Chapter 3

A body of mass \(2 \mathrm{~kg}\) moves vertically downwards with an acceleration \(a=19.6 \mathrm{~m} / \mathrm{s}^{2}\). The force acting on the body simultaneously with the force of gravity is \(\left(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\right.\), neglect air resistance) (A) \(19.6 \mathrm{~N}\) (B) \(19.2 \mathrm{~N}\) (C) \(59.2 \mathrm{~N}\) (D) \(58.8 \mathrm{~N}\)

A girl of mass \(50 \mathrm{~kg}\) stands on a measuring scale in a lift. At an instant, it is detected that the reading reduces to \(40 \mathrm{~kg}\) for a while and then returns to original value. It can be said that (A) The lift was in constant motion upwards (B) The lift was in constant motion downwards (C) The lift was suddenly started in downward motion (D) The lift was suddenly started in upward motion

Two masses \(m\) and \(M\) are connected by a light string passing over a smooth pulley. When set free, \(m\) moves up by \(1.4 \mathrm{~m}\) in \(2 \mathrm{~s}\). The ratio \(\frac{m}{M}\) is \(\left(g=9.8 \mathrm{~ms}^{-2}\right.\) ) (A) \(\frac{13}{15}\) (B) \(\frac{15}{13}\) (C) \(\frac{9}{7}\) (D) \(\frac{7}{9}\)

A block of mass \(m\) is attached to a massless spring of spring constant \(K\). This system is accelerated upward with acceleration \(a\). The elongation in spring will be (A) \(\frac{m g}{K}\) (B) \(\frac{m(g-a)}{K}\) (C) \(\frac{m(g+a)}{K}\) (D) \(\frac{m a}{K}\)

A body of mass \(1.5 \mathrm{~kg}\) is thrown vertically upwards with an initial velocity of \(40 \mathrm{~m} / \mathrm{s}\) reaches its highest point after \(3 \mathrm{~s}\). The air resistance acting on the body during the ascent is (assuming air resistance to be uniform, \(g=10 \mathrm{~m} / \mathrm{s}^{2}\) ) (A) \(35 \mathrm{~N}\) (B) \(25 \mathrm{~N}\) (C) \(15 \mathrm{~N}\) (D) \(5 \mathrm{~N}\)

A string of length \(L\) and mass \(M\) are lying on a horizontal table. A force \(F\) is applied at one of its ends. Tension in the string at a distance \(x\) from the ends at which force is applied is (A) Zero (B) \(F\) (C) \(F(L-x) / L\) (D) \(F(L-x) / M\)

Three blocks \(m_{1}, m_{2}\) and \(m_{3}\) of masses \(8 \mathrm{~kg}, 3 \mathrm{~kg}\) and \(1 \mathrm{~kg}\) are placed in contact on a smooth surface. Forces \(F_{1}=140 \mathrm{~N}\) and \(F_{2}=20 \mathrm{~N}\) are acting on blocks \(m_{1}\) and \(m_{3}\), respectively, as shown. The reaction between blocks \(m_{2}\) and \(m_{3}\) is (A) \(2.5 \mathrm{~N}\) (B) \(7.5 \mathrm{~N}\) (C) \(22.5 \mathrm{~N}\) (D) \(30 \mathrm{~N}\)

A fireman wants to slide down a rope. The breaking load for the rope is \(\frac{3}{4}\) th of the weight of the fireman. The acceleration of the fireman to prevent the rope from breaking will be (Acceleration due to gravity is \(g\) ) (A) \(g / 4\) (B) \(g / 2\) (C) \(3 g / 4\) (D) Zero

An elevator starts from rest with a constant upward acceleration. It moves \(2 \mathrm{~m}\) in the first \(0.6\) second. \(\mathrm{A}\) passenger in the elevator is holding a \(3 \mathrm{~kg}\) package by a vertical string. When the elevator is moving, what is the tension in the string? (A) \(4 \mathrm{~N}\) (B) \(62.7 \mathrm{~N}\) (C) \(29.4 \mathrm{~N}\) (D) \(20.6 \mathrm{~N}\)

A block of \(10 \mathrm{~kg}\) is pulled by a constant speed on a rough horizontal surface by a force of \(19.6 \mathrm{~N}\). The co-efficient of friction is (A) \(0.1\) (B) \(0.2\) (C) \(0.3\) (D) \(0.4\)

