Deducing Mathematically

Q.

A wheel rotating with an angular speed of $600rpm$ is given a constant acceleration of $1800rp{m}^2$ for $10sec$. The number of revolutions revolved by the wheel is.

1. $125$

2. $100$

3. $75$

4. $50$

Q. A wheel rotates with a constant angular acceleration of $3.6{\text{rad/s}}^2$. If angular velocity of the wheel is $4.0\text{rad/s}$ at time $t_0=0$, what angle does the wheel rotate in $1\text{s}$? What will be its angular velocity at $t=1\text{s}$?

1. $\theta =5.8\text{rad},\omega =0.4\text{rad/s}$
2. $\theta =2.2\text{rad},\omega =0.4\text{rad/s}$
3. $\theta =5.8\text{rad},\omega =7.6\text{rad/s}$
4. $\theta =2.2\text{rad},\omega =7.6\text{rad/s}$

Q. A pulley wheel of diameter $10\text{cm}$ has a $1\text{m}$ long cord wrapped around its periphery. The wheel is initially at rest. Assuming angular acceleration of wheel to be uniform and the time taken for the cord to unwind completely to be $18\text{s}$, find its tangential acceleration.

1. $6.17{\text{mm/s}}^2$
2. $2{\text{mm/s}}^2$
3. $0.1{\text{mm/s}}^2$
4. $0.01{\text{mm/s}}^2$

Q. A solid body rotates about a stationary axis with an angular deceleration $\beta \propto \sqrt{\omega }$ where $\omega$ is its angular velocity. If at the initial moment of time, its angular velocity was equal to ${\omega }_0$, then the mean angular velocity of the body averaged over the whole time of rotation till it comes to rest is,

1. $\frac{{\omega }_0}{\sqrt{3}}$
2. $\frac{{\omega }_0}{3}$
3. $\frac{{\omega }_0}{2}$
4. $\frac{{\omega }_0}{\sqrt{2}}$

Q. A body is moving down into a well through a rope passing over a fixed pulley of radius $20\text{cm}$. Assume that there is no slipping between the rope and pulley. Calculate the angular velocity and angular acceleration of the pulley at an instant when the body is going down at a speed of $40\text{cm/s}$ and has an acceleration of $4{\text{m/s}}^2$?

1. $\omega =2\text{rad/s},\alpha =20{\text{rad/s}}^2$
2. $\omega =2\text{rad/s},\alpha =40{\text{rad/s}}^2$
3. $\omega =1\text{rad/s},\alpha =20{\text{rad/s}}^2$
4. $\omega =1\text{rad/s},\alpha =40{\text{rad/s}}^2$

Q.

A wheel rotates with a constant angular acceleration of 2.0 $\frac{rad}{s^2}$. If the wheel starts from rest, how many revolutions will it make in the first 10 seconds?

1. 32

2. 8

3. 16

4. 4

Q. Figure below shows a small wheel fixed coaxially on a bigger one of double the radius. The system rotates about the common axis. The strings supporting A and B do not slip on the wheels. If 'x' and 'y' be the distances travelled by A and B in the same time interval, then

1. x = 2y
2. x = y
3. y = 2x
4. None of these

Q. A wheel is making revolutions about its axis with uniform angular acceleration. Starting from rest, it reaches 100 rev/sec in 4 secs. Its angular acceleration is.

1. $25\frac{rev}{s^2}$
2. $500\frac{rev}{s^2}$
3. $100\frac{rev}{s^2}$
4. $625\frac{rev}{s^2}$

Q. In a continuous printing process, paper roll is wrapped on a cylindrical core which is free to rotate about a fixed horizontal axis. The paper is drawn into the process at a constant speed $v$, as shown in the figure. If $r$ is the radius of the paper on the roll at any given time and $b$ is the thickness of the paper, then the angular acceleration of the roll at this instant is

1. $\frac{v^{2}b}{2{\pi }^2{r}^3}$
2. Zero because $v$ is constant
3. $\frac{v^{2}b}{2\pi r^3}$
4. Cannot be calculated from the given information

Q.

Two particle A and B move on a circle. Initially Particle A and B are diagonally opposite to each other. Particle A moves with angular velocity $\pi \frac{rad}{sec}$, angular acceleration $\frac{\pi }{2}\frac{rad}{se{c}^2}$ and particle B moves with constant angular velocity $2\pi \frac{rad}{sec}$. Find the time after which both the particle A and B will collide.

1. $t=2sec$

2. $t=1sec$

3. $t=4sec$

4. $t=\frac{4\pm \sqrt{14}}{2}sec$

Q.

A fan is rotating with angular velocity 100 $\frac{rev}{sec}$. Then it is switched off. It takes 5 minutes to stop. (i) Find the total number of revolution made before it stops. (Assume uniform angular retardation) (ii)) Find the value of angular retardation (iii) Find the average angular velocity during this interval.

