Chapter 5

The bob of a pendulum of mass \(m\) and length \(L\) is displaced, \(90^{\circ}\) from the vertical and gently released. In order that the string may not break upon passing through the lowest point, its minimum strength must be (a) \(m g\) (b) \(2 \mathrm{mg}_{\mathrm{J}}\) (c) \(3 \mathrm{mg}\) (d) \(4 \mathrm{mg}\)

A car-wheel is rotated to uniform angular acceleration about its axis., Initially its angular velocity is zero. It rotates through an angle \(\theta_{1}\) in the first \(2 \mathrm{~s}\), in the next \(2 \mathrm{~s}\), it rotates through an additional angle \(\theta_{2}\), the ratio of \(\frac{\theta_{2}}{\theta_{1}}\) is (a) 1 (b) 2 (c) 3 (d) 5

A body is acted upon by a constant force directed towards a fixed point. The magnitude of the force varies inversely as the square of the distance from the fixed point. What is the nature of the path? (a) Straight line (b) Parabola (c) Circle (d) Hyperbola

A sphere of mass \(0.2 \mathrm{~kg}\) is attached to an inextensible string of length \(0.5 \mathrm{~m}\) whose upper end is fixed to the ceilling. The sphere is made to describe a horizontal circle of radius \(0.3 \mathrm{~m}\). The speed of the sphere will be (a) \(1.5 \mathrm{~ms}^{-1}\) (b) \(2.5 \mathrm{~ms}^{-1}\) (c) \(3.2 \mathrm{~ms}^{-1}\) (d) \(4.7 \mathrm{~ms}^{-1}\)

The kinetic energy \(K\) of a particle moving along a circle of radius \(R\) depends on the distance covered \(s\) as \(K=a s^{2}\), where \(a\) is a constant. The force acting on the particle is (a) \(2 a \frac{5^{2}}{R}\) (b) \(2 \operatorname{as}\left(1+\frac{s^{2}}{R^{2}}\right)^{1 / 2}\) (c) 2 as (d) \(2 a \frac{R^{2}}{s}\)

A particle of mass \(m\) is moving in circular path of constant radius \(r\) such that its centripetal acceleration \(a_{c}\) is varying with time \(t\) as \(a_{c}=k^{2} r t^{2}\). The power delivered to the particle by the forces acting on it is (a) \(2 \pi m k^{2} r^{2} t\) (b) \(m k^{2} r^{2} t\) (c) \(\frac{m k^{4} r^{2} t^{5}}{3}\) (d) zero

Two particles of equal mass are connected to a rope \(A B\) of negligible mass such that one is at end \(A\) and other dividing the length of rope in the ratio \(1: 2\) from \(B\). The rope is rotated about end \(B\) in a horizontal plane. Ratio of tensions in the smaller part to the other is (ignore the effect of gravity) (a) \(4: 3\) (b) \(1: 4\) (c) \(1: 2\) (d) \(1: 3\)

When the road is dry and coefficient of friction is \(\mu\), the maximum speed of a car in a circular path is \(10 \mathrm{~ms}^{-1}\). If the road becomes wet and \(\mu^{\prime}=\mu / 2\), what is the maximum speed permitted? (a) \(5 \mathrm{~ms}^{-1}\) (b) \(10 \mathrm{~ms}^{-1}\) (c) \(10 \sqrt{2} \mathrm{~ms}^{-1}\) (d) \(5 \sqrt{2} \mathrm{~ms}^{-1}\)

A coin is placed on a gramophone record rotating at a speed of \(45 \mathrm{rpm}\). It flies away when the rotational speed is \(50 \mathrm{rpm}\). If two such coins are placed over the other on the same record, both of them will fly away when rotational speed is (a) \(100 \mathrm{rpm}\) (b) \(25 \mathrm{rpm}\) (c) \(12.5 \mathrm{rpm}\) (d) \(50 \mathrm{rpm}_{1}\)