A body of mass \(m\) is kept stationary on a rough inclined plane of inclination \(\theta .\) The magnitude of force acting on the body by the inclined plane is (A) \(m g\) (B) \(m g \sin \theta\) (C) \(m g \cos \theta\) (D) \(m g \sqrt{1+\cos ^{2} \theta}\)

With what minimum acceleration mass \(M\) must be moved on frictionless surface so that \(m\) remains stick to it as shown. The co-efficient of friction between \(M\) and \(m\) is \(\mu\). (A) \(\mu g\) (B) \(\frac{g}{\mu}\) (C) \(\frac{\mu m g}{M+m}\) (D) \(\frac{\mu m g}{M}\)

A block of mass \(0.1 \mathrm{~kg}\) is held against a wall by applying a horizontal force of \(5 \mathrm{~N}\) on the block. If the co-efficient of friction between the block and the wall is \(0.5\), the magnitude of the frictional force acting on the block is (A) \(2.5 \mathrm{~N}\) (B) \(0.98 \mathrm{~N}\) (C) \(4.9 \mathrm{~N}\) (D) \(0.49 \mathrm{~N}\)

A body of mass \(60 \mathrm{~kg}\) is dragged with just enough force to start moving on a rough surface with co-efficient of static and kinetic friction \(0.5\) and \(0.4\), respectively. On applying the same force, what is the acceleration \(\left(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\right)\) ? (A) \(0.98 \mathrm{~m} / \mathrm{s}^{2}\) (B) \(9.8 \mathrm{~m} / \mathrm{s}^{2}\) (C) \(.54 \mathrm{~m} / \mathrm{s}^{2}\) (D) \(5.292 \mathrm{~m} / \mathrm{s}^{2}\)

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

A lift is moving downwards with an acceleration equal to acceleration due to gravity. A body of mass \(M\) kept on the floor of the lift is pulled horizontally. If the co-efficient of friction is \(\mu\), then the frictional resistance offered by the body is (A) \(M g\) (B) \(\mu \mathrm{Mg}\) (C) \(2 \mu \mathrm{Mg}\) (D) Zero

Consider the system shown in Fig. 3.78. The wall is smooth, but the surface of blocks \(A\) and \(B\) in contact are rough. The friction on \(B\) due to \(A\) in equilibrium is (A) Upward (B) Downward (C) Zero (D) The system cannot remain in equilibrium.

A block of mass \(m\) is placed on the top of another block of mass \(M\) as shown in the Fig. \(3.81\). The co-efficient of friction between them is \(\mu .\) The maximum acceleration with which the block \(M\) may move so that \(m\) also moves along with it is (A) \(\mu g\) (B) \(g / \mu\) (C) \(\mu^{2} / g\) (D) \(g / \mu^{2}\)

A block \(A\) of mass \(m\) is placed over a plank \(B\) of mass \(2 \mathrm{~m}\). Plank \(B\) is placed over a smooth horizontal surface. The co-efficient of friction between \(A\) and \(B\) is \(\frac{1}{2} .\) Block \(A\) is given a velocity \(v_{0}\) towards right. Acceleration of \(B\) relative to \(A\) is (A) \(\frac{g}{2}\) (B) \(g\) (C) \(\frac{3 g}{4}\) (D) Zero

The upper half of an incline plane with inclination \(\phi\) is perfectly smooth, while the lower half is rough. A body starting from rest at the top will again come to rest at the bottom if the co-efficient of friction for the lower half is given by (A) \(2 \tan \phi\) (B) \(\tan \phi\) (C) \(2 \sin \phi\) (D) \(2 \cos \phi\)

A pebble of mass \(0.05 \mathrm{~kg}\) is thrown vertically upwards. The direction and magnitude of the net force on the pebble is given below, choose the incorrect option. (A) During its upward motion, force is \(0.5 \mathrm{~N}\) in vertically upward. (B) During its downward motion, force is \(0.5 \mathrm{~N}\) in vertically downward. (C) At the highest point, where it is momentarily at rest, force is \(0.5 \mathrm{~N}\) in vertically downward. (D) If the pebble was thrown at an angle of say \(45^{\circ}\) with the horizontal direction, force is \(0.5 \mathrm{~N}\) in vertically downward (Ignoring air resistance).