1. No. of revolutions = 15000

2. $\alpha$ = $\frac{-2}{3}\pi \frac{rad}{s^2}$

3. ${\omega }_{av}$ = $100\pi$ $\frac{rad}{s}$

4. ${\omega }_{av}$ = $10\pi$ $\frac{rad}{s}$

Q. A wheel starting from rest is uniformly accelerated at $2{\text{rad/s}}^2$ for 20 seconds. It rotates uniformly for the next 20 seconds and is finally brought to rest in the next 20 seconds. Total angular displacement of the wheel in radians is

1. $400\text{rad}$
2. $800\text{rad}$
3. $1600\text{rad}$
4. $1200\text{rad}$

Q. A disc is rotating with an angular velocity ${\omega }_o$. A constant retarding torque is applied on it to stop the disc. The angular velocity becomes $\frac{{\omega }_o}{2}$ after $n$ rotations. How many more rotations will it make before coming to rest?

1. $\frac{n}{4}$
2. $\frac{n}{5}$
3. $\frac{n}{2}$
4. $\frac{n}{3}$

Q. A wheel of radius $30cm$ has $8$ spokes in it. It is mounted on a fixed axle and spins at $5rev/s$. If an arrow of length $24cm$ is to be shot parallel to the axis of the wheel without hitting rim or the spokes of the wheel. What minimum speed the arrow must have? (Ignore the effect of gravity)

1. $9.6m/s$
2. $4.8m/s$
3. $2.4m/s$
4. $19.2m/s$

Q. The angular velocity of a point on the rim of a rotating wheel is given by $\omega =(4-6t+3t^2)\text{rad/s}$. What is the average angular acceleration for the time interval $t=2\text{s}$ to $t=4\text{s}$ ?

1. $12{\text{rad/s}}^2$
2. $6{\text{rad/s}}^2$
3. $24{\text{rad/s}}^2$
4. $3{\text{rad/s}}^2$

Q. The angular position of a rotating disc is given as $\theta =6t^2+5t-4$ where $\theta$ is in radians, $t$ is in seconds. What is the average angular velocity for the first time interval from $t=0\text{(s)}$ to $t=4\text{(s)}$?

1. $29\text{rad/s}$
2. $18\text{rad/s}$
3. $4\text{rad/s}$
4. $27\text{rad/s}$

Q.

A block is moving the rim of a pulley which is fixed in a horizontal position. If the angular speed of the disc is 10 rad/s and its diameter is 20 cm, then speed of the block at same instant is

1. 2 m/s
2. 0.2 m/s
3. 20 m/s
4. 0.02 m/s

Q. Starting from rest, a fan takes four seconds to attain the maximum speed of $800\text{rpm}$ (revolutions per minute). Assuming uniform acceleration, calculate the time taken by the fan in attaining half the maximum speed.

1. $1\text{second}$
2. $4\text{seconds}$
3. $1.5\text{seconds}$
4. $2\text{seconds}$

Q.

The wheel of a motor, accelerated uniformly from rest, rotates through 2.5 radian during the first second. Find the angle rotated during the next second.

1. 10 rad

2. 7.5 rad

3. 5 rad

4. 12.5 rad

Q. The moment of inertia of a body about a given axis is $1.2\text{kg}-{\text{m}}^2$. Initially, the body is at rest. In order to produce a rotational kinetic energy of $1500\text{J}$, an angular acceleration of $25{\text{rad/s}}^2$ must be applied about that axis for a duration of

1. $4\text{s}$
2. $2\text{s}$
3. $8\text{s}$
4. $10\text{s}$

Q. Two particles $A$ and $B$ are moving in same direction on a circular path. Initially particle $A$ and $B$ are diametrically opposite to each other. Particle $A$ moves with angular velocity $\pi rad/s$ and angular acceleration $\frac{\pi }{2}rad/s^2$ and particle $B$ moves with constant angular velocity $2\pi rad/s$. Find the time after which both particles $A$ and $B$ collides.

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

Q. A clock is in x- y plane. Find angular velocity of second hand of a clock (in rad/s).

1. $\frac{\pi }{30}\left(\hat{i}\right)$
2. $\frac{\pi }{30}(-\hat{j})$
3. $\frac{\pi }{60}\left(\hat{k}\right)$
4. $\frac{\pi }{30}(-\hat{k})$

Q. A body is moving down into a well through a rope passing over a fixed pulley of radius $20\text{cm}$. Assume that there is no slipping between the rope and pulley. Calculate the angular velocity and angular acceleration of the pulley at an instant when the body is going down at a speed of $40\text{cm/s}$ and has an acceleration of $4{\text{m/s}}^2$?

1. $\omega =2\text{rad/s},\alpha =20{\text{rad/s}}^2$
2. $\omega =2\text{rad/s},\alpha =40{\text{rad/s}}^2$
3. $\omega =1\text{rad/s},\alpha =20{\text{rad/s}}^2$
4. $\omega =1\text{rad/s},\alpha =40{\text{rad/s}}^2$