Q. No.A solid sphere is in rolling motion. In rolling motion, a body possesses translational kinetic energy (Kt) as well as rotational kinetic energy (Kr) simultaneously. The ratio Kt : (Kt + Kr) for the sphere will be:
1. 7:10
2. 5:7
3. 10:7
4. 2:5
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 disc and a solid sphere of the same radius but different masses roll off on two inclined planes of the same altitude and length. Which one of the two objects gets to the bottom of the plane first?
1. Sphere
2. Both reach at the same time
3. Depends on their masses
4. Disc
A uniform circular disc of radius \(50~\text{cm}\) 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 \(2.0~\text{rad/s}^2.\) Its net acceleration in \(\text{m/s}^2\) at the end of \(2.0~\text s\) 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 a point \(P\) on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity \(\omega_0\) 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. | \(x = \frac{m_2L}{m_1+m_2}\) |
| 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 in the figure.
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. \( \frac{1}{2} m v_0^2\)
Three identical spherical shells, each of mass \(m\) and radius \(r\) are placed as shown in the figure. Consider an axis \(XX',\) which is touching two shells and passing through the diameter of the third shell. The moment of inertia of the system consisting of these three spherical shells about the \(XX'\) axis is:
| 1. | \(\frac{11}{5}mr^2\) | 2. | \(3mr^2\) |
| 3. | \(\frac{16}{5}mr^2\) | 4. | \(4mr^2\) |
A force \(\vec{F}=\alpha \hat{i}+3 \hat{j}+6 \hat{k}\) is acting at a point \(\vec{r}=2 \hat{i}-6 \hat{j}-12 \hat{k}\). The value of \(\alpha\) for which angular momentum about the origin is conserved is:
1. \(-1\)
2. \(2\)
3. zero
4. \(1\)
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