The magnitude and direction of the net force acting on a stone of mass \(0.1 \mathrm{~kg}\) is given below, choose the incorrect statement. (A) Just after it is dropped from the window of a stationary train, \(F=1.0 \mathrm{~N}\) (Vertically down ward) (B) Just after it is dropped from the window of a train running at a constant velocity of \(36 \mathrm{~km} / \mathrm{h}\), \(F=1.0 \mathrm{~N}\) (Vertically downward). (C) Just after it is dropped from the window of a train accelerating with \(1 \mathrm{~ms}^{-2}, F=1.0 \mathrm{~N}\) (Vertically downward). (D) Just after it is dropped from the window of a train accelerating with \(1 \mathrm{~ms}^{-2}, F=2.0 \mathrm{~N}\) (Vertically downward).

One end of a string of length \(l\) is connected to a particle of mass \(m\) and the other to a small peg on a smooth horizontal table. If the particle moves in a circular motion with speed \(v\), the net force on the particle (directed towards the centre) is: (A) \(T\) (B) \(T-\frac{m v^{2}}{l}\) (C) \(T+\frac{m v^{2}}{l}\) (D) 0

A constant retarding force of \(50 \mathrm{~N}\) is applied to a body of mass \(20 \mathrm{~kg}\) moving initially with a speed of \(15 \mathrm{~ms}^{-1}\). How long does the body take to stop? (A) \(2 \mathrm{~s}\) (B) \(4 \mathrm{~s}\) (C) \(6 \mathrm{~s}\) (D) \(8 \mathrm{~s}\)

A constant force acting on a body of mass \(3.0 \mathrm{~kg}\) changes its speed from \(2.0 \mathrm{~ms}^{-1}\) to \(3.5 \mathrm{~ms}^{-1}\) in \(25 \mathrm{~s}\). The direction of the motion of the body remains unchanged. What is the magnitude and direction of the force? (A) \(0.18 \mathrm{~N}\) (B) \(0.36 \mathrm{~N}\) (C) \(0.9 \mathrm{~N}\) (D) None of these

A body of mass \(5 \mathrm{~kg}\) is acted upon by two perpendicular forces \(8 \mathrm{~N}\) and \(6 \mathrm{~N}\). Give the magnitude of the acceleration of the body. (A) \(2 \mathrm{~m} / \mathrm{s}^{2}\) (B) \(4 \mathrm{~m} / \mathrm{s}^{2}\) (C) \(6 \mathrm{~m} / \mathrm{s}^{2}\) (D) \(8 \mathrm{~m} / \mathrm{s}^{2}\)

The driver of three-wheeler moving with a speed of \(36 \mathrm{~km} / \mathrm{h}\) sees a child standing in the middle of the road and brings his vehicle to rest in \(4.0 \mathrm{~s}\) just in time to save the child. What is the average retarding force on the vehicle? The mass of the three-wheeler is \(400 \mathrm{~kg}\) and the mass of the driver is \(65 \mathrm{~kg}\). (A) \(1162.5 \mathrm{~N}\) (B) \(116.25 \mathrm{~N}\) (C) \(1112 \mathrm{~N}\) (D) None of these

A man of mass \(70 \mathrm{~kg}\) stands on a weighing scale in lift which is moving. Choose the correct statement. (A) Reading of weighing scale is \(700 \mathrm{~N}\) upwards with a uniform speed of \(10 \mathrm{~ms}^{-1}\). (B) Reading of weighing scale is \(700 \mathrm{~N}\) downwards with a uniform acceleration of \(5 \mathrm{~ms}^{-2}\) (C) Reading of weighing scale is \(700 \mathrm{~N}\) upwards with a uniform acceleration of \(5 \mathrm{~ms}^{-2}\) (D) Reading of weighing scale is \(700 \mathrm{~N}\) if the lift mechanism failed and it fall down freely under gravity.

A ball is travelling with uniform translatory motion. This means that (A) it is at rest. (B) the path can be a straight line or circular and the ball travels with uniform speed. (C) all parts of the ball have the same velocity (magnitude and direction) and the velocity is constant. (D) the centre of the ball moves with constant velocity and the ball spins about its centre uniformly.

A metre scale is moving with uniform velocity. This implies (A) the force acting on the scale is zero, but a torque about the centre of mass can act on the scale. (B) the force acting on the scale is zero and the torque acting about centre of mass of the scale is also zero. (C) the total force acting on it need not be zero but the torque on it is zero. (D) neither the force nor the torque needs to be zero.

A hockey player is moving northward and suddenly turns westward with the same speed to avoid an opponent. The force that acts on the player is (A) frictional force along westward. (B) muscle force along southward. (C) frictional force along south-west. (D) muscle force along south-west.

