Q. No.A bob of mass \(m\) attached to an inextensible string of length \(l\) is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed \(\omega\) rad/s about the vertical. About the point of suspension:
| 1. | angular momentum changes in magnitude but not in direction. |
| 2. | angular momentum changes in direction but not in magnitude. |
| 3. | angular momentum changes in both direction and magnitude. |
| 4. | angular momentum is conserved. |
1. \(14.4~\text{kg m}^2\text{s}^{-1}\)
2. \(8.64~\text{kg m}^2\text{s}^{-1}\)
3. \(20.16~\text{kg m}^2\text{s}^{-1}\)
4. \(11.52~\text{kg m}^2\text{s}^{-1}\)
A particle of mass \(m\) is moving along side of a square of side '\(a\)', with a uniform speed \(v\) in the x-y plane as shown in the figure:
Which of the following statements is false for the angular momentum \(\vec L\) about the origin?
| 1. | \(\vec{L}=-\frac{m vR}{\sqrt{2}} \hat{k}\) when the particle is moving from \(A\) to \(B\). |
| 2. | \(\vec{L}=m v\left[\frac{R}{\sqrt{2}}+a\right] \hat{k} \) when the particle is moving from \(C\) to \(D\). |
| 3. | \(\vec{L}=m v\left[\frac{R}{\sqrt{2}}+a\right] \hat{k}\) when the particle is moving from \(B\) to \(C\). |
| 4. | \(\vec{L}=\frac{m vR}{\sqrt{2}} \hat{k}\) when the particle is moving from \(D\) to \(A\). |
A thin smooth rod of length \(L\) and mass \(M\) is rotating freely with angular speed \(\omega_0\)
1. \(
\frac{M \omega_0}{M+3 m}
\)
2. \(\frac{M \omega_0}{M+2 m}
\)
3. \(\frac{M \omega_0}{M+m}
\)
4. \(\frac{M \omega_0}{M+6 m}\)
Two coaxial discs, having moments of inertia \(I_1\) and \(\frac{I_1}{2}\) are rotating with respective angular velocities \(\omega_1\) and \(\frac{\omega_1}{2}\), about their common axis. They are brought in contact with each other and there after they rotate with a common angular velocity. If \(E_f\) and \(E_i\) are the final and initial total energies, then (\(E_f-E_i\)) is:
1. \( \frac{I_1 \omega_1^2}{6} \)
2. \( \frac{3}{8} I_1 \omega_1^2 \)
3. \( \frac{I_1 \omega_1^2}{12} \)
4. \( \frac{I_1 \omega_1^2}{24}\)
The time dependence of the position of a particle with mass \(m=2~\text{kg}\) is given by:
\(\vec{r}(t)=(2 t \hat{i}-3 t^2 \hat{j})~\text{m}.\)
Its angular momentum, with respect to the origin, at time \(t=2~\text{s}\) is:
1. \( 36 ~\hat{k} \) kg-m2/s
2. \( -48~\hat{k} \) kg-m2/s
3. \( -34(\hat{k}-\hat{i}) \) kg-m2/s
4. \( 48(\hat{i}+ \hat{j})\) kg-m2/s
Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centres. The moment of inertia and angular velocity of the first disc are \(0.1\) kg-m2 and \(10\) rad s–1 respectively while those for the second one are \(0.2\) kg-m2 and \(5\) rad s–1 respectively. At some instant, they get stuck together and start rotating as a single system about their common axis with some angular speed. The Kinetic energy of the combined system is:
1. \(\frac{10}{3}~\text{J}\)
2. \(\frac{2}{3}~\text{J}\)
3. \(\frac{5}{3}~\text{J}\)
4. \(\frac{20}{3}~\text{J}\)
A person with a mass of \(80~\text{kg}\) is standing on the rim of a circular platform with a mass of \(200~\text{kg}\) and rotating about its axis at a speed of \(5\) revolutions per minute (rpm). As the person moves toward the centre of the platform, what will be the platform's new rotational speed (in rpm) once the person reaches its centre?
1. \(3\)
2. \(6\)
3. \(9\)
4. \(12\)
A circular disc of mass \(M\) and radius \(R\) is rotating about its axis with angular speed \(\omega_1\). 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 \(\omega_2\). The energy lost in the process is \(p\%\) of the initial energy. Value of \(p\) is:
1. \(10\)
2. \(20\)
3. \(30\)
4. \(40\)
1. \(0\)
2. \(\frac{1}{4}\)
3. \(\frac{3}{4}\)
4. \(\frac{5}{6}\)
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