A body of mass \(1 \mathrm{~kg}\) is moving in a vertical circular path of radius \(1 \mathrm{~m} .\) The difference between the kinetic energies at its highest and lowest point is (a) \(20 \mathrm{~J}\) (b) \(10 \mathrm{~J}\) (c) \(4 \sqrt{5} \mathrm{~J}\) (d) \(10 \sqrt{5} \mathrm{~J}\)

The maximum and minimum tension in the string whirling in a circle of radius \(2.5 \mathrm{~m}\) with constant velocity are in the ratio \(5: 3\), then its velocity is (a) \(\sqrt{98} \mathrm{~ms}^{-1}\) (b) \(7 \mathrm{~ms}^{-1}\) (c) \(\sqrt{490} \mathrm{~ms}^{-1}\) (d) \(\sqrt{4.9} \mathrm{~ms}^{-1}\)

A particle moves along a circle of radius \(\left(\frac{20}{\pi}\right) \mathrm{m}\) with constant tangential acceleration. If the velocity of the particle is \(80 \mathrm{~ms}^{-1}\), at the end of seconds revolution after motion has begun, the tangential acceleration is (a) \(40 \mathrm{~ms}^{-2}\) (b) \(640 \pi \mathrm{ms}^{-2}\) (c) \(1609 \pi \mathrm{ms}^{-2}\) (d) \(40 \pi \mathrm{ms}^{-2}\)

A long horizontal rod has a bead, which can slide along its length and initially placed at a distance \(L\) from one end \(A\) of the rod. The rod is set in angular acceleration \(\alpha\). If the coefficient of friction, between the rod and the bead is \(\mu\) and gravity is neglected, then the time after which the bead starts slipping is (a) \(\sqrt{\mu / \alpha}\) (b) \(\mu / \sqrt{\alpha}\) (c) \(1 / \sqrt{\mu \alpha}\) (d) infinitesimal

The distance \(r\) from the origin of a particle moving in \(x y\)-plane varies with time as \(r=2 t\) and the angle made by the radius vector with positive \(x\)-axis is \(\theta=4 t\). Here, \(t\) is in second, \(r\) in metre and \(\theta\) in radian. The speed of the particle at \(t=1 \mathrm{~s}\) is (a) \(10 \mathrm{~ms}^{-1}\) (b) \(16 \mathrm{~ms}^{-1}\) (c) \(20 \mathrm{~ms}^{-1}\) (d) \(12 \mathrm{~ms}^{-1}\)

A car moving on a circular path and takes a turn. If \(R_{1}\) and \(R_{2}\) be the reactions on the inner and outer wheels respectively, then (a) \(R_{1}=R_{2}\) (b) \(R_{1}R_{2}\) (d) \(R_{1} \geq R_{2}\)

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

The length of second's hand in a watch is \(1 \mathrm{~cm}\). The change in velocity of its tip in \(15 \mathrm{~s}\) is (a) zero (b) \(\frac{\pi}{30 \sqrt{2}} \mathrm{~cm} / \mathrm{s}\) (c) \(\frac{\pi}{30} \mathrm{~cm} / \mathrm{s}\) (d) \(\frac{\pi \sqrt{2}}{30} \mathrm{~cm} / \mathrm{s}\)

A wheel rotates with a constant angular velocity of \(300 \mathrm{rpm}\). The angle through which the wheel rotates in one second is (a) \(\pi \mathrm{rad}\) (b) \(5 \pi \mathrm{rad}\) (c) \(10 \pi \mathrm{rad}\) (d) \(20 \pi \mathrm{rad}\)

A string is would round the rim of a mounted fly wheel of mass \(20 \mathrm{~kg}\) and radius \(20 \mathrm{~cm}\). A steady ball of \(25 \mathrm{~N}\) is applied on the cord. Neglecting friction and mass of the string, the angular acceleration of the wheel is (a) \(50 \mathrm{rad} \mathrm{s}^{-2}\) (b) \(25 \mathrm{rad} \mathrm{s}^{-2}\) (c) \(6.25 \mathrm{rad} \mathrm{s}^{-2}\) (d) \(12.5 \mathrm{rad} \mathrm{s}^{-2}\)

