Radial & Tangential Acceleration for Non Uniform Circular Motion

Q.

A coin placed on uniformly rotating turntable just slips if it is placed at a distance of 4 cm from the centre. If the angular velocity of the turntable is doubled, it will just slip at a distance of

1. 2 cm

2. 1 cm

3. 8 cm

4. 4 cm

Q.

Let 1.00 kg of liquid water at ${100}^{\circ }C$ be converted to steam at ${100}^{\circ }C$ by boiling at standard atmospheric pressure (which is 1.00 atm or 1.01 $\times$ 105 Pa) in the arrangement as shown. The volume of that water changes from an initial value of $1.00\times {10}^{-3}$ $m^3$ as a liquid to 1.671 $m^3$ as steam.

1. How much work is done by the system during this process?

1. 4200 kJ

2. 100 kJ

3. 169 kJ

4. 4.2 kJ

Q. A small ball of mass $m$ is attached with a light rigid rod of length $l$ and hinged at one of its ends such that it can rotate in vertical plane. Minimum speed $v$ that should be given to the ball at lowest point to complete the vertical circular path is

1. $\sqrt{7gl}$
2. $\sqrt{5gl}$
3. $\sqrt{3gl}$
4. $\sqrt{4gl}$

Q.

A pendulum clock loses 12 seconds a day if the temperature is ${40}^{\circ }C$ and goes fast by 4 seconds a day if the temperature is ${20}^{\circ }C$. Find the temperature at which the clock will show correct time, and the coefficient of linear expansion of the metal of the pendulum shaft.

1. ;

2. ;

3. ;

4. ;

Q. A particle of mass $m$ just completes the vertical circular motion. What will be the difference in tension at the lowest and highest point?

1. $2mg$
2. $4mg$
3. $8mg$
4. $6mg$

Q. A heavy mass is attached to a thin wire and is whirled in a vertical circle with constant speed. The wire is most likely to break.

1. When the mass is at height B.
2. When the mass is at the lowest point.
3. When the wire is horizontal.
4. At an angle of ${30}^{\circ }$ from the upward vertical.

Q. A $40\text{kg}$ mass at the end of a rope of length $l$, oscillates in a vertical plane with angular amplitude ${\theta }_o$. What is the tension $T$ in the rope when it makes an angle $\theta$ with the vertical? If the breaking strength of the rope is $80\text{kgf}$, what is the maximum angular amplitude ${\theta }_{max}$ with which the mass can oscillate without the rope breaking?

1. $T=mg(2\cos \theta -3\cos {\theta }_0),{\theta }_{max}={30}^{\circ }$
2. $T=mg(3\cos \theta -2\cos {\theta }_0),{\theta }_{max}={60}^{\circ }$
3. $T=mg(3\cos \theta -2\cos {\theta }_0),{\theta }_{max}={30}^{\circ }$
4. $T=mg(2\cos \theta -3\cos {\theta }_0),{\theta }_{max}={60}^{\circ }$

Q. A particle of mass $2\text{kg}$ is moving in a circular path of constant radius $1\text{m}$ such that its centripetal acceleration $a_c$ is varying with time $t$ as $a_c=100t^2$. The power delivered to the particle by the force acting on it at $t=5\text{sec}$ is:

1. $400\text{W}$
2. $1000\text{W}$
3. $500\text{W}$
4. $0\text{W}$

Q. A spherical ball of mass $m$ begins to slide down a fixed smooth sphere from the top with negligible initial velocity. What is its tangential acceleration when it breaks off the sphere?

1. $\frac{2g}{3}$
2. $\frac{g}{3}$
3. $\frac{\sqrt{5}}{3}g$
4. Ball never leaves contact

Q. A particle starting from rest moves with constant tangential acceleration on a circular path of radius $\frac{10}{\pi }\text{m}$. After $2.5$ rotation the velocity of the particle is $50\text{m/s}$, its tangential acceleration is

1. $25{\text{m/s}}^2$
2. $25{\text{rad/s}}^2$
3. $2500{\pi }^2{\text{m/s}}^2$
4. $2500{\pi }^2{\text{rad/s}}^2$

