Assertion (A): For the purpose of calculation of moment of inertia, body’s mass can be assumed to be concentrated at its centre of mass.
Reason (R): Moment of inertia of a rigid about an axis passing through its centre of mass 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
Moment of inertia depends critically on the distribution of mass relative to the axis of rotation, so mass cannot generally be assumed concentrated at the center of mass (A is false). Also, the moment of inertia of a rigid body about an axis passing through its center of mass is generally not zero (e.g., a disc has \(I = \frac{1}{2}MR^2\)). Thus, (R) is false. Both statements are incorrect.
Assertion (A): It will be much easier to accelerate a merry-go-round full of children if they stand close to its axis then if they all stand at the outer edge.
Reason (R): For larger moment of inertia, the angular acceleration is small for given torque.
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
From the relation \(\tau = I \alpha\), where \(\tau\) is torque, \(I\) is moment of inertia, and \(\alpha\) is angular acceleration. If children stand closer to the axis, \(I\) decreases. For a given \(\tau\), a smaller \(I\) leads to a larger \(\alpha\), making it easier to accelerate. So, (A) is true. Reason (R) states that for larger \(I\), \(\alpha\) is small for given \(\tau\), which is also true and explains (A).
Assertion (A): Inertia and moment of inertia are same quantities.
Reason (R): Moment of inertia represents the capacity of a rigid body to oppose its state of oscillatory motion.
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
Inertia (mass) measures resistance to translational motion, while moment of inertia measures resistance to rotational motion. They are distinct quantities. Moment of inertia opposes changes in a body's state of \(\text{rotational}\) motion, not oscillatory motion. Therefore, both Assertion (A) and Reason (R) are false.
Assertion (A): If two different axes are at same distance from the centre of mass of a rigid body then moment of inertia of the given rigid body about both the axes will always be equal.
Reason (R): According to perpendicular axis theorem \(\text{I} = \text{I}_{\text{cm}} + \text{Md}^2\) where symbols have their usual meaning.
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 false. Moment of inertia depends on both the distance and the orientation of the axis. Reason (R) is false. The given formula is for the parallel axis theorem, not the perpendicular axis theorem.
Assertion (A): A wheel moving down a perfectly frictionless inclined plane will undergo slipping (not rolling).
Reason (R): For pure rolling, work done against frictional force 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; friction provides the torque for rolling. Without friction, the wheel slips. Reason (R) is true; in pure rolling, the contact point is stationary, so static friction does no work. However, (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).
If moment of inertia of a spinning object drops to \( \left(\frac{1}{4}\right)^{\text{th}} \) of its initial value, the ratio of new rotational kinetic energy to initial rotational kinetic energy will be (Assume net external torque about the axis of rotation is zero)
1. 1 : 4
2. 4 : 1
3. 2 : 1
4. 1 : 2
View Answer
Under zero external torque, angular momentum is conserved: \( I_1 \omega_1 = I_2 \omega_2 \). If \( I_2 = I_1/4 \), then \( \omega_2 = 4\omega_1 \). The ratio of kinetic energy is \( \frac{K_2}{K_1} = \frac{\frac{1}{2}I_2\omega_2^2}{\frac{1}{2}I_1\omega_1^2} = \frac{1}{4} \times 16 = 4 \implies 4 : 1 \).