If a particle covers half the circle of radius \(R\) with constant speed, then (a) change in momentum is \(m \mathrm{vr}\) (b) change in \(\mathrm{KE}\) is \(\frac{1}{2} m v^{2}\) (c) change in \(\mathrm{KE}\) is \(\mathrm{mv}^{2}\) (d) change in \(\mathrm{KE}\) is zero

An aeroplane flying at a velocity of \(900 \mathrm{kmh}^{-1}\) loops the loop. If the maximum force pressing the pilot against the seat is five times its weight, the loop radius should be (a) \(1594 \mathrm{~m}\) (b) \(1402 \mathrm{~m}\) (c) \(1315 \mathrm{~m}\) (d) \(1167 \mathrm{~m}\)

The string of a pendulum of length \(l\) is displaced through \(90^{\circ}\) from the vertical and released. Then, the minimum strength of the string in order to withstand the tension as the pendulum passes through the mean position is (a) \(\overline{m g}\) (b) \(6 \mathrm{mg}\) (c) \(3 \mathrm{mg}\) (d) \(5 \mathrm{mg}\)

An object is being weighed on a spring balance moving around a curve of radius \(100 \mathrm{~m}\) at a speed \(7 \mathrm{~ms}^{-1}\). The object has a weight of 60 kg-wt. The reading registered on the spring balance would be (a) \(60.075 \mathrm{~kg}-\mathrm{wt}\) (b) \(60.125 \mathrm{~kg}\)-wt (c) \(60.175 \mathrm{~kg}-\mathrm{wt}\) (d) \(60.225 \mathrm{~kg}-\mathrm{wt}\)

An object of mass \(10 \mathrm{~kg}\) is whirled round a horizontal circle of radius \(4 \mathrm{~m}\) by a revolving string inclined \(30^{\circ}\) to the vertical. If the uniform speed of the object is \(5 \mathrm{~ms}^{-1}\), the tension in the string (approximately) is (a) \(720 \mathrm{~N}\) (b) \(960 \mathrm{~N}\) (c) \(114 \mathrm{~N}\) (d) \(125 \mathrm{~N}\)

A stone of mass \(1 \mathrm{~kg}\) is tied to a string \(4 \mathrm{~m}\) long and is rotated at constant speed of \(40 \mathrm{~ms}^{-1}\) in a vertical circle. The ratio of the tension at the top and the bottom is (a) \(11: 12\) (b) \(39: 41\) (c) \(41: 39\) (d) \(12: 11\)

A particle moves along a circle with a constant speed. If \(a\) is acceleration and \(E\) is kinetic energy of the particle, then (a) \(a\) is constant (b) \(E\) is constant (c) \(a\) is variable (d) \(E\) is variable

A weightless thread can bear tension upto \(3.7 \mathrm{~kg}\)-wt. A stone of mass \(500 \mathrm{~g}\) is tied to it and revolved in a circular path of radius \(4 \mathrm{~m}\) in a vertical plane. If \(g=10 \mathrm{~ms}^{-2}\), then the maximum angular velocity of the stone will be (a) \(4 \mathrm{rad} / \mathrm{s}\) (b) \(16 \mathrm{rad} / \mathrm{s}\) (c) \(\sqrt{21} \mathrm{rad} / \mathrm{s}\) (d) \(2 \mathrm{rad} / \mathrm{s}\)

A body of mass \(m\) is moving in a circle of radius \(r\) with a constant speed \(v\). The force on the body is \(m v^{2} / r\) and is directed towards the centre. What is the work done by this force in moving the body over half the circumference of the circle? (a) \(\frac{m v^{2}}{r} \times \pi r\) (b) \(\frac{m v^{2}}{r^{2}}\) (c) zero (d) \(\frac{\pi r}{m v^{2}}\)

