Radial & Tangential Acceleration for Non Uniform Circular Motion

Q.

A proton, a deuteron, and an $\alpha$-particle are projected perpendicular to the direction of a uniform magnetic field with the same kinetic energy. The ratio of the radii of the circular paths described by them is

Q. A particle moves along a circle of radius $\left(\frac{20}{\pi }\right)m$ with constant tangential acceleration. If velocity of the particle is $80\text{m/s}$ at the end of the $2^{nd}$ revolution after the motion has begun, the tangential acceleration is

1. $640{\text{m/s}}^2$
2. $40{\text{m/s}}^2$
3. $40\pi {\text{m/s}}^2$
4. $160{\text{m/s}}^2$

Q. A particle of mass $m$ is fixed to one end of a light rigid rod of length $l$ and rotated in a vertical circular path about its other end. The minimum speed of the particle at its highest point must be:

1. Zero
2. $\sqrt{gl}$
3. $\sqrt{1.5gl}$
4. $\sqrt{2gl}$

Q. A particle is moving along a circle of radius $1\text{m}$ with an angular speed $\omega =2t^2+5$, the tangential acceleration of the particle after $2\text{seconds}$ is

1. $4{\text{m/s}}^2$
2. $2{\text{m/s}}^2$
3. $8{\text{m/s}}^2$
4. $13{\text{m/s}}^2$

Q. The maximum and minimum tension in the string whirling in a circle of radius 2.5 m with constant velocity are in the ratio 5 : 3 then its velocity is

1. $\sqrt{98}$ m/s
2. 7 m/s
3. $\sqrt{490}$ m/s
4. $\sqrt{4.9}$

Q.

Give an example of an accelerated body, moving with a uniform speed.

Q. A particle tied to a string describes a vertical circular motion of radius $R$ continuously. If it has velocity $\sqrt{3gR}$ at the highest point, then the ratio of respective tensions in the string at the highest and lowest point is

1. $4:3$
2. $3:2$
3. $5:4$
4. $1:4$

Q. A particle is kept at rest at the top of a sphere of diameter 42 m. When disturbed slightly, it slides down. At what height ‘h’ from the bottom, the particle will leave the sphere

1. 14 m
2. 28 m
3. 35 m
4. 7 m

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

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 uniform cylinder of mass $M$ and radius $R$ is to be pulled over a step of height $a(a<R)$ by applying a force $F$ at its centre $O$ perpendicular to the plane through the axes of the cylinder on the edge of the step (see figure). The minimum value of $F$ required is :

1. $Mg\sqrt{1-{\left(\frac{R-a}{R}\right)}^2}$
2. $Mg\sqrt{{\left(\frac{R}{R-a}\right)}^2-1}$
3. $Mg\frac{a}{R}$
4. $Mg\sqrt{1-\frac{a^2}{R^2}}$

Q. A particle is given an initial speed $u$ inside a smooth spherical shell of radius $R=1\text{m}$ and it is just able to complete the circle. Acceleration of the particle when its velocity is vertical is:

1. $g\sqrt{10}$
2. $\frac{g}{\sqrt{6}}$
3. $g$
4. $\frac{g}{\sqrt{2}}$

Q. An object is tied to a string and rotated in a vertical circle of radius $r$. Constant speed $\left(v\right)$ is maintained along the trajectory. If $\frac{T_{max}}{T_{min}}=2$, then $\frac{v^2}{rg}=$

Q. A particle moves with deceleration along the circle of radius $R$ so that at any moment of time its tangential and normal accelerations are equal in moduli. At the initial moment $t=0$ the speed of the particle equals $v_0$, then the speed of the particle as a function of the distance covered $S$ will be

1. $v=v_0{e}^{-\frac{S}{R}}$
2. $v=v_0{e}^{-\frac{S}{R}}$
3. $v=v_0{e}^{\frac{R}{S}}$
4. $v=v_0{e}^{-\frac{R}{S}}$

Q. A particle moves in a circle of radius $4\text{m}$ at a speed given by $v=4t$, where $v$ is in $\text{m/s}$ and $t$ in seconds. The total acceleration of particle at $t=1\text{s}$ is :

1. $4{\text{m/s}}^2$
2. $8{\text{m/s}}^2$
3. 
   $4\sqrt{2}{\text{m/s}}^2$
4. $4\sqrt{3}{\text{m/s}}^2$

Q.

