Introduction to Radial & Tangential Acceleration

Q.

The angular velocity of the second hand of a clock is

Q. A stone tied to the end of a string of length $1\text{m}$ is whirled in a horizontal circle with a constant speed. If stone makes $22$ revolution in $44$ $\text{second}$, what is the magnitude and direction of the acceleration of stone?

1. Acceleration will be ${\pi }^2{\text{m/s}}^2$ radially outwards.
2. Acceleration will be ${\pi }^2{\text{m/s}}^2$ along the tangent to the circle.
3. Acceleration will be $\frac{\pi }{4}{\text{m/s}}^2$ radially inwards.
4. Acceleration will be ${\pi }^2{\text{m/s}}^2$ radially inwards.

Q.

A body is moving in a circular path with a constant speed. It has

1. A constant velocity

2. A constant acceleration

3. An acceleration of constant magnitude

4. An acceleration which varies with time

Q. What is uniform circular motion? Which physical quantities remain constant for a uniform circular motion?

Q. Wheel A of radius $r_A=10\text{cm}$ is coupled by belt B to a wheel C of radius $r_c=25\text{cm}$. The angular speed of the wheel A is increased from rest at a constant rate of $1.6{\text{rad/s}}^2.$ Find the time needed for wheel C to reach an angular speed of $100\text{rev/min},$ assuming that the belt does not slip.

1. 9 seconds
2. 16 seconds
3. 4 seconds
4. 25 seconds

Q.

In the Bohr model of hydrogen atom, the electron is treated as a particle going in a circle with the centre at the proton. The proton itself is assumed to be fixed in an inertial frame. The centripetal force is provided by the Coloumb attraction. In the ground state, the electron goes round the proton in a circle of radius $5.3\times {10}^{-11}m.$ Find the speed of the electron in the ground state. Mass of the electron = $9.1\times {10}^{-31}$ kg and charge of the electron = 1.6 \times 10^{-19}C.

Q.

For a particle in uniform circular motion the acceleration $\overrightarrow{a}$ at a point $P(R,\theta )$on the circle of radius R is (here $\theta$ is measured from the x-axis)

1. $-\frac{v^2}{R}cos\theta \hat{i}+\frac{v^2}{R}sin\theta \hat{j}$

2. $-\frac{v^2}{R}sin\theta \hat{i}+\frac{v^2}{R}cos\theta \hat{j}$

3. $-\frac{v^2}{R}cos\theta \hat{i}+\frac{v^2}{R}sin\theta \hat{j}$

4. $\frac{v^2}{R}+\frac{v^2}{R}\hat{j}$

Q. A pendulum consists of a wooden bob of mass $m$ and length $l$. A bullet of mass $m_1$ is fired towards the pendulum with a speed $V_1$ and it emerges from the bob with speed $\frac{V_1}{3}.$ The bob just completes motion along a vertical circle. Then $v_1$ is

1. $\frac{m}{m_1}\sqrt{5gl}$
   
2. $\frac{3m}{2m_1}\sqrt{5gl}$
   
3. $\frac{2m}{3m_1}\sqrt{5gl}$
   
4. $\frac{m_1}{m}\sqrt{5gl}$
   

Q. Justify the statement uniform circular motion is accelerated motion.

Q. A $10\text{kg}$ ball attached at the end of a rigid massless rod of length $(L=1\text{m})$ rotates at constant speed in a horizontal circle of radius $0.5\text{m}$ with a period of $1.57\text{s}$, as shown in the figure. The force exerted by the rod on the ball is (Take $g=10{\text{ms}}^{-2}$ & $\pi =3.14$)

1. $20\sqrt{41}\text{N}$
2. $10\sqrt{41}\text{N}$
3. $20\sqrt{20}\text{N}$
4. $10\sqrt{20}\text{N}$

Q. Tangential acceleration of a particle moving in a circle of radius $1\text{m}$ varies with time t as shown in figure (initial velocity of particle is zero). Time after which total acceleration of the particle makes an angle of ${30}^{\circ }$ with radial acceleration is:

1. $4\text{sec}$
2. $\sqrt{2}\text{sec}$
3. $2^{2/3}\text{sec}$
4. $\frac{4}{3}\text{sec}$

Q. A particle of mass 2kg is moving along a circular path of radius 1m .if its angular speed is 2πrad/s , the centripetal force will be

Q. A small block is placed on a turn table. The coefficient of static friction between the block and the table is ${\mu }_s=0.5$. If the turn table is rotating with constant angular acceleration $\alpha =3{\text{rad/s}}^2$, what will be the angular speed at which the block begins to slip on the table ?
Given $r=1\text{m},g=10{\text{m/s}}^2.$

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

Q. The tangential acceleration of a particle in a circular motion of radius $2\text{m}$ is $a_t=\alpha t{\text{m/s}}^2$ (where $\alpha$ is a constant) . Initially, the particle is at rest. Net acceleration vector of the particle makes ${45}^{\circ }$ with the radial acceleration after 2 sec. The value of constant $\alpha$ is:

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

Q. A simple pendulum is oscillating without damping. When the displacement of the bob is less than maximum, its acceleration vector $\overrightarrow{a}$ is correctly shown in which of the following options ?