Q. A $2\text{kg}$ ball is swinging in a vertical circle at the end of an inextensible string $2\text{m}$ long. The angular speed of the ball if the string can sustain a maximum tension of $119.6\text{N}$ is (Take $g=9.8{\text{m/s}}^2$)

1. $5\sqrt{2}\text{rad/s}$
2. $5\text{rad/s}$
3. $2\sqrt{2}\text{rad/s}$
4. $2\text{rad/s}$

Q. The bob of the pendulum shown in the figure describes an arc of a circle in a vertical plane. If the tension in the cord is $3$ times of the weight of the bob for the position shown, the velocity of the bob at this position is

1. $6.47\text{m/s}$
2. $10\text{m/s}$
3. $0\text{m/s}$
4. $15\text{m/s}$

Q. The angle turned by a body undergoing circular motion depends on time as $\theta =(2t^2-{\theta }_0{t}^3)\text{rad}$, where ${\theta }_0$ is constant. Then the angular acceleration of the body at $t=2\text{s}$ is

1. $-12{\theta }_0{\text{rad/s}}^2$
2. $2-6{\theta }_0{\text{rad/s}}^2$
3. $4-12{\theta }_0{\text{rad/s}}^2$
4. $4-3{\theta }_0{\text{rad/s}}^2$

Q. Find the angle between the net acceleration vector and tangential acceleration vector for a particle moving in a circle of radius $4m$ and its speed is increasing at $3m/s$ every second, when the speed of the particle is $4m/s$.

1. ${37}^{\circ }$
2. ${53}^{\circ }$
3. ${60}^{\circ }$
4. ${45}^{\circ }$

Q. A stone of mass $1\text{kg}$ tied to a light string of length $\frac{10}{3}\text{m}$ is whirled in a vertical circle. If the ratio of the maximum tension to minimum tension is 4 and $g=10{\text{m/s}}^2$, then the speed of the stone at the highest point of the circle is

1. $0\text{m/s}$
2. $20\text{m/s}$
3. $10\text{m/s}$
4. $5\text{m/s}$

Q.

Match the block diagrams of blocks of mass m with their corresponding free body diagram

(Bodies are at rest w.r.t. ground)

1. P - (i); Q - (iv); R - (vi); S - (vii)

2. P - (ii); Q - (iv); R - (vi); S - (viii)

3. P - (iii); Q - (v); R - (v); S - (viii)

4. P - (iii); Q - (iv); R - (vi); S - (v)

Q. A solid body starts from rest. Angular acceleration of the body spinning about a stationary axis is given by $\alpha =2\cos \theta$ (in $rad/s^2$) where $\theta$ is the angle of rotation from initial position. Then which of the following graphs correctly represents the variation of angular velocity with respect to $\theta$?

1. 
2. 
3. 
4. 

Q. A car is moving on a circular road of diameter $50\text{m}$ with a speed $5{\text{ms}}^{-1}.$ It is suddenly accelerated at a rate $1{\text{ms}}^{-2}.$ If its mass is $500\text{kg}$, net force acting on the car at that instant is,

1. $500\text{N}$
2. $1000\text{N}$
3. $500\sqrt{2}\text{N}$
4. $\frac{500}{\sqrt{2}}\text{N}$

Q. A $1\text{kg}$ stone at the end of a $1\text{m}$ long string is whirled in a vertical circle at a constant speed of $4\text{m/s}$. The tension in the string is $6\text{N}$ when the stone is (Take $g=10{\text{m/s}}^2$)

1. At the top of the circle.
2. At the bottom of the circle.
3. Half way down
4. None of above

Q. A long horizontal rod has a bead which can slide along its length and is initially placed at a distance $L$ from one end $A$ of the rod. The rod is set in angular motion about $A$ with a constant angular acceleration $\alpha$. If the coefficient of static friction between the rod and bead is $\mu$, and gravity is neglected, then the time after which the bead starts slipping is:

1. $\text{infinitesimal}$
2. $\frac{\mu }{\sqrt{\alpha }}$
3. $\frac{1}{\sqrt{\mu \alpha }}$
4. $\sqrt{\frac{\mu }{\alpha }}$

