Chapter 7

The bob of a pendulum is released from a horizontal position \(A\) as shown in the figure. If the length of the pendulum is \(1.5 \mathrm{~m}\), then the speed with which the bob arrives at the lower most point \(B\). Given that it dissipated \(5 \%\) of its initial energy against air resistance? (a) \(6.0 \mathrm{~m} / \mathrm{s}\) (b) \(6.5 \mathrm{~m} / \mathrm{s}\) (c) \(4.5 \mathrm{~m} / \mathrm{s}\) (d) \(5.3 \mathrm{~m} / \mathrm{s}\)

Under the action of a force, a \(2 \mathrm{~kg}\) body moves such that its position \(x\) as a function of time \(t\) is given by \(x=t^{3} / 3\), where \(x\) is in metre and \(t\) in second. The work done by the force in the first two seconds is (a) \(1.6 \mathrm{~J}\) (b) \(16 \mathrm{~J}\) (c) \(160 \mathrm{~J}\) (d) \(1600 \mathrm{~J}\)

If a man speeds up by \(1 \mathrm{~ms}^{-1}\), his kinetic energy increases by \(44 \%\). His original speed in \(\mathrm{ms}^{-1}\) is (a) 1 (b) 2 (c) 5 (d) 4

The work done in pulling up a block of wood weighing \(2 \mathrm{kN}\) for a length of \(10 \mathrm{~m}\) on a smooth plane inclined at an angle of \(15^{\circ}\) with the horizontal is \(\left[\sin 15^{\circ}=0.2588\right]\) (a) \(4.36 \mathrm{~kJ}\) (b) \(5.13 \mathrm{k}]\) (c) \(8.91 \mathrm{~kJ}\) (d) \(9.82 \mathrm{~kJ}\)

A mass \(M\) is lowered with the help of a string by a distance \(h\) at a constant acceleration \(g / 2 .\) The work done by the string will be (a) \(\frac{M g h}{2}\) (b) \(\frac{-M g h}{2}\) (c) \(\frac{3 M g h}{2}\) (d) \(\frac{-3 M g h}{2}\)

A bullet fired from a gun with a velocity of \(10^{4} \mathrm{~ms}^{-1}\) goes through a bag full of straw. If the bullet loses half of its kinetic energy in the bag, its velocity when it comes out of the bag will be (a) \(7071.06 \mathrm{~ms}^{-1}\) (b) \(707 \mathrm{~ms}^{-1}\) (c) \(70.71 \mathrm{~ms}^{-1}\) (d) \(707.06 \mathrm{~ms}^{-1}\)

An electron and a proton are moving under the influence of mutual forces. In calculating the change in the kinetic energy of the system during motion, one ignores the magnetic force of one on another. This is because, \(\quad\) [NCERT Exemplar] (a) the two magnetic forces are equal and opposite, so they produce no net effect (b) the magnetic forces do no work on each particle (c) the magnetic forces do equal and opposite (but non-zero) work on each particle (d) the magnetic forces are necessarily negligible

A ball of mass \(0.2 \mathrm{~kg}\) is thrown vertically upwards by applying a force by hand. If the hand moves \(0.2 \mathrm{~m}\) while applying the force and the ball goes upto \(2 \mathrm{~m}\) height further. Find the magnitude of force (Consider \(g=10 \mathrm{~m} / \mathrm{s}^{2}\) ) (a) \(22 \mathrm{~N}\) (b) \(4 \mathrm{~N}\) (c) \(16 \mathrm{~N}\) (d) \(20 \mathrm{~N}\)

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

A \(5 \mathrm{~kg}\) brick of \(20 \mathrm{~cm} \times 10 \mathrm{~cm} \times 8 \mathrm{~cm}\) dimensionless lying on the largest base. It is now made to stand with length vertical. If \(g=10 \mathrm{~ms}^{-2}\), then the amount of work done is (a) \(3 \mathrm{~J}\) (b) \(5 \mathrm{~J}\) (c) 7\(]\) (d) \(9 \mathrm{~J}\)

A block of mass \(10 \mathrm{~kg}\) slides down a rough slope which is inclined at \(45^{\circ}\) to the horizontal. The coefficient of sliding friction is \(0.30\). When the bloek has slide \(5 \mathrm{~m}\), the work done on the block by the force of friction is nearly (a) \(115 \mathrm{~J}\) (b) \(75 \sqrt{2} \mathrm{~J}\) (c) \(321.4 \mathrm{~J}\) (d) \(-321.4 \mathrm{~J}\)

