The instantaneous power delivered to the body as a function of time is given by [AIEEE 2005] (a) \(\frac{1}{2} \frac{m v^{2}}{T^{2}} t^{2}\) (b) \(\frac{1}{2} \frac{m v^{2}}{T^{2}} t\) (c) \(\frac{m v^{2}}{T^{2}} t^{2}\) (d) \(\frac{m v^{2}}{T^{2}} t\)

In a shotput event an athlete throws the shotput of mass \(10 \mathrm{~kg}\) with an inital speed of \(1 \mathrm{~m} \mathrm{~s}^{-1}\) at \(45^{\circ}\) from a height \(1.5 \mathrm{~m}\) above ground. Assuming air resistance to be negligible and acceleration due to gravity to be \(10 \mathrm{~ms}^{-2}\), the kinetic energy of the shotput when it just reaches the ground will be [NCERT Exemplar] (a) \(2.5 \mathrm{~J}\) (b) \(5.0 \mathrm{~J}\) (c) \(52.5 \mathrm{~J}\) (d) \(155.0 \mathrm{~J}\)

A particle is placed at the origin and a force \(F=k x\) is acting on it (where \(k\) is a positive constant). If \(U(0)=0\), the graph of \(U(x)\) versus \(x\) will be, figure (where \(U\) is the potential energy function) [UP SEE 2004]

A machine which is \(75 \%\) efficient uses \(12 \mathrm{~J}\) of energy in lifting up a \(1 \mathrm{~kg}\) mass through a certain distance. The mass is then allowed to fall through that distance. The velocity of the ball at the end of its fall is (a) \(\sqrt{24} \mathrm{~ms}^{-1}\) (b) \(\sqrt{32} \mathrm{~ms}^{-1}\) (c) \(\sqrt{18} \mathrm{~ms}^{-1}\) (d) \(3 \mathrm{~ms}^{-1}\)

A body of mass \(4 \mathrm{~kg}\) is moving with momentum of \(8 \mathrm{~kg}-\mathrm{ms}^{-1}\). A force of \(0.2 \mathrm{~N}\) acts on it in the direction of motion of the body for 10 s. The increase in kinetic energy in joule is (a) 10 (b) \(8.5\) (c) \(4.5\) (d) 4

A body of mass \(M\) is dropped from a height \(h\) on a sand floor. If the body penetrates \(x \mathrm{~cm}\) into the sand, the average resistance offered by the sand to the body is (a) \(M g\left(\frac{h}{x}\right)\) (b) \(M g\left(1+\frac{h}{x}\right)\) (c) \(M g h+M g x\) (d) \(M g\left(1-\frac{h}{x}\right)\)

A mass of \(50 \mathrm{~kg}\) is raised through a certain height by a machine whose efficiency is \(90 \%\), the energy is \(5000 \mathrm{~J}\). If the mass is now released, its kinetic energy on hitting the ground shall be (a) \(5000 \mathrm{~J}\) (b) \(4500 \mathrm{~J}\) (c) \(4000 \mathrm{~J}\) (d) \(5500 \mathrm{~J}\)

Given that the position of the body in metre is a function of time as follows $$ x=2 t^{4}+5 t+4 $$ The mass of the body is \(2 \mathrm{~kg}\). What is the increase in its kinetic energy, one second after the start of motion? (a) \(168 \mathrm{~J}\) (b) \(169 \mathrm{~J}\) (c) 32\(\rfloor\) (d) \(144 \mathrm{~J}\)

A bomb of mass \(9 \mathrm{~kg}\) explodes into 2 pieces of mass \(3 \mathrm{~kg}\) and \(6 \mathrm{~kg}\). The velocity of mass \(3 \mathrm{~kg}\) is \(1.6 \mathrm{~m} / \mathrm{s}\), the kinetic energy of mass \(6 \mathrm{~kg}\) is (a) \(3.84 \mathrm{~J}\) (b) \(9.6 \mathrm{~J}\) (c) \(1.92 \mathrm{~J}\) (d) \(2.92 \mathrm{~J}\)

An engine pumps water continuously through a hole. Speed with which water passes through the hole nozzle is \(v\) and \(k\) is the mass per unit length of the water jet as it leaves the nozzle. Find the rate at which kinetic energy is being imparted to the water. (a) \(\frac{1}{2} k v^{2}\) (b) \(\frac{1}{2} k v^{3}\) (c) \(\frac{v^{2}}{2 k}\) (d) \(\frac{v^{3}}{2 k}\)