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

For a particle performing uniform circular motion, choose the correct statement (s) from the following [NCERT Exemplar] (a) Magnitude of particle velocity (speed) remains coonstant (b) Particle velocity remains directed perpendicular to radius vector (c) Direction of acceleration keeps changing as particle moves (d) Angular momentum is constant in magnitude but direction keeps changing

An object is tied to a string and rotated in a vertical circle of radius \(r .\) Constant speed is maintained along the trajectory. If \(T_{\max } / T_{\min }=2\), then \(v^{2} / r g\) is (a) 1 (b) 2 (c) 3 (d) 4

The speed of revolution of a particle moving round a circle is doubled and its angular speed is halved. What happens to the centripetal acceleration? (a) Unchanged (b) Halved (c) Doubled (d) 4 times

If \(a_{r}\) and \(a_{t}\) represent radial and tangential acceleration respectively, the motion of a particle will be circular if (a) \(a_{r}=0\) and \(a_{t}=0\) (b) \(a_{r}=0\) but \(a_{t} \neq 0\) (c) \(a_{r} \neq 0\) and \(a_{t}=0\) (d) \(a_{r} \neq 0\) and \(a_{t} \neq 0\)

A simple pendulum oscillates in a vertical plane. When it passes through the mean position the tension in the string is 3 times the weight of pendulum bob. What is the maximum displacement of the pendulum with respect to the vertical? (a) \(30^{\circ}\) (b) \(45^{\circ}\) (c) \(60^{\circ}\) (d) \(90^{\circ}\)

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 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) \(\sqrt{2\left(u^{2}-g L\right)}\)

Read each of the following statements carefully and state with reasons, chose the correct statement (s) (i) The net acceleration of a particle in the circular motion is always along the radius of the circle towards the centre. (ii) The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point. (iii) The acceleration vector of a particle in uniform circular motion averaged over one cycle is a null vector. (a) (i) and (iii) (b) (ii) and (iii) (c) (iii) Only (d) All the three

A body of mass \(1 \mathrm{~kg}\) is rotating in a vertical circle of radius \(1 \mathrm{~m}\). What will be the difference in its kinetic energy at the top and bottom of the circle? (Take \(g=10 \mathrm{~ms}^{-2}\) ) (a) \(10 \mathrm{~J}\) (b) \(20 \mathrm{~J}\) (c) \(30 \mathrm{~J}\) (d) \(50 \mathrm{~J}\)

A fan is making 600 revolution per minute. If after some time it makes 1200 revolution per minute, then increase in its angular velocity is (a) \(10 \pi \mathrm{rad} / \mathrm{s}\) (b) \(20 \pi \mathrm{rad} / \mathrm{s}\) (c) \(40 \pi \mathrm{rad} / \mathrm{s}\) (d) \(60 \pi \mathrm{rad} / \mathrm{s}\)

A body moves along a circular path of radius \(5 \mathrm{~m}\). The coefficient of friction between the surface of path and the body is \(0.5\). The angular velocity, in \(\mathrm{rad} / \mathrm{s}\), with which the body should move so that it does not leave the path is \(\left(g=10 \mathrm{~ms}^{-2}\right)\) (a) 4 (b) 3 (c) 2 (d) 1

A car is moving on a circular level road of radius of curvature \(300 \mathrm{~m}\). If the coefficient of friction is \(0.3\) and acceleration due to gravity \(10 \mathrm{~ms}^{-2}\), the maximum speed of the car can have is (in \(\mathrm{kmh}^{-1}\) ) (a) 30 (b) 81 (c) 108 [d) 162

A railway carriage has its centre of gravity at a height of \(1 \mathrm{~m}\) above the rails, which are \(1.5 \mathrm{~m}\) apart. The maximum safe speed at which it could travel round an unbanked curve of radius \(100 \mathrm{~m}\) is (a) \(12 \mathrm{~ms}^{-1}\) (b) \(18 \mathrm{~ms}^{-1}\) (c) \(22 \mathrm{~ms}^{-1}\) (d) \(27 \mathrm{~ms}^{-1}\)

Assertion As the frictional force increases the safe velocity limit for taking a turn on an unbanked road also increases. Reason Banking of roads will increase the value of limiting velocity.