In a children's park, there is a slide which has a total length of \(10 \mathrm{~m}\) and a height of \(8.0 \mathrm{~m}\). A vertical ladder is provided to reach the top. A boy weighing \(200 \mathrm{~N}\) climbs up the ladder to the top of the slide and slides down to the ground. The average friction offered by the slide is three-tenth of his weight. The work done by the slide on the boy as he comes down is \(\begin{array}{lll}\text { (a) zero } & \text { (b) }+600 \mathrm{~J} & \text { (c) }-600 \mathrm{~J}\end{array}\) (d) \(+1600 \mathrm{~J}\)

A body of mass \(3 \mathrm{~kg}\) acted upon by a constant force is displaced by \(s\) metre, given by relation \(s=\frac{1}{3} t^{2}\), where \(t\) is in second. Work done by the force in \(2 \mathrm{~s}\) (a) \(\frac{8}{3} \mathrm{~J}\) (b) \(\frac{19}{5} \mathrm{~J}\) (c) \(\frac{5}{19} \mathrm{~J}\) (d) \(\frac{3}{8} \mathrm{~J}\)

A body of mass \(3 \mathrm{~kg}\) is under a force which causes a displacement is given by \(s=\frac{t^{3}}{3}(\) in \(\mathrm{m})\). Find the work done by the force in first 2 seconds. (a) 2\(]\) (b) \(3.8 \mathrm{~J}\) (c) \(5.2 \mathrm{~J}\) (d) \(24 \mathrm{~J}\)

A man squatting on the ground gets straight up and stand. The force of reaction of ground on the man during the process is \(\quad\) [NCERT Exemplar] (a) constant and equal to \(m g\) in magnitude (b) constant and greater than \(m g\) in magnitude (c) variable but always greater than \(\mathrm{mg}\) (d) at first greater than \(m g\), and later becomes equal to \(m g\)

A gun of mass \(20 \mathrm{~kg}\) has bullet of mass \(0.1 \mathrm{~kg}\) in it. The gun is free to recoil \(804 \mathrm{~J}\) of recoil energy are released on firing the gun. The speed of bullet \(\left(\mathrm{ms}^{-1}\right)\) is (a) \(\sqrt{804 \times 2010}\) (b) \(\sqrt{\frac{2010}{804}}\) (c) \(\sqrt{\frac{804}{2010}}\) (d) \(\sqrt{804 \times 4 \times 10^{3}}\)

A ball is released from the top of a tower. The ratio of work done by force of gravity in Ist second, 2nd second and 3 rd second of the motion of ball is (a) \(1: 2: 3\) (b) \(1: 4: 16\) (c) \(1: 3: 5\) (d) \(1: 9: 25\)

Power supplied to a particle of mass \(2 \mathrm{~kg}\) varies with time as \(P=\frac{3 t^{2}}{2}\) watt. Here \(t\) is in second. If the velocity of particle at \(t=0\) is \(v=0\), the velocity of particle at time \(t=2\) s will be (a) \(1 \mathrm{~ms}^{-1}\) (b) \(4 \mathrm{~ms}^{-1}\) (c) \(2 \mathrm{~ms}^{-1}\) (d) \(2 \sqrt{2} \mathrm{~ms}^{-1}\)

A plate of mass \(m\), length \(b\) and breadth \(a\) is initially lying on a horizontal floor with length parallel to the floor and breadth perpendicular to the floor. The work done to erect it on its breadth is (a) \(m g\left[\frac{b}{2}\right]\) (b) \(m g\left[a+\frac{b}{2}\right]\) (c) \(m g\left[\frac{b-a}{2}\right]\) (d) \(m g\left[\frac{b+a}{2}\right]\)

A body of mass \(3 \mathrm{~kg}\) is under a force which causes a displacement in it, given by \(s=t^{2} / 3\) (in m). Find the work done by the force in \(2 \mathrm{~s}\) (a) \(2 \mathrm{~J}\) (b) \(3.8 \mathrm{~J}\) (c) \(5.2 \mathrm{~J}\) (d) \(2.6 \mathrm{~J}\)

Power supplied to a particle of mass \(2 \mathrm{~kg}\) varies with time as \(P=t^{2} / 2\) watt, where \(t\) is in second. If velocity of particle at \(t=0\) is \(v=0\), the velocity of particle at \(t=2 \mathrm{~s}\) will be (a) \(1 \mathrm{~ms}^{-1}\) (b) \(4 \mathrm{~ms}^{-1}\) (c) \(2 \sqrt{\frac{2}{3}} \mathrm{~ms}^{-1}\) (d) \(2 \sqrt{2} \mathrm{~ms}^{-1}\)

