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 :
1. 24 rad/sec
2. 36 rad/sec
3. 15 rad/sec
4. 26 rad/sec
View Answer
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 \(\omega_2 = 24\text{ rad/sec}\).
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.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
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.
Assertion (A): If there is no external torque on a body about its centre of mass, then the velocity of the center of mass remains constant.
Reason (R): The angular momentum of a system always remains constant.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Assertion (A) is false; constant COM velocity requires zero net external force, not zero external torque.
Reason (R) is false; angular momentum is conserved only when net external torque is zero.
Assertion (A): A ballet dancer increases or decreases the angular velocity of spin, about the vertical axis by pulling in or extending out her limbs.
Reason (R): \(L = I\omega\) which is constant about rotational axis where symbols have their usual meaning.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Assertion (A) is true: A ballet dancer changes her moment of inertia \(I\) by adjusting her body posture. Reason (R) is true: Angular momentum \(L = I\omega\) is conserved in the absence of external torque. Therefore, as \(I\) changes, \(\omega\) must change inversely to keep \(L\) constant. R is the correct explanation of A.
Assertion (A): Angular momentum of a body may remain conserved even when moment of inertia of body changes.
Reason (R): Angular momentum of a body does not depend upon moment of inertia of the body.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Assertion (A) is true: If no external torque acts, angular momentum \(L\) is conserved. If moment of inertia \(I\) changes, angular velocity \(\omega\) adjusts to keep \(L = I\omega\) constant.
Reason (R) is false: Angular momentum \(L = I\omega\) explicitly depends on the moment of inertia \(I\). Thus, (A) is true, (R) is false.
Assertion (A): In case of rolling without sliding, friction force can act in forward and backward direction both.
Reason (R): The angular momentum of a system will be conserved only about that point about which external angular impulse is zero.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
Assertion (A) is true: Friction acts to prevent slipping, which can be forward or backward depending on the situation (e.g., accelerating or braking).
Reason (R) is true: Angular momentum is conserved when the net external torque (and thus angular impulse) about the point is zero. Both statements are true, but R does not explain A.
Assertion (A): If total external torque on a rigid system is zero, its angular momentum remains constant.
Reason (R): The change in angular momentum is equal to the angular impulse of the resultant torque.
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
View Answer
Assertion (A) is true, stating the conservation of angular momentum. Reason (R) is true, defining the angular impulse-momentum theorem \(\Delta \vec{\text{L}} = int \vec{\tau} \text{dt}\). If \(\vec{\tau}_{ext} = 0\), then \(\Delta \vec{\text{L}} = 0\), so \(\vec{\text{L}}\) is constant. (R) correctly explains (A).
Assertion (A): For a system of particles under central force field, the total angular momentum is conserved.
Reason (R): The torque acting on such a system is zero.
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
View Answer
Assertion (A) is true, angular momentum is conserved when net torque is zero. Reason (R) is true. For a central force \(\vec{F}\) acting along \(\vec{r}\) (position vector), the torque \(\vec{\tau} = \vec{r} \times \vec{F} = 0\). Since \(\vec{\tau}=0\), \(\frac{\text{d}\vec{\text{L}}}{\text{dt}} = 0\), hence \(\vec{\text{L}}\) is conserved. (R) correctly explains (A).