1. 
2. 
3. 
4. 

Q.

A vertical spring of force constant $100\mathrm{N}/\mathrm{m}$ is attached to a hanging mass of $10\mathrm{kg}$. Now an external force is applied on mass so that the spring is stretched (maximum) by additional $2\mathrm{m}$. The work by the force F is ($\mathrm{g}=10\mathrm{m}/{\mathrm{s}}^2$)

Q. A body is revolving with a constant speed along a circle. If its direction of motion is reversed but the speed remains the same then?
(a) the centripetal force will not suffer any change in magnitude
(b) the centripetal force will have its direction reversed
(c) the centripetal force will not suffer any change in direction
(d) the centripetal force is doubled.

1. B, C
2. A, B
3. C, D
4. A, C

Q. A particle is moving in a circular path. The acceleration and momentum of the particle at a certain moment are $\overrightarrow{a}=(4\hat{i}+3\hat{j})m/s^2$ and $\overrightarrow{p}=(8\hat{i}-6\hat{j})kgm/s$. The motion of the particle is

1. Uniform circular motion
2. Accelerated circular motion
3. Decelerated circular motion
4. We cannot say anything with $\overrightarrow{a}$ and $\overrightarrow{p}$ only.

Q.

The motion of an object moving in a circle with constant speed is an example of accelerated motion.

1. True
2. False

Q. A block of mass $m$ moves on a horizontal circular path against the wall of a cylindrical room of radius $R$. The floor of the room on which the block moves is smooth, but the friction coefficient between the wall and the block is $\mu$. Take ${\mu }_s={\mu }_k=\mu$. The block is given an initial speed $v_{\circ }$. Obtain the speed of the block '$v$' after one revolution.

1. $v_{\circ }{e}^{-\pi \mu }$
2. $v_{\circ }{e}^{\frac{-\pi \mu }{2}}$
3. $v_{\circ }{e}^{\frac{-3\pi \mu }{2}}$
4. $v_{\circ }{e}^{-2\pi \mu }$

Q.

Which of the following is not a unit of angular displacement?

1. Radians

2. Degrees

3. Metres

4. Revolutions

Q.

Solve the previous problem if the paperweight is inverted at its place so that the spherical surface touches the paper.

Q. A stone tied to one end of string of $1m$ long is whirled in a horizontal circle with a constant speed. If the stone makes $10$ revolutions in $20s$, the magnitude of the acceleration of the stone is close to

1. $98.6\frac{cm}{s^2}$
2. $490\frac{cm}{s^2}$
3. $987\frac{cm}{s^2}$
4. $996\frac{cm}{s^2}$

Q. A particle moves in a circle of radius $2\text{cm}$ at a speed given by $v=4t$, where $v$ is in $\text{cm/s}$ and $t$ is in seconds. Then

1. tangential acceleration at $t=1\text{s}$ is $4{\text{cm/s}}^2$
2. total acceleration at $t=1\text{s}$ is $4\sqrt{2}{\text{cm/s}}^2$
3. total acceleration at $t=1\text{s}$ is $4\sqrt{5}{\text{cm/s}}^2$
4. tangential acceleration at $t=1\text{s}$ is $5{\text{cm/s}}^2$

Q. The angular velocity of a particle moving in a circle of radius $50cm$ is increased in $5$ min from $100$ revolutions per minute to $400$ revolutions per minute. Find the tangential acceleration of the particle.

1. $60m/s^2$
2. $\frac{\pi }{30}m/s^2$
3. $\frac{\pi }{15}m/s^2$
4. $\frac{\pi }{60}m/s^2$

Q.

A body goes round with a constant speed in a circular orbit. Is the motion uniform or accelerated?

Q.

A body moves around the sun with constant speed in a circular orbit. Is the motion of the body uniform or accelerated?

Q. The length of the seconds hand in a wall clock is $10\text{cm}$. The magnitude of velocity and acceleration of its tip are

1. ${10}^{-2}\text{m/s}$ and ${10}^{-3}{\text{m/s}}^2$
2. ${10}^{-2}\text{m/s}$ and ${10}^{-4}{\text{m/s}}^2$
3. ${10}^{-1}\text{m/s}$ and ${10}^{-2}{\text{m/s}}^2$
4. ${10}^{-2}\text{m/s}$ and ${10}^{-2}{\text{m/s}}^2$

Q. A small block slides down from the top of a inverted hemisphere of radius $r$. It is assumed that there is no friction between the block and the hemisphere. At what height $h$ from ground, will the block lose contact with the surface of sphere ?

1. $\frac{r}{3}$
2. $\frac{2r}{3}$
3. $\frac{r}{2}$
4. $\frac{r}{4}$

Q. The body is moving in a circular path, the component of velocity in the radial direction is

1. $0$
2. speed of the body
3. $\frac{v}{R}$
4. none of the above