Select the correct option based on statements below:   Assertion (A): The center of mass of an isolated system of particles remains at rest if it is initially at rest. Reason (R): Internal forces acting within a system cannot change the velocity of the center of mass which is proportional to the total momentum of the system.   1. (A) is true but (R) is false. 2. (A) is false but (R) is true. 3. Both (A) and (R) are true and (R) is the correct explanation of (A). 4. Both (A) and (R) are true but (R) is not the correct explanation of (A).

Subtopic:  Center of Mass |
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A uniform rod of mass $$M$$ and length $$L$$ lies at rest on a smooth horizontal plane, as shown in the figure. A particle of mass $$m,$$ moving with an initial velocity $$u$$ strikes one end $$(A)$$ of the rod and stops. The initial velocity $$u$$ is perpendicular to the length $$(AB)$$ of the rod.

Consider the following statements:
(P) Momentum of the system is conserved.
(Q) Kinetic energy of the system does not change after the collision.
(R) Angular momentum of the system is conserved.

1.  P, Q, R are true.
2.  P, R are true.
3.  Only R is true.
4.  Only P is true.
Subtopic:  Angular Momentum |
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A block of mass $$m$$ is placed atop another block of mass $$M,$$ and the combination is at rest on a smooth horizontal table. A force $$F_1$$ is applied to $$m$$ and another force $$F_2$$ is applied to $$M,$$ the two acting horizontally and in opposite directions. Consider the following statements about the acceleration $$(a_{cm})$$ of the centre of mass of the system). (Take right as positive)

(A) $$a_{cm}=\frac{F_1-F_2}{m+M},$$ if there is no friction acting between $$m$$ and $$M$$
(B) $$a_{cm}=\frac{F_1-F_2}{m+M},$$ if there is static friction between $$m$$ and $$M$$
(C) $$a_{cm}=\frac{F_1-F_2}{m+M},$$ in all situations

1.  only A is true.
2.  only B is true.
3.  C is true.
4.  A, B are true but C is false.
Subtopic:  Center of Mass |
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Assertion: The angular momentum of system always remains constant.
Reason: For a system, $$\tau_{ext} = {dL \over dt} = 0$$
1. Both assertion and reason are correct and reason is the  correct explanation of assertion
2. Both Assertion and Reason are correct but reason is not the correct explanation of assertion
3. Assertion is correct and the reason is incorrect
4. Assertion is incorrect and the reason is correct

Subtopic:  Angular Momentum |
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Assertion: The total kinetic energy of a rolling solid sphere is the sum of translational and rotational kinetic energies
Reason: For all solid bodies, total kinetic energy is always twice of translational kinetic energy
1. Both assertion and reason are true and the reason is the correct explanation of assertion
2. Both assertion and reason are true but reason is not the correct explanation of assertion
3. Assertion is true but reason is false
4. Both assertion and reason are false
Subtopic:  Rolling Motion |
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Assertion: A ladder is more apt to slip when you are high up on it than when you just begin to climb.
Reason: At high up on a ladder, the torque is large and on climbing up the torque is small.
1. Both assertion and reason are true and the reason is the correct explanation of the assertion.
2. Both assertion and reason are true but the reason is not the correct explanation of the assertion.
3. Assertion is true but the reason is false.
4. Both assertion and reason are false.

Subtopic:  Torque |
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Assertion: If there is no external torque on a body about its centre of mass, then the velocity of the centre of mass remains constant.
Reason: The linear momentum of an isolated system remains constant.
1. Both assertion and reason are true and the reason is the correct explanation of assertion
2. Both assertion and reason are true but the reason is not the correct explanation of assertion
3. Assertion is true but the reason is false
4. Assertion is false but the reason is true

Subtopic:  Torque |
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Assertion: In case of pure rolling, the force of friction becomes zero.
Reason: The speed at the point of contact is zero.
1. Both the assertion and reason are true and reason explains the assertion.
2. Both the assertion and reason are true but reason does not explain the assertion.
3. Assertion is true but reason is false.
4. Assertion is false but reason is true.

Subtopic:  Rolling Motion |
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Assertion:  When a body accelerates down an incline rolling purely the static friction force acts on the body.
Reason: Point of contact of the body with incline remains at rest.
1. Both the assertion and reason are true and reason explains the assertion.
2. Both the assertion and reason are true but reason does not explain the assertion.
3. Assertion is true but reason is false.
4. Assertion is false but reason is true.

Subtopic:  Rolling Motion |
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Assertion: The velocity of a body at the bottom of an inclined plane of a given height is more when it slides down the plane, compared to when it is rolling down the same plane.
Reason: In rolling down, a body acquires both kinetic energy of translation and rotation.
1. Both assertion and reason are true and the reason is the correct explanation of the assertion.
2. Both assertion and reason are true but the reason is not the correct explanation of the assertion.
3. Assertion is true but the reason is false.
4. Both assertion and reason are false.

Subtopic:  Rolling Motion |
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