Radial & Tangential Acceleration for Non Uniform Circular Motion

Q. In a beyblade game, the beyblade has an initial angular speed of $180\text{rpm}$. It slows down and eventually comes to rest in a time of 5 minutes. The average angular acceleration of the beyblade is

1. $\frac{\pi }{30}{\text{rad/s}}^2$
2. $\frac{\pi }{40}{\text{rad/s}}^2$
3. $\frac{\pi }{50}{\text{rad/s}}^2$
4. $\frac{\pi }{70}{\text{rad/s}}^2$

Q.

Force of $50N$ acting on a body at an angle $\theta$ with horizontal. If $150J$ work is done by displacing it $3m$, then $\theta$is

1. $60°$

2. $30°$

3. $0°$

4. $45°$

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

Q. A metallic bob of mass ‘$m$’ attached to a string, is raised through a height of $50\text{cm}$ and released. At what distance along the vertical from the point of suspension should a nail be placed so that the bob just completes a circle with the nail as the centre?

1. $0.2\text{m}$
2. $2\text{m}$
3. $1\text{m}$
4. $1.8\text{m}$

Q. A small stone of man $0.2\text{kg}$ tied to a massless inextensible string is rotated in a vertical circle of radius of $2\text{m}$. If the particle is just able to complete a vertical circle, what is the speed at the highest point of the circular path? Also, calculate speed if the mass of stone is increased by $50$% (Take $g=10m/s^2)$

1. $4.47\text{m/s},6.70\text{m/s}$
2. $4.47\text{m/s},10.05\text{m/s}$
3. $10.05\text{m/s},4.47\text{m/s}$
4. $4.47\text{m/s},4.47\text{m/s}$

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:
(Here, $\mu$ in the options represents micro)

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

Q. A particle is moving in a circle of radius $R$ in such a way that at any instant, the radial 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 particle initially at rest is moving along a circle of radius 3m with constant angular acceleration of $2\mathrm{rad}/{\mathrm{s}}^2$. Determine its linear velocity and angular velocity at $t=5s$. Also determine its radial, tangential and total acceleration at $t=5s$.

Q. A small block of mass $m$ slides along a smooth frictionless track as shown in the figure.
(i) If it starts from rest at $P$, what is the resultant force acting on it at $Q$?
(ii) At what height above the bottom of the loop should the block be released so that the force it exerts against the track at the top of the loop equals its weight?

1. $\sqrt{75}\text{mg},3R$
2. $\sqrt{65}\text{mg},2R$
3. $\sqrt{75}\text{mg},2R$
4. $\sqrt{65}\text{mg},3R$

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

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. 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 ball of mass $m$ is moving on a circular path with angular velocity $\omega$ and tangential acceleration $a_t$ as shown in the figure. What will be the direction of net force acting on the body at point $C$?

1. Along CF
2. Along CA
3. Along CG
4. Along CB

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. A ball of mass $m$ is moving on a circular path with angular velocity $\omega$ and tangential acceleration $a_t$ as shown in the figure. What will be the direction of net force acting on the body at point $C$?

1. Along CF
2. Along CA
3. Along CG
4. Along CB

Q. A particle is moving on a circular path of radius $3cm$ with a time varying speed of $v=3tcm/s$, where $t$ is in seconds. The magnitude of tangential acceleration and the total acceleration of the particle at $t=1s$ is

1. $3cm/s^2,3cm/s^2$
2. $3\sqrt{2}cm/s^2,3\sqrt{2}cm/s^2$
3. $\frac{3}{2}cm/s^2,3cm/s^2$
4. $3cm/s^2,3\sqrt{2}cm/s^2$

Q. When a ceiling fan is switched off its angular velocity reduces to 50% while it makes 36 rotations. How many more rotations will it make before coming to rest (Assume uniform angular retardation)

1. 18
2. 12
3. 36
4. 48

Q. A particle is suspended from a fixed point by a string of length $5\text{m}$. It is projected from equilibrium position with such a velocity that the string slackens after the particle has reached a height $8\text{m}$ above the lowest point. Choose the correct option(s)
[Take $g=10{\text{m/s}}^2$]

1. Velocity of particle just before the string slackens is $\sqrt{30}\text{m/s}$
2. Velocity of particle just before the string slackens is $\sqrt{20}\text{m/s}$
3. Particle can rise further to a vertical height of $1.96\text{m}$
4. Particle can rise further to a vertical height of $0.96\text{m}$

Q. A small ring is attached to the vertical ring as shown in the figure. The small ring is given velocity $v_B=\sqrt{4gl}$ by a sharp hit, where $l$ is the radius of the vertical ring. Find the normal force acting on the small ring at the topmost point, if the mass of the small ring is $m$.

1. $2mg$
2. $2Mg$
3. $mg$
4. $Mg$

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

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. Figure shows a smooth vertical circular track of radius $R$. A block slides along the surface $AB$ when it is given a velocity equal to $\sqrt{6gR}$ at point $A$. The ratio of the force exerted by the track on the block at point $A$ to that at point $B$ is

1. $0.25$
2. $0.35$
3. $0.45$
4. $0.55$

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 small block of mass $m$ slides along a smooth frictionless track as shown in the figure.
(i) If it starts from rest at $P$, what is the resultant force acting on it at $Q$?
(ii) At what height above the bottom of the loop should the block be released so that the force it exerts against the track at the top of the loop equals its weight?

1. $\sqrt{75}\text{mg},3R$
2. $\sqrt{65}\text{mg},2R$
3. $\sqrt{75}\text{mg},2R$
4. $\sqrt{65}\text{mg},3R$

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. The bob of the pendulum shown in the figure is projected with a velocity $v=\sqrt{gl}$. Which of the following statement is correct?

1. The pendulum completes the circular path and returns back to point $A$
2. The pendulum oscillates between $B$and $D$, always reaching to the points $B$ and $D$
3. The pendulum oscillates between $B$ and $D$, but never reaches $B$ or $D$
4. The pendulum crosses point $B$ and travels in parabolic path onwards.

Q. In the graph shown below, variation of angular speed with respect to time is given. The number of revolutions completed during the entire motion is

1. 3600
2. 1200
3. 2400
4. 1600

Q. The bob of a pendulum at rest is given a sharp hit to impart a horizontal velocity $u=\sqrt{10gl}$, where $l$ is the length of the pendulum. Find the velocity of the bob when the string is horizontal.

1. $\sqrt{2gl}$
2. $\sqrt{4gl}$
3. $\sqrt{6gl}$
4. $\sqrt{8gl}$

Q. The bob of a pendulum at rest is given a sharp hit to impart a horizontal velocity $u=\sqrt{10gl}$, where $l$ is the length of the pendulum. Then, the tension in the string when it is horizontal is $xmg$ where $x=$