Consider the situation shown in figure (6-E2). Calculate (a) the acceleration of the 1.0 kg blocks, (b) the tension inthe string ocnecting the 1.0 kg blocks and (c) the tensionin th tring attached to 0.50 kg.

Q. A small block of mass $m$ slides along a smooth frictionless track as shown in the figure. If the block starts from rest at $P$ at height $h$ from the bottom, then which of the following statements are true?

1. For $h=5R$, the resultant force acting on the block at $Q$ is $\sqrt{75}mg$
2. If the force exerted by the block against the track at the top of the loop equals its weight, then $h=3R$
3. For $h=5R$, the resultant force acting on the block at $Q$ is $\sqrt{65}mg$
4. If the block should not fall off at the top of the circular track, it must be released from a minimum height of $h=\frac{5R}{2}$

Q.

Is acceleration possible in uniform motion? Give example.

Q. A pulley of $10cm$ diameter is wrapped with a cord of length $2m$ around its periphery. If pulley is initially at rest and is given an angular acceleration of $0.5rad/s^2$, then total angular displacement of the pulley for the cord to unwind completely and the time taken by the cord to unwind completely are

1. $20rad,4\sqrt{10}s$
2. $40rad,\sqrt{10}s$
3. $40rad,4\sqrt{10}s$
4. $20rad,\sqrt{10}s$

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. The kinetic energy $K$ of a particle moving along a circle of radius $R$ depends on the distance covered as $K=a{s}^2,$ where $a$ is a constant. The force acting on the particle is

1. $2a\frac{s^2}{R}$
2. $2as{(1+\frac{s^2}{R^2})}^{\frac{1}{2}}$
3. $2as$
4. $2a\frac{R^2}{s}$

Q. A particle is moving in a circle of radius $R$ in such a way that at any instant, the normal and tangential components of acceleration are equal. If its speed at $t=0$ is $v_0$, the time taken to complete the first two revolutions is:

1. $\frac{R}{v_0}$
2. $\frac{R}{v_0}{e}^{-4\pi }$
3. $\frac{R}{v_0}(1-e^{-4\pi })$
4. $\frac{R}{v_0}(1+e^{-4\pi })$

Q. A body of mass m hangs at one end of a string of length l, the other end of which is fixed. It is given a horizontal velocity so that the string would just reach where it makes an angle of ${60}^{\circ }$ with the vertical. The tension in the string at mean position is

1. 2 mg
2. mg
3. 3 mg
4. √3mg

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

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

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

Q. A simple pendulum is oscillating in a vertical plane. If resultant acceleration of bob of mass $m$ at a point $A$ is in horizontal direction, find the tangential force at this point in terms of tension $\left(T\right)$ and weight of pendulum $\left(mg\right)$.

1. $T+mg$
2. $\frac{T}{mg}\sqrt{(mg{)}^2+T^2}$
3. $\frac{mg}{T}\sqrt{T^2-(mg{)}^2}$
4. $\frac{mg}{T}\sqrt{(mg{)}^2+T^2}$

Q. The velocity of a particle moving in a curvilinear path varies with time as $\overrightarrow{v}=2t^2\hat{i}+t^3\hat{j}m/s$ where $t$ is in seconds. At the end of $t=1s$, the value of tangential acceleration is

1. $\frac{11}{\sqrt{5}}m/s^2$
2. $\frac{11}{\sqrt{10}}m/s^2$
3. $\sqrt{5}m/s^2$
4. $11\sqrt{5}m/s^2$

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

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

Q. An aircraft executes a vertical turn of radius $R=500\text{m}$ with a constant velocity $v=360\text{km/h}$. The normal reaction on the pilot of mass $m=70\text{kg}$ at the lower, upper and middle points of the loop will respectively be :-

1. $1400\text{N},700\text{N},2100\text{N}$
2. $2100\text{N},700\text{N},1400\text{N}$
3. $2100\text{N},700\text{N},700\sqrt{5}\text{N}$
4. $700\text{N},700\sqrt{5}\text{N},2100\text{N}$

Q. A small particle of mass $0.36\text{g}$ rests on a horizontal turntable at a distance $25\text{cm}$ from the axis of the spindle. The turntable is accelerated at a rate of $\alpha =\frac{1}{3}{\text{rad s}}^{-2}$. The frictional force that the table exerts on the particle $2\text{s}$ after the startup is

1. $30\mu \text{N}$
2. $50\mu \text{N}$
3. $40\mu \text{N}$
4. $60\mu \text{N}$