A car of mass \(2000 \mathrm{~kg}\) is moving with a speed of \(10 \mathrm{~ms}^{-1}\) on a circular path of radius \(20 \mathrm{~m}\) on a level road. What must be the frictional force between the car and the road so that the car does not slip? (a) \(10^{4} \mathrm{~N}\) (b) \(10^{3} \mathrm{~N}\) (c) \(10^{5} \mathrm{~N}\) (d) \(10^{2} \mathrm{~N}\)

A stone tied to the end of a string \(80 \mathrm{~cm}\) long is whirled in a horizontal circle with a constant speed. If the stone makes 14 revolutions in \(25 \mathrm{~s}\), what is the magnitude and direction of acceleration of the stone? (a) \(9.9 \mathrm{~m} / \mathrm{s}^{2}\) along the tangent (b) \(7.9 \mathrm{~m} / \mathrm{s}^{2}\) along the radius (c) \(9.9 \mathrm{~m} / \mathrm{s}^{2}\) along the radius (d) None of the above

An aircraft executes a horizontal loop of radius \(1 \mathrm{~km}\) with a speed of \(900 \mathrm{~km} / \mathrm{h}\). Compare its centripetal acceleration with the acceleration due to gravity. (a) 6 (b) 7 (c) \(\underline{8}\) (d) 5

A body of mass \(5 \mathrm{~kg}\) is moving in a circle of radius \(1 \mathrm{~m}\) with angular velocity of \(2 \mathrm{rad} / \mathrm{s}\). The centripetal force is [Orissa JEE 2011] (a) \(10 \mathrm{~N}\) (b) \(20 \mathrm{~N}\) (c) \(30 \mathrm{~N}\) (d) \(40 \mathrm{~N}\)

If a car is to travel with a speed \(v\) along the frictionless banked circular track of radius \(r\), the required angle of banking so that the car does skid is [J\&K 2010] (a) \(\theta=\tan ^{-1}\left(\frac{v^{2}}{r g}\right)\) (b) \(\theta=\tan ^{-1}\left(\frac{v}{r g}\right)\) (c) \(\theta=\tan ^{-1}\left(\frac{r^{2}}{v g}\right)\) (d) \(\theta<\tan ^{-1}\left(\frac{\partial^{2}}{r g}\right)\)

What should be the coefficient of friction between the tyres and the road, when a car travelling at \(60 \mathrm{kmh}^{-1}\) makes a level turn of radius \(40 \mathrm{~m} ?\) (a) \(0.5\) (b) \(0.66\) (c) \(0.71\) (d) \(0.80\)

A car is moving along a circular path of radius \(500 \mathrm{~m}\) with a speed of \(30 \mathrm{~ms}^{-1}\). If at some instant, its speed increases at the rate of \(2 \mathrm{~ms}^{-1}\), then at that instant the magnitude of resultant acceleration will be [UP SEE 2009] (a) \(4.7 \mathrm{~ms}^{-2}\) (b) \(3.8 \mathrm{~ms}^{-2}\) (c) \(3 \mathrm{~ms}^{-2}\) (d) \(2.7 \mathrm{~ms}^{-2}\)

The maximum speed with which a car is driven round a curve of radius \(18 \mathrm{~m}\) without skidding (where, \(g=10 \mathrm{~ms}^{-2}\) and the coefficient of friction between rubber tyres and the roadway is \(0.2\) ) is (a) \(36.0 \mathrm{kmh}^{-1}\) (b) \(18.0 \mathrm{kmh}^{-1}\) (c) \(21.6 \mathrm{kmh}^{-1}\) (d) \(14.4 \mathrm{kmh}^{-1}\)