A position-dependent force \(F=3 x^{2}-2 x+7\) acts on a body of mass \(7 \mathrm{~kg}\) and displaces it from \(x=0 \mathrm{~m}\) to \(x=5 \mathrm{~m}\). The work done on the body is \(x^{\prime}\) joule. If both \(F\) and \(x\) are measured in SI units, the value of \(x^{\prime}\) is (a) 135 (b) 235 (c) 335 (d) 935

Power applied to a particle varies with time as \(P=\left(3 t^{2}-2 t+1\right)\) watt, where \(t\) is in second. Find the change in its kinetic energy between \(t=2 \mathrm{~s}\) and \(t=4 \mathrm{~s}\). (a) \(32 \mathrm{~J}\) (b) \(46 \mathrm{~J}\) (c) \(61 \mathrm{~J}\) (d) \(100 \mathrm{~J}\)

A body of mass \(0.5 \mathrm{~kg}\) travels in a straight line with velocity, \(v=a x^{3 / 2}\), where \(a=5 \mathrm{~m}^{-1 / 2} / \mathrm{s}\). What is the work done by the net force during its displacement from \(x=0\) to \(x=2 \mathrm{~m} ?\) (a) \(30 \mathrm{~J}\) (b) \(40 \mathrm{~J}\) (c) \(20 \mathrm{~J}\) (d) \(50 \mathrm{~J}\)

A car of mass \(1000 \mathrm{~kg}\) moves at a constant speed of \(20 \mathrm{~ms}^{-1}\) up an incline. Assume that the frictional force is \(200 \mathrm{~N}\) and that \(\sin \theta=1 / 20\), where, \(\theta\) is the angle of the incline to the horizontal. The \(g=10 \mathrm{~ms}^{-2}\). Find the power developed by the engine? (a) \(14 \mathrm{~kW}\) (b) \(4 \mathrm{~kW}\) (cl \(10 \mathrm{~kW}\) (d) \(28 \mathrm{~kW}\)

A uniform chain of length \(L\) and mass \(M\) overhangs a horizontal table with its two-third part on the table. The friction coefficient between the table and the chain is \(\mu\). The work done by the friction during the period the chain slips off the table is (a) \(-\frac{1}{4} \mu \mathrm{Mg} \mathrm{L}\) (b) \(-\frac{2}{9} \mu \mathrm{Mgl}\) (c) \(-\frac{4}{9} \mu \mathrm{Mg} L\) (d) \(-\frac{6}{7} \mu \mathrm{Mg} L\)

The human heart discharges 75 cc of blood through the arteries at each beat against an average pressure of \(10 \mathrm{~cm}\) of mercury. Assuming that the pulse frequency is 72 per minute the rate of working of heart in watt, is (Density of mercury \(=13.6 \mathrm{~g} / \mathrm{cc}\) and \(\left.g=9.8 \mathrm{~ms}^{-2}\right)\) (a) \(11.9\) (b) 1,19 (c) \(0.119\) (d) 119

A \(5 \mathrm{~kg}\) stone of relative density 3 is resting at the bed of a lake. It is lifted through a height of \(5 \mathrm{~m}\) in the lake. If \(g=10 \mathrm{~ms}^{-2}\), then the work done is (a) \(\frac{500}{3} \mathrm{~J}\) (b) \(\frac{350}{3} \mathrm{~J}\) (c) \(\frac{750}{3} \mathrm{~J}\) (d) zero

A particle of mass \(2 \mathrm{~kg}\) starts moving in a straight line with an initial velocity of \(2 \mathrm{~ms}^{-1}\) at a constant acceleration of \(2 \mathrm{~ms}^{-2}\). Then rate of change of kinetic energy (a) is four times the velocity at any moment (b) is two times the displacement at any moment (c) is four times the rate of change of velocity at any moment (d) is constant through out

A force acts on a \(30 \mathrm{~g}\) particle in such a way that the position of the particle as function of time is given by \(x=3 t-4 t^{2}+t^{3}\), where \(x\) is in metre and \(t\) is in second. The work done during the first 4 seconds is (a) 5.28 J (b) \(450 \mathrm{~mJ}\) (c) \(490 \mathrm{~m} \mathrm{~J}\) (d) \(530 \mathrm{~mJ}\)

