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 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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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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Given below are two statements: 
Assertion (A): The centre of mass of an electron and proton when released moves faster towards the proton.
Reason (R): This is because the proton is heavier.
  
1. Both (A) and (R) are true and (R) is the correct explanation of (A).
2. Both (A) and (R) are true but (R) is not the correct explanation of (A).
3. (A) is true but (R) is false.
4. Both (A) and (R) are false.
Subtopic:  Center of Mass |
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Assertion: The centre of mass of a two-particle system lies on the line joining the two particles, being closer to the heavier particle.
Reason: The product of the mass of one particle and its distance from the centre of mass is numerically equal to the product of the mass of another particle and its distance from the centre of mass.

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. Both assertion and reason are false.
Subtopic:  Center of Mass |
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Assertion: The center of mass of a body or a system of bodies is the point that moves as though all the mass were concentrated there and all external forces were applied there.
Reason: The centre of mass will not move due to mutual attractive forces between two particles.

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:  Center of Mass |
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