A body with mass \(5 \mathrm{~kg}\) is acted upon by a force \(F=(-3 \hat{i}+4 \hat{j}) N .\) If its initial velocity at \(t=0\) is \(v=(6 \hat{i}-12 \hat{j}) \mathrm{ms}^{-1}\), the time at which it will just have a velocity along the \(y\)-axis is \(\begin{array}{lll}\text { (A) Never } & \text { (B) } 10 \mathrm{~s} & \text { (C) } 2 \mathrm{~s}\end{array}\) (D) \(15 \mathrm{~s}\)

A car of mass \(m\) starts from rest and acquires a velocity along east, \(v=v \hat{i}(v>0)\) in two seconds. Assuming the car moves with uniform acceleration, the force exerted on the car is (A) \(\frac{m v}{2}\) eastward and is exerted by the car engine. (B) \(\frac{m v}{2}\) eastward and is due to the friction on the tyres exerted by the road. (C) more than \(\frac{m v}{2}\) eastward exerted due to the engine and overcomes the friction of the road. (D) \(\frac{m v}{\text { exerted by the engine. }}\)

Two bodies of masses \(10 \mathrm{~kg}\) and \(20 \mathrm{~kg}\), respectively, kept on a smooth, horizontal surface are tied to the ends of a light string. A horizontal force \(F=600 \mathrm{~N}\) is applied to body of mass \(10 \mathrm{~kg}\). What is the tension in the string in each case? (A) \(200 \mathrm{~N}\) (B) \(100 \mathrm{~N}\) (C) \(400 \mathrm{~N}\) (D) \(600 \mathrm{~N}\)

Two masses \(8 \mathrm{~kg}\) and \(12 \mathrm{~kg}\) are connected at the two ends of a light inextensible string that goes over a frictionless pulley. Find the tension the string when the masses are released. (A) \(96 \mathrm{~N}\) (B) \(80 \mathrm{~N}\) (C) \(100 \mathrm{~N}\) (D) None of these

At a curved path of the road, the roadbed is raised a little on the side away from the center of the curved path. The slope of the roadbed is given by (A) \(\tan ^{-1} \frac{v^{2} g}{r}\) (B) \(\tan ^{-1} \frac{r g}{v^{2}}\) (C) \(\tan ^{-1} \frac{r}{g v^{2}}\) (D) \(\tan ^{-1} \frac{v^{2}}{r g}\)

A car of mass \(m\) is being driven on a circular path of radius \(R\). In which of the following circumstances it will not slip ( \(\mu\) is coefficient of friction between surface and road) (A) \(\frac{m v^{2}}{R} \geq \mu m g\) (B) \(\frac{m v^{2}}{R}=4 \mu \mathrm{mg}\) (C) \(\frac{m v^{2}}{R}>m g\) (D) None

For a particle rotating in a vertical circle with uniform speed, the maximum and minimum tension in the string are in the ratio \(5: 3\). If the radius of vertical circle is \(2 \mathrm{~m}\), the speed of revolving body is \(\left(g=10 \mathrm{~m} / \mathrm{s}^{2}\right)\) (A) \(\sqrt{5} \mathrm{~m} / \mathrm{s}\) (B) \(4 \sqrt{5} \mathrm{~m} / \mathrm{s}\) (C) \(5 \mathrm{~m} / \mathrm{s}\) (D) \(10 \mathrm{~m} / \mathrm{s}\)

A stone of mass \(1.5 \mathrm{~kg}\) is tied at the end of \(0.5 \mathrm{~m}\) long string and whirled in a vertical circular path at constant speed of \(2 \mathrm{~ms}^{-1}\). The maximum tension in the string is \(\left(g=10 \mathrm{~ms}^{-2}\right)\) (A) \(27 \mathrm{~N}\) (B) \(3 \mathrm{~N}\) (C) \(90 \mathrm{~N}\) (D) \(15 \mathrm{~N}\)

A particle of mass \(0.1 \mathrm{~kg}\) is whirled at the end of a string in a vertical circle of radius \(1.0 \mathrm{~m}\) at a constant speed of \(5 \mathrm{~m} / \mathrm{s}\). The tension in the string at the highest point of its path is \(\left(g=10 \mathrm{~m} / \mathrm{s}^{2}\right)\) (A) \(0.5 \mathrm{~N}\) \(\begin{array}{lll}\text { (B) } 1.0 \mathrm{~N} & \text { (C) } 1.5 \mathrm{~N} & \text { (D) } 2.0 \mathrm{~N}\end{array}\)