A car weighing \(1400 \mathrm{~kg}\) is moving at a speed of \(54 \mathrm{kmh}^{-1}\) up a hill when the motor stops. If it is just able to reach the destination which is at a height of \(10 \mathrm{~m}\) above the point, then the work done against friction (negative of the work done by the friction) is (Take \(g=10 \mathrm{~ms}^{-2}\) ) (a) \(10 \mathrm{~kJ}\) (b) \(15 \mathrm{~kJ}\) (c) \(17.5 \mathrm{~kJ}\) (d) \(25 \mathrm{~kJ}\)

A cord is used to lower vertically a block of mass \(M\) by a distance \(d\) with constant downward acceleration \(g / 4\) work done by the cord on the block is (a) \(M g \frac{d}{4}\) (b) \(3 M g \frac{d}{4}\) (c) \(-3 M g \frac{d}{4}\) (d) \(\mathrm{Mgd}\)

Water is drawn from a well in a \(5 \mathrm{~kg}\) drum of capacity 55 L by two ropes connected to the top of the drum. The linear mass density of each rope is \(0.5 \mathrm{kgm}^{-1}\). The work done in lifting water to the ground from the surface of water in the well \(20 \mathrm{~m}\) below is \(\left(g=10 \mathrm{~ms}^{-2}\right)\) (a) \(1.4 \times 10^{4} \mathrm{~J}\) (b) \(1.5 \times 10^{4} \mathrm{~J}\) (c) \(9.8 \times 10 \times 6 \mathrm{~J}\) (d) \(18 . \mathrm{J}\)

A bullet when fired at a target with velocity of \(100 \mathrm{~ms}^{-1}\) penetrates \(1 \mathrm{~m}\) into it. If the bullet is fired at a similar target with a thickness \(0.5 \mathrm{~m}\), then it will emerge from it with a velocity of (a) \(50 \sqrt{2} \mathrm{~m} / \mathrm{s}\) (b) \(\frac{50}{\sqrt{2}} \mathrm{~m} / \mathrm{s}\) (c) \(50 \mathrm{~m} / \mathrm{s}\) (d) \(10 \mathrm{~m} / \mathrm{s}\)

A wire of length \(L\) suspended vertically from a rigid support is made to suffer extension \(l\) in its length by applying a force \(F\). The work is (a) \(\frac{F l}{2}\) (b) \(F l\) (c) \(2 \mathrm{Fl}\) (d) \(\mathrm{Fl}\)

A bicyclist comes to a skidding stop in \(10 \mathrm{~m}\). During this process, the force on the bicycle due to the road is \(200 \mathrm{~N}\) and is directly opposed to the motion. The work done by the cycle on the road is \(\quad\) [NCERT Exemplar] (a) \(+2000 \mathrm{~J}\) (b) - \(200 \mathrm{~J}\) (c) zero (d) \(-20,000 \mathrm{~J}\)

A particle moves on a rough horizontal ground with some initial velocity \(v_{0}\). If \(\frac{3}{4}\) th of its kinetic energy is lost due to friction in time \(t_{0}\), the coefficient of friction between the particle and the ground is (a) \(\frac{v_{0}}{2 g t_{0}}\) (b) \(\frac{v_{0}}{4 g t_{0}}\) (c) \(\frac{3 v_{0}}{4 g t_{0}}\) (d) \(\frac{v_{0}}{g t_{0}}\)

A uniform chain of length \(L\) and mass \(M\) is lying on a smooth table and one third of its length is hanging vertically down over the edge of the table. If \(g\) is acceleration due to gravity, the work required to pull the hanging part on the the table is (a) \(M g L\) (b) \(\mathrm{Mgl} / 3\) (c) \(\mathrm{Mg} \mathrm{L} / 9\) (d) \(\mathrm{Mgl} / 18\)

During inelastic collision between two bodies, which of the following quantities always remain conserved? [NCERT Exemplar] (a) Total kinetic energy (b) Total mechanical energy (c) Total linear momentum (d) Speed of each body

A man running has half the kinetic energy of a boy of half his mass. The man speeds up by \(1 \mathrm{~ms}^{-1}\) and then has kinetic energy as that of the boy. What were the original speeds of man and the boy? (a) \(\sqrt{2} \mathrm{~ms}^{-1} ; 2 \sqrt{2}-1 \mathrm{~ms}^{-1}\) (b) \((\sqrt{2}-1) \mathrm{ms}^{-1}, 2(\sqrt{2}-1) \mathrm{ms}^{-1}\) \((c)(\sqrt{2}+1) \mathrm{ms}^{-1} ; 2(\sqrt{2}+1) \mathrm{ms}^{-1}\) (d) None of the above

