Question 11:
easy
If torque on a body is zero, then which is conserved
Explanation:
- Torque (
where
- If
, then:
This means angular momentum is conserved.
Question 12:
easy
A uniform disc of radius R rotates about an axis through its centre and perpendicular to its plane with angular velocity \(\omega\). A stationary disc of the same mass but half the radius is placed on it axially. The final angular velocity of the system is
Using conservation of angular momentum: \(I_1\omega = (I_1 + I_2)\omega_f\). Since \(I_1 = \frac{1}{2}MR^2\) and \(I_2 = \frac{1}{2}M(R/2)^2 = \frac{1}{8}MR^2\), we get \(\omega_f = \frac{1/2}{1/2+1/8}\omega = \frac{4}{5}\omega\).
Question 13:
easy
If the ice on the polar caps of the Earth melts, the duration of day will
As polar ice melts and water flows towards the equator, mass is distributed further from the rotational axis, increasing the moment of inertia \(I\). Due to conservation of angular momentum, the angular velocity \(\omega\) decreases, which increases the duration of the day.
Question 14:
moderate
A uniform rod of mass \(m\) and length \(\ell\) is pivoted about one end and hung vertically. Another mass \(m\) hits it perpendicular to its length with a velocity \(v\) at its midpoint and sticks to it. The initial angular velocity of the rod is:
Conserving angular momentum about pivot: \(L_i = mv\frac{\ell}{2}\). Total moment of inertia is \(I = \frac{1}{3}m\ell^2 + m(\ell/2)^2 = \frac{7}{12}m\ell^2\). Equating \(I\omega = L_i\) yields \(\omega = \frac{6v}{7\ell}\).
Question 15:
easy
A particle of mass \(m\) is moving on a circle of radius \(R\) with kinetic energy \(K\). Then angular momentum of particle about centre of circle will be:
Kinetic energy \(K = \frac{p^2}{2m}\) gives momentum \(p = \sqrt{2mK}\) . Angular momentum is given by \(L = pR = \sqrt{2mK} R\).
Question 16:
easy
Assertion: In the absence of external torque kinetic energy of a system remains conserved.
Reason: In the absence of external torque angular momentum of a system remains conserved.
No external torque means angular momentum is conserved (Reason is true). Kinetic energy may not be conserved as internal forces can change it (Assertion is false).
Question 17:
easy
A particle is moving along a straight line parallel to x-axis with constant velocity. Its angular momentum about the origin:
The angular momentum is \(L = m v d\), where \(d\) is the constant perpendicular distance of the line of motion from the origin. Since \(m\), \(v\), and \(d\) are all constant, \(L\) remains constant.
Question 18:
easy
If the rotational kinetic energy of a body increased by 300% then determine the percentage increase in its angular momentum:
Since rotational kinetic energy \(K = \frac{L^2}{2I}\), we have \(L \propto \sqrt{K}\). An increase of 300% means \(K_f = 4K_i\), so \(L_f = 2L_i\). The percentage increase in angular momentum is 100%.
Question 19:
easy
A disc of mass \(2\text{ kg}\) and radius \(0.2\text{ m}\) is rotating with angular velocity \(30\text{ rad/sec}\). If a mass of \(0.25\text{ kg}\) is put gently on periphery of disc then angular velocity of disc is :
By conservation of angular momentum, \(I_1 \omega_1 = I_2 \omega_2\). Here, \(I_1 = \frac{1}{2} M R^2\) and \(I_2 = \frac{1}{2} M R^2 + m R^2\). Substituting the values: \(1 \times 30 = (1 + 0.25) \omega_2\), which gives \(\omega_2 = 24\text{ rad/sec}\).
Question 20:
easy
Assertion (A): A solid copper and solid aluminium sphere of same masses are spinning about their axes with same angular velocities copper sphere has more angular momentum than aluminium.
Reason (R): Both copper and aluminium sphere have same radius.
Angular momentum \( L = I\omega = \frac{2}{5} MR^2 \omega \). If mass (M) and angular velocity (\( \omega \)) are the same, L depends on \( R^2 \). Copper is denser than aluminium (\( \rho_{Cu} > \rho_{Al} \)). For the same mass, \( V_{Cu} < V_{Al} \), implying \( R_{Cu} < R_{Al} \). Therefore, \( L_{Cu} < L_{Al} \), making (A) false.
Also, \( R_{Cu} < R_{Al} \), so (R) is false. Both (A) and (R) are false.