Q. No.
A rope is wrapped around a hollow cylinder with a mass of \(3\) kg and a radius of \(40\) cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of \(30\) N?
1. \(0.25 ~\text{rad/s}^2 \)
2. \(25 ~\text{rad/s}^2 \)
3. \(5 ~\text{m/s}^2 \)
4. \(25 ~\text{m/s}^2 \)
1. \(0.25 ~\text{rad/s}^2 \)
2. \(25 ~\text{rad/s}^2 \)
3. \(5 ~\text{m/s}^2 \)
4. \(25 ~\text{m/s}^2 \)
Two discs of the same moment of inertia are rotating about their regular axis passing through centre and perpendicular to the plane of the disc with angular velocities \(\omega_1\) and \(\omega_2\). They are brought into contact face to face with their axis of rotation coinciding. The expression for loss of energy during this process is:
1. \(\frac{1}{4}I\left(\omega_1-\omega_2\right)^2\)
2. \(I\left(\omega_1-\omega_2\right)^2\)
3. \(\frac{1}{8}I\left(\omega_1-\omega_2\right)^2\)
4. \(\frac{1}{2}I\left(\omega_1-\omega_2\right)^2\)
Which of the following statements are correct?
| (a) | centre of mass of a body always coincides with the centre of gravity of the body . |
| (b) | centre of gravity of a body is the point about which the total gravitational torque on the body is zero. |
| (c) | a couple on a body produce both translational and rotation motion in a body. |
| (d) | mechanical advantage greater than one means that small effort can be used to lift a large load. |
| 1. | (a) and (b) | 2. | (b) and (c) |
| 3. | (c) and (d) | 4. | (b) and (d) |
The moment of the force, \(\overset{\rightarrow}{F} = 4 \hat{i} + 5 \hat{j} - 6 \hat{k}\) at point (\(2,\) \(0,\) \(-3\)) about the point (\(2,\) \(-2,\) \(-2\)) is given by:
1. \(- 8 \hat{i} - 4 \hat{j} - 7 \hat{k}\)
2. \(- 4 \hat{i} - \hat{j} - 8 \hat{k}\)
3. \(- 7 \hat{i} - 8 \hat{j} - 4 \hat{k}\)
4. \(- 7 \hat{i} - 4 \hat{j} - 8 \hat{k}\)
A solid sphere is rotating freely about its axis of symmetry in free space. The radius of the sphere is increased keeping its mass the same. Which of the following physical quantities would remain constant for the sphere?
| 1. | angular velocity |
| 2. | moment of inertia |
| 3. | rotational kinetic energy |
| 4. | angular momentum |
1. \(\frac{13}{32}MR^2\)
2. \(\frac{11}{32}MR^2\)
3. \(\frac{9}{32}MR^2\)
4. \(\frac{15}{32}MR^2\)
A uniform circular disc of radius \(50\) cm at rest is free to turn about an axis that is perpendicular to its plane and passes through its centre. It is subjected to a torque that produces a constant angular acceleration of \(2.0\) rad s–2. Its net acceleration in ms–2 at the end of \(2.0\) s is approximately:
1. \(7\)
2. \(6\)
3. \(3\)
4. \(8\)
Point masses \(m_1\) and \(m_2\), are placed at the opposite ends of a rigid rod of length \(L\) and negligible mass. The rod is set into rotation about an axis perpendicular to it. The position of point \(P\) on this rod through which the axis should pass so that the ork required to set the rod rotating with angular velocity is minimum is given by:
| 1. | \(x = \frac{m_1L}{m_1+m_2}\) | 2. | \(x= \frac{m_1}{m_2}L\) |
| 3. | \(x= \frac{m_2}{m_1}L\) | 4. | \(x = \frac{m_2L}{m_1+m_2}\) |
A rod of weight \(w\) is supported by two parallel knife edges, A and B, and is in equilibrium in a horizontal position. The knives are at a distance \(d\) from each other. The centre of mass of the rod is at a distance \(x \) from A. The normal reaction on A is:
| 1. | \(wx \over d\) | 2. | \(wd \over x\) |
| 3. | \(w(d-x) \over x\) | 4. | \(w(d-x) \over d\) |
A mass \(m\) moves in a circle on a smooth horizontal plane with velocity \(v_0\) at a radius \(R_0\). The mass is attached to a string that passes through a smooth hole in the plane, as shown.
The tension in the string is increased gradually and finally, \(m\) moves in a circle of radius \(\frac{R_0}{2}\). The final value of the kinetic energy is:
1. \( m v_0^2 \)
2. \( \frac{1}{4} m v_0^2 \)
3. \( 2 m v_0^2 \)
4. \( \frac{1}{2} m v_0^2\)
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