A spring of spring constant \(5 \times 10^{3} \mathrm{Nm}^{-1}\) is stretched initially by \(5 \mathrm{~cm}\) from the unstretched position. Then the work required to stretch it further by another \(5 \mathrm{~cm}\) is (a) \(12.50 \mathrm{~N}-\mathrm{m}\) (b) \(18.75 \mathrm{~N}-\mathrm{m}\) (c) \(25.00 \mathrm{~N}-\mathrm{m}\) (d) \(6.25 \mathrm{~N}-\mathrm{m}\)

An engine pumps up \(100 \mathrm{~kg}\) of water through a height of \(10 \mathrm{~m}\) in 5 s. Given that the efficiency of the engine is \(60 \%\). If \(g=10 \mathrm{~m} / \mathrm{s}^{2}\), the power of the engine is (a) \(3.3 \mathrm{~kW}\) (b) \(0.33 \mathrm{~kW}\) (c) \(0.033 \mathrm{~kW}\) (d) \(33 \mathrm{~kW}\)

A rod \(A B\) of mass \(10 \mathrm{~kg}\) and length \(4 \mathrm{~m}\) rests on a horizontal floor with end \(A\) fixed so as to rotate it in vertical. Work done on the rod is \(100 \mathrm{~J} .\) The height to which the end \(B\) be raised vertically above the floor is (a) \(1.5 \mathrm{~m}\) (b) \(2.0 \mathrm{~m}\) (c) \(1.0 \mathrm{~m}\) (d) \(2.5 \mathrm{~m}\)

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 unstretched. Then the maximum extension in the spring is (a) \(\frac{4 \mathrm{Mg}}{k}\) (b) \(\frac{2 M g}{k}\) (c) \(\frac{\mathrm{Mg}}{\mathrm{k}}\) (d) \(\frac{M g}{2 k}\)

A \(0.5 \mathrm{~kg}\) ball is thrown up with an initial speed \(14 \mathrm{~ms}^{-1}\) and reaches a maximum height of \(8 \mathrm{~m} .\) How much energy is dissipate by air drag acting on the ball during the ascent? (a) \(19.6 \mathrm{~J}\) (b) \(4.9 \mathrm{~J}\) (c) \(10 \mathrm{~J}\) (d) \(9.8 \mathrm{~J}\)

A force \(F=A y^{2}+B y+C\) acts on a body in the \(y\)-direction. The work done by this force during a displacement from \(y=-a\) to \(y=a\) is (a) \(\frac{2 A a^{5}}{3}\) (b) \(\frac{2 A a^{5}}{3}+2 C a\) (c) \(\frac{2 A a^{5}}{3}+\frac{B a^{2}}{2}+C a\) (d) None of these

The kinetic energy \(k\) of a particle moving along a circle of radius \(R\) depends upon the distance \(s\) as \(k=a s^{2}\). The force acting on the particle is (a) \(2 a \frac{s^{2}}{R}\) (b) \(2 a s\left[1+\frac{s^{2}}{R^{2}}\right]^{t / 2}\) (c) \(2 a\) (d) \(2 \underline{a}\)

A man of mass \(m\), standing at the bottom of the staircaseof height \(L\) climbs it and stands at its top. [NCERT Exemplar] (a) Work done by all forces on man is equal to the rise in potential energy \(m g L\) (b) Work done by all forces on man is zero (c) Work done by the gravitational force on man is \(\mathrm{mgL}\) (d) The reaction force from a step does not do work because the point of application of the force does not move while the force exists

A \(2.0 \mathrm{~kg}\) block is dropped from a height of \(40 \mathrm{~cm}\) onto a spring of spring constant \(k=1960 \mathrm{Nm}^{-1}\), Find the maximum distance the spring is compressed. (a) \(0.080 \mathrm{~m}\) (b) \(0.20 \mathrm{~m}\) (c) \(0.40 \mathrm{~m}\) (d) \(0.10 \mathrm{~m}\)

If the kinetic energy of a body is directly proportional to time \(t\), the magnitude of the force acting on the body is (a) directly proportional to \(\sqrt{t}\) (b) inversely proportional to \(\sqrt{t}\) (c) directly proportional to the speed of the body (d) inversely proportional to the speed of the body