Q. The bob of the pendulum shown in the figure describes an arc of a circle in a vertical plane. If the tension in the cord is $3$ times of the weight of the bob for the position shown, the velocity of the bob at this position is

1. $6.47\text{m/s}$
2. $10\text{m/s}$
3. $0\text{m/s}$
4. $15\text{m/s}$

Q. A spinning top has an angular retardation of $\alpha =k{\omega }^2$ (in ${\text{rad/s}}^2$), where $\omega$ is angular velocity of the top and $k=1{\text{rad}}^{-1}$. At ${\theta }_0=0\text{rad}$, if ${\omega }_0=120\text{rad/s}$, then the relation between angular displacement ($\theta$) and angular velocity is

1. $\omega =120e^{-\theta }$
2. $\omega =120e^{\theta }$
3. $\omega =60e^{-\theta }$
4. $\omega =60e^{\theta }$

Q. The angular velocity of a particle moving along positive direction varies as $\omega =\beta \sqrt{\theta }$ (in $rad/s$) where $\beta$ is a positive constant. Assuming that $\theta =0$ (at time $t=0$), the value of angular acceleration of the particle

1. Is directly proportional to time
2. Is constant with time
3. Is inversely proportional with time
4. Cannot be determined as data is insufficient

Q. A wheel starting from rest, rotates with $5{\text{rad/s}}^2$ angular acceleration along its axis. After $2\text{s}$ of motion, the magnitude of radial and tangential component of acceleration of the particle at a distance $2\text{cm}$ from the axis are ________ and ________ ${\text{cm s}}^{-2}$.

1. $200,5$
2. $50,10$
3. $100,5$
4. $200,10$

Q. In a simple pendulum, the breaking strength of the string is double the weight of the bob. The bob is released from rest when the string is horizontal. The string breaks when it makes an angle $\theta$ with the vertical. Then

1. $\theta ={\cos }^{-1}\left(\frac{1}{3}\right)$
2. $\theta ={60}^{\circ }$
3. $\theta ={\cos }^{-1}\left(\frac{2}{3}\right)$
4. $\theta =0^{\circ }$

Q.

A proton of mass $1.6\times {10}^{-27}$ kg goes round in a circular orbit of radius 0.10 m under a centripetal force of $4\times {10}^{-13}N$. then the frequency of revolution of the proton is about

1. $0.08\times {10}^{8}cyclespersec$

2. $4\times {10}^{8}cyclespersec$

3. $8\times {10}^{8}cyclespersec$

4. $12\times {10}^{8}cyclespersec$

Q. A particle is going in a spiral path as shown in figure with constant speed.
Figure

(a) The velocity of the particle is constant.
(b) The acceleration of the particle is constant.
(c) The magnitude of acceleration is constant.
(d) The magnitude of acceleration is decreasing continuously.

Q. A pulley wheel of diameter $2cm$ has a $1m$ long cord strapped across its periphery. The wheel is initially at rest. If it is given an angular acceleration of $0.1rad/s^2$, then the time taken for the cord to unwind completely is

1. $20s$
2. $20\sqrt{3}s$
3. $20\sqrt{5}s$
4. $10\sqrt{1}0s$

Q. A particle starts from rest and moves on a circular path with constant tangential acceleration of $0.6{\text{m/s}}^2$. If the particle slips when its total acceleration becomes $1{\text{m/s}}^2$, then the angle moved by it before it starts slipping is

1. $\frac{2}{9}\text{rad}$
2. $\frac{2}{3}\text{rad}$
3. $\frac{2}{5}\text{rad}$
4. $\frac{2}{7}\text{rad}$

Q. A spinning top has an angular retardation of $\alpha =k{\omega }^2$ (in ${\text{rad/s}}^2$), where $\omega$ is angular velocity of the top and $k=1{\text{rad}}^{-1}$. At ${\theta }_0=0\text{rad}$, if ${\omega }_0=120\text{rad/s}$, then the relation between angular displacement ($\theta$) and angular velocity is

1. $\omega =120e^{-\theta }$
2. $\omega =120e^{\theta }$
3. $\omega =60e^{-\theta }$
4. $\omega =60e^{\theta }$