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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Assertion: Angular momentum of an isolated system of particles is conserved.

Reason: The net torque on an isolated system of particles is zero and the rate of change of angular momentum equals the torque.

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. Assertion is false but the reason is true.

Subtopic:  Torque |
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Assertion: If the sun were to 'suddenly' be removed, then the earth would continue to move around in its orbit.

Reason: Angular momentum of a system of particles is conserved when there is no external torque.

1. Both assertion and reason are true, and reason is the correct explanation of the assertion.

2. Both assertion and reason are true, but reason is not the correct explanation of the assertion.

3. Assertion is true but reason is false.

4. Assertion is false but reason 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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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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Consider the following statements-
(a) A body is in translational equilibrium if net force on it is zero.
(b) A body is in rotational equilibrium if net torque about any point is zero.
Choose the correct statements:
1.  (a) only
2.  (b) only
3.  both (a) and (b)
4.  neither (a) nor (b)
Subtopic:  Rotational Motion: Kinematics |
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Consider the following two statements :

(A): Linear momentum of a system of particles is zero.
(B): Kinetic energy of a system of particles is zero.

1. A implies B and B implies A

2. A does not imply B and B does not imply A

3. A implies B but B does not imply A

4. B implies A but A does not imply B

Subtopic:  Center of Mass |
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Consider the following two statements:

(A): Linear momentum of the system remains constant.
(B): Centre of mass of the system remains at rest.

1. A implies B and B implies A

2. A does not imply B and B does not imply A

3. A implies B but B does not imply A

4. B implies A but A does not imply B

Subtopic:  Center of Mass |
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Given the following statements:
a. The centre of gravity (C.G.) of a body is the point at which the weight of the body acts.
b. If the earth is assumed to have an infinitely large radius, the centre of mass coincides with the centre of gravity.
c. To evaluate the gravitational field intensity due to any body at an external point, the entire mass of the body can be considered to be concentrated at its C.G.
d. The radius of gyration of any body rotating about an axis is the length of the perpendicular dropped from the C.G. of the body to the axis.
Which one of the following pairs of statements is correct?
1. (a) and (b)
2. (b) and (c)
3. (c) and (d)
4. (d) and (a)

Subtopic:  Center of Mass |
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Which of the following statements are correct? 
(a) Centre of mass of a body always coincides with the centre of gravity of the body 
(b) Centre of gravity of a body is the point about which the total gravitational torque on the body is zero 
(c) A couple on a body produce both translational and rotation motion in a body 
(d) Mechanical advantage greater than one means that small effort can be used to lift a large load

1. (a) and (b)

2. (b) and (c)

3. (c) and (d)

4. (b) and (d)

Subtopic:  Center of Mass |
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