Deducing Mathematically

Q. The instantaneous angular position of a point on a rotating wheel is given by the equation $\theta \left(t\right)=2t^3-6t^2$. The torque on the wheel becomes zero at

1. $t=0.5\text{s}$
2. $t=0.25\text{s}$
3. $t=2\text{s}$
4. $t=1\text{s}$

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

Q.

Consider two uniform discs of the same thickness and different radii $R_1=R$ and $R_2=\alpha R$ made of the same material. If the ratio of their moments of inertia $I_1$ and $I_2$, respectively, about their axes is $I_1:I_2=1:16$ then the value of $\alpha$ is

1. $\sqrt{2}$

2. $2$

3. $2\sqrt{2}$

4. $4$

Q. A circular disc of mass $M$ and radius $R$ is rotating about its axis with angular speed $\omega .$ If another stationary disc having radius $\frac{R}{2}$ and same mass $M$ is dropped co-axially on to the rotating disc. Gradually, both discs attain constant angular speed ${\omega }_f.$ The energy lost in the process is $p\%$ of the initial energy. The value of $p$ is (integer only)

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

Q. A fan is running at $300\text{rpm}$. It is switched off. It comes to rest by uniformly decreasing its angular speed in $10$ seconds. The total number of revolutions in this period are :-

1. $15$
2. $25$
3. $35$
4. $30$

Q. The angular position of a point on the rim of a rotating wheel is given by $\theta =4t-3t^2+t^3$, where $\theta$ is in radians and $t$ is in seconds. What is the instantaneous angular acceleration at $t=2\text{sec}$?

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

Q.

A point initially at rest moves along $X-axis$. Its acceleration varies with time as $a=(6t+5)m{s}^{-2}$. If it starts from origin, the distance covered in $2s$ is

1. $20m$

2. $18m$

3. $16m$

4. $25m$

Q.

A wheel of mass 10 kg and radius 20 cm is rotating at an angular speed of 100 rev/min when the motor is turned off. Neglecting the friction at the axle, calculate the force that must be applied tangentially to the wheel to bring it to rest in 10 revolutions.

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

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 track is mounted on a large wheel that is free to turn with negligible friction about a vertical axis. Consider a toy train of mass M placed on the track with the system initially at rest, the trains electrical power is turned on. The train reaches speed v with respect to the track. What is the wheels angular speed if its mass is m and its radius is R? (Treat it as a hoop and neglect the mass of the spokes and hub).

1. $\frac{\mathrm{v}}{\left({\displaystyle \frac{\mathrm{m}}{\mathrm{M}}+1}\right)\mathrm{R}}$

2. $\frac{\mathrm{v}}{\left({\displaystyle \frac{\mathrm{m}}{\mathrm{M}}+2}\right)\mathrm{R}}$

3. $\frac{\mathrm{v}}{\left({\displaystyle \frac{\mathrm{M}}{\mathrm{m}}-1}\right)\mathrm{R}}$

4. $\frac{\mathrm{v}}{\left({\displaystyle \frac{\mathrm{M}}{\mathrm{m}}-2}\right)\mathrm{R}}$

Q. A wheel initially at rest, is now rotated with a uniform angular acceleration. The wheel rotates through an angle ${\theta }_1$ in first one second and through an additional angle ${\theta }_2$ in the next one second. The ratio $\frac{{\theta }_2}{{\theta }_1}$ is

1. 4
2. 3
3. 1
4. 2

Q. Two tunnels are dug from one side of the earth's surface to the other side, one along a diameter and the other along a chord, now two partcles are dropped from one end to reach the other end of tunnels. Both the particles oscillate simple harmonically along the tunnels. Let $T_1$ and $T_2$ be the time periods and $V_1$ and $V_2$ be the maximum speeds of the particles in these two tunnels. Then

1. $T_1=T_2$
2. $T_1>T_2$
3. $V_1=V_2$
4. $V_1>V_2$

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. if the angular velocity of a body rotating about an axis is doubled and its moment of inertia is halved, the rotational kinetic energy will change by a factor of:

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

Q. A block hangs from a string wrapped on a disc of radius $20\text{cm}$ free to rotate about its axis which is fixed in a horizontal position. If the angular speed of the disc is $10\text{rad/s}$ at some instant, with what speed is the block going down at that instant? (Answer in $\text{m/s}$)

Q.

Two blocks of masses 10 kg and 20 kg are placed on the X-axis.The first mass is moved on the axis by a distance of 2 cm.By what distance should the second mass be moved to keep the position of the centre of mass unchanged?

Q.

A particle starting from rest travels a distance x in the first 2 seconds and a distance y in the next two seconds with constant acceleration, then.

1. y = x

2. y = 2x

3. y = 3x.

4. y = 4x

Q.

A shell acquires the initial velocity $v$ = 320 m/s, having made $n$ = 2.0 turns inside the barrel whose length is equal to $l$ = 2.0 m. Assuming that the shell moves inside the barrel with a uniform acceleration, find the angular velocity of its axial rotation at the moment when the shell escapes the barrel.

1. 

2. 

3. 

4. Data insufficient

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. A rough disc of mass $m$ rotates freely with an angular velocity $\omega$. If another rough disc of mass $\frac{m}{2}$ and of same radius but spinning in opposite sense with angular speed $\omega$ is kept on the first disc. Then:

1. The final angular speed of the disc is $\frac{\omega }{3}$.
2. The net work done by friction is zero.
3. The fiction does a positive work on the lighter disc.
4. The net wok done by friction is $\frac{-m{r}^2{\omega }^2}{3}$.

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. 

2. 

3. 

4. 

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

Q. As a part of a maintenance inspection the compressor of a jet engine is made to spin according to the graph as shown. The number of revolutions made by the compressor during the test is

1. 9000
2. 11250
3. 16570
4. 12750

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

Q.

Diameter of a wheel is 6 m. The linear speed of a point on its rim is 33 $m{s}^{-1}$. Then its angular velocity in rpm.

1. 100

2. 102

3. 104

4. 105

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. $4\text{s}$