In the stable equilibrium position, a body has (a) maximum potential energy (b) minimum potential energy (c) minimum kinetic energy (d) maximum kinetic energy

A stone is dropped from the top of a tall tower. The ratio of the kinetic energy of the stone at the end of three seconds to the increase in the kinetic energy of the stone during the next three seconds is (a) \(1: 1\) (b) \(1: 2\) (c) \(1: 3\) (d) \(1: 9\)

A rectangular plank of mass \(m_{1}\) and height \(a\) is kept on a horizontal surface. Another rectangular plank of mass \(m_{2}\) and height \(b\) is placed over the first plank. The gravitational potential energy of the system is (a) \(\left[m_{1}+m_{2}(a+b)\right] g\) (b) \(\left[\left(\frac{m_{1}+m_{2}}{2} a+m_{2} \frac{b}{2}\right)\right] g\) (c) \(\left[\left(\frac{m_{1}}{2}+m_{2}\right) a+m_{2} \frac{b}{2}\right] g\) (d) \(\left[\left(\frac{m_{1}}{2}+m_{2}\right) a+m_{1} \frac{b}{2}\right] g\)

A body is initially at rest. It undergoes one-dimensional motion with constant acceleration. The power delivered to it at time \(t\) is proportional to [NCERT] (a) \(t^{12}\) (b) \(t\) (c) \(t^{3 / 2}\) (d) \(t^{2}\)

A \(10 \mathrm{~m}\) long iron chain of linear mass density \(0.8 \mathrm{~kg} \mathrm{~m}^{-1}\) is hanging freely from a rigid support. If \(g=10 \mathrm{~ms}^{-2}\), then the power required to lift the chain upto the point of support in \(10 \mathrm{~s}\) is (a) \(10 \mathrm{~W}\) (b) \(20 \mathrm{~W}\) (c) \(30 \mathrm{~W}\) (d) \(40 \mathrm{~W}\)

A 10 HP motor pump out water from a well of depth \(20 \mathrm{~m}\) and falls a water tank of volume 22380 litre at a height of \(10 \mathrm{~m}\) from the ground the running time of the motor to fill the empty water tank is \(\left(g=10 \mathrm{~ms}^{-2}\right.\) ) (a) \(5 \mathrm{~min}\) (b) \(10 \mathrm{~min}\) (c) \(15 \mathrm{~min}\) (d) \(20 \mathrm{~min}\)

An engine of power \(7500 \mathrm{~W}\) makes a train move on a horizontal surface with constant velocity of \(20 \mathrm{~ms}^{-1} .\) The force involved in the problem is (a) \(375 \mathrm{~N}\) (b) \(400 \mathrm{~N}\) (c) \(500 \mathrm{~N}\) (d) \(600 \mathrm{~N}\)

A one kilowatt motor is used to pump water from a well \(10 \mathrm{~m}\) deep. The quantity of water pumped out per second is nearly (a) \(1 \mathrm{~kg}\) (b) \(10 \mathrm{~kg}\) (c) \(100 \mathrm{~kg}\) (d) \(1000 \mathrm{~kg}\)

A dam is situated at a height of \(550 \mathrm{~m}\) above sea level and supplies water to a power house which is at a height of \(50 \mathrm{~m}\) above sea level. \(2000 \mathrm{~kg}\) of water passes through the turbines per second. What would be the maximum electrical power output of the power house if the whole system were \(80 \%\) efficient? (a) \(8 \mathrm{MW}\) (b) \(10 \mathrm{MW}\) (c) \(12.5 \mathrm{MW}\) (d) \(16 \mathrm{MW}\)

An automobile weighing \(1200 \mathrm{~kg}\) climbs up a hill that rises \(1 \mathrm{~m}\) in 20 s. Neglecting frictional effects, the minimum power developed by the engine is \(9000 \mathrm{~W}\). If \(g=10 \mathrm{~ms}^{-2}\), then the velocity of the automobile is (a) \(36 \mathrm{kmh}^{-1}\) (b) \(54 \mathrm{kmh}^{-1}\) (c) \(72 \mathrm{kmh}^{-1}\) (d) \(90 \mathrm{kmh}^{-1}\)