A motor car is traveling at \(60 \mathrm{~m} / \mathrm{s}\) on a circular road of radius \(1200 \mathrm{~m}\). It is increasing its speed at the rate of \(4 \mathrm{~m} / \mathrm{s}^{2}\). The acceleration of the car is (A) \(3 \mathrm{~ms}^{-2}\) (B) \(4 \mathrm{~ms}^{-2}\) (C) \(5 \mathrm{~ms}^{-2}\) (D) \(7 \mathrm{~ms}^{-2}\)

A \(2 \mathrm{~kg}\) stone at the end of a string \(1 \mathrm{~m}\) long is whirled in a vertical circle at a constant speed. The speed of the stone is \(4 \mathrm{~m} / \mathrm{s}\). The tension in the string will be \(52 \mathrm{~N}\), when the stone is (A) at the top of the circle (B) at the bottom of the circle (C) halfway down (D) none of the above

A particle is moving with a constant angular acceleration of \(4 \mathrm{rad} / \mathrm{s}^{2}\) in a circular path. At \(t=0\), particle was at rest. Find the time at which the magnitudes of centripetal acceleration and tangential acceleration are equal. \(\begin{array}{llll}\text { (A) } 1 \mathrm{~s} & \text { (B) } 2 \mathrm{~s} & \text { (C) } \frac{1}{2} \mathrm{~s} & \text { (D) } \frac{1}{4} \mathrm{~s}\end{array}\)

A particle is rotating in a circle of radius \(R\) with constant angular velocity \(\omega .\) Its average velocity during \(t\) seconds after start of motion is (A) \(\frac{2 R}{t} \sin \left(\frac{\omega t}{2}\right)\) (B) \(\frac{2 R}{t} \cos \left(\frac{\omega t}{2}\right)\) (C) \(\frac{R}{t} \sin \left(\frac{\omega t}{2}\right)\) (D) \(\frac{R}{t} \cos \left(\frac{\omega t}{2}\right)\)

A particle is moving along the circular path with a speed \(v\) and tangential acceleration is \(g\) at an instant. If the radius of the circular path be \(r\), then the net acceleration of the particle at that instant is (A) \(\frac{v^{2}}{r}+g\) (B) \(\frac{v^{2}}{r^{2}}+g^{2}\) (C) \(\left[\frac{v^{4}}{r^{2}}+g^{2}\right]^{\frac{1}{2}}\) (D) \(\left[\frac{v^{2}}{r}+g^{2}\right]^{\frac{1}{2}}\)

A particle of mass \(m\) is fixed to one end of a light spring of force constant \(k\) and unstretched length \(l\). The other end of the spring is fixed and it is rotated in horizontal circle with an angular velocity \(\omega\), in gravity free space. The increase in length of the spring will be (A) \(\frac{m \omega^{2} l}{k}\) (B) \(\frac{m \omega^{2} l}{k-m \omega^{2}}\) (C) \(\frac{m \omega^{2} l}{k+m \omega^{2}}\) (D) none of these

The angular velocity of a wheel increases from 1200 rpm to 4500 rpm in \(10 \mathrm{~s}\). The number of revolutions made during this time is (A) 950 (B) 475 (C) \(237.5\) (D) \(118.75\)

A disc is rotating with an angular velocity \(\omega_{0} .\) A constant retarding torque is applied on it to stop the disc. The angular velocity becomes \(\omega_{0} / 2\) after \(n\) rotations. How many more rotations will it make before coming to rest? (A) \(n\) (B) \(2 n\) (C) \(\frac{n}{2}\) (D) \(\frac{n}{3}\)

An insect crawls up a hemispherical surface very slowly (Fig. \(3.88\) ). The coefficient of friction between the insect and the surface is \(1 / 3 .\) If the line joining the centre of hemispherical surface to the insect makes an angle \(\alpha\) with the vertical, the maximum possible value of \(\alpha\) is given by (A) \(\cot \alpha=3\) (B) \(\tan \alpha=3\) (C) \(\sec \alpha=3\) (D) \(\operatorname{cosec} \alpha=3\)

Starting from rest, a particle rotates in a circle of radius \(R=\sqrt{2} \mathrm{~m}\) with an angular acceleration \(\alpha=(\pi / 4) \mathrm{rad} / \mathrm{s}^{2}\). The magnitude of average velocity of the particle over the time it rotates a quarter circle is (A) \(1.5 \mathrm{~m} / \mathrm{s}\) (B) \(2 \mathrm{~m} / \mathrm{s}\) (C) \(1 \mathrm{~m} / \mathrm{s}\) (D) \(1.25 \mathrm{~m} / \mathrm{s}\)