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
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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): When a sphere is rolls on a horizontal table it slows down and eventually stops.
Reason (R): When the sphere rolls on the table, both the sphere and the surface deform near the contact. As a result, the normal force does not pass through the centre and provide an angular deceleration.
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
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A rolling sphere stops due to rolling friction. This friction arises from deformations at the contact point, causing the normal force to produce a torque that opposes the rolling motion, leading to angular deceleration.
Assertion (A): When the disc rolls without slipping, friction is required because condition of pure rolling is velocity of point of contact is zero.
Reason (R): The force of friction in the case of a disc rolling without slipping down an inclined plane 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
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Assertion (A) is true: For pure rolling, the point of contact velocity is zero, and friction provides the necessary torque.
Reason (R) is false: For a disc rolling without slipping down an inclined plane, friction is present and acts up the incline to provide the torque for rotation. Thus, (A) is true, (R) is false.
Assertion (A): It is more difficult to open the door by applying the force near the hinge.
Reason (R): Torque is maximum at hinge.
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
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Assertion (A) is true: Torque \(\tau = rF\sin\theta\) requires a larger force \(F\) for a smaller lever arm \(r\) (near the hinge). Reason (R) is false: Torque is zero at the hinge (pivot point) as \(r=0\). Thus, (A) is true, (R) is false.
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
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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
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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): A body is rolling without slipping on a surface. There must be frictional force to start such a motion.
Reason (R): In rolling without slipping, work done against the frictional force is zero on rolling 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
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Assertion (A) is true: Friction provides the necessary torque to initiate the angular acceleration required for rolling.
Reason (R) is true: In pure rolling, the point of contact is instantaneously at rest, so the work done by static friction is zero. Both statements are true, but R does not explain A.
Assertion (A): If the moment of inertia of a non-uniform thin circular ring is same about two different axes parallel to each other and lying in the plane of ring, then both the axis can be at same distance from geometrical centre of the ring.
Reason (R): From parallel axis theorem \(I = I_{cm} + md^2\), (where terms have usual meaning). Moment of inertia of a body about two axes parallel to each other and at a same distance from centre of mass of the body is same.
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
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Assertion (A) is true: If two axes have the same moment of inertia, they must be equidistant from the center of mass. It is possible for them to also be equidistant from the geometrical center (e.g., if the CM coincides with the geometrical center).
Reason (R) is true: The parallel axis theorem states \(I = I_{cm} + md^2\), so if two parallel axes are at the same distance \(d\) from the CM, their moments of inertia will be equal. Both statements are true, but R does not directly explain A.
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
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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): 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
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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).