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Motion of Center of Mass

Practice Questions:

Question 1:

Two bodies of masses 2 kg and 4 kg, initially at rest, start moving towards each other due to mutual gravitational attraction. At a certain instant their speeds are 2 m/s and 1 m/s respectively. The speed of their centre of mass at that instant is

1. 5 m/s

2. 6 m/s

3. 8 m/s

4. zero (0)

View Answer

As Centre of mass was initially at rest and no external force act on it, Center of mass will remain at rest.

Note: Gravitational force here is internal force acting between two objects.

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Question 2: easy

A boat of mass 40 kg is at rest. A dog of mass 4 kg starts moving in the boat with velocity of 10 m/s. What is the speed of boat ?

1. 1 m/s

2. 2 m/s

3. 4 m/s

4. 8 m/s

View Answer

\[ v_{cm}=\frac{(m_{1}v_{1}+ m_{2}v_{2})}{(m_{1}+m_{2})} \]

As Center of mass will remain at rest \[ (m_{1}v_{1}+ m_{2}v_{2}) = 0 \]

4 × 10 + 40 × v = 0

v = -1 m/s

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Question 3: easy

A system consists of two identical particles. One particle is at rest and the other particle has an acceleration a. The centre of mass of the system has an acceleration

1. 2a

2. a

3. a/2

4. a/4

View Answer

\[ a_{cm} = \frac{(m_{1}a_{1}+ m_{2}a_{2})}{(m_{1}+ m_{2})} \]

\[ a_{cm} = \frac{(m\times a+ m\times 0)}{(m+ m)}= \frac{a}{2} \]

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Question 4: moderate

Two persons of masses 55 kg and 65 kg are at the opposite ends of a boat. The length of the boat is 3.0 m and its mass is 100 kg. The 55 kg man walks upto the 65 kg man and sits with him. If the boat is in still water, the centre of mass of the system shifts by

1. 0.75 m

2. 2.3 m

3. 3.0 m

4. zero

View Answer

As the boat was initially at rest and no external force acts on it centre of mass will remain at rest.

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Question 5: easy

Two particles \(A\) and \(B\) initially at rest, move towards each other under mutual force of attraction. At an instance when the speed of \(A\) is \(v\) and speed of \(B\) is \(3v\), the speed of centre of mass is

1. \(v\)

2. \(4v\)

3. \(2v\)

4. Zero

View Answer

Since there is no external force acting on the system, the acceleration of the centre of mass is zero. Since the system was initially at rest, the velocity of the centre of mass remains zero.

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Question 6: moderate

A boat of length \( 12\text{ m} \) and mass \( 840\text{ kg} \) is floating without motion in still water. A man of mass \( 60\text{ kg} \) standing at one end of it walks to the other end of it and stops. The magnitude of displacement of the boat relative to the ground is:

1. \( 50\text{ cm} \)

2. \( 80\text{ cm} \)

3. \( 120\text{ cm} \)

4. \( 150\text{ cm} \)

View Answer

Since no external horizontal force acts on the boat-man system, the center of mass does not move. The displacement of the boat is \( x = \frac{m L}{m + M} = \frac{60 \times 12}{60 + 840} = 0.8\text{ m} = 80\text{ cm} \).

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Question 7: moderate

Two bodies with masses \( m_1 \) and \( m_2 \) (\( m_1 > m_2 \)) are joined by a string passing over a fixed pulley. Assuming masses of the pulley and thread are negligible. Then the acceleration of the centre of mass of the system is:

1. \( \left(\frac{m_1 - m_2}{m_1 + m_2}\right) g \)

2. \( \left(\frac{m_1 - m_2}{m_1 + m_2}\right)^2 g \)

3. \( \frac{m_1 g}{(m_1 + m_2)} \)

4. \( \frac{m_2 g}{(m_1 + m_2)} \)

View Answer

The acceleration of each block is \( a = \left(\frac{m_1 - m_2}{m_1 + m_2}\right) g \). The acceleration of the center of mass is \( a_{\text{cm}} = \frac{m_1 a_1 + m_2 a_2}{m_1 + m_2} = \left(\frac{m_1 - m_2}{m_1 + m_2}\right) a = \left(\frac{m_1 - m_2}{m_1 + m_2}\right)^2 g \).

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Question 8: easy

Consider a system of two identical particles. One of the particles is at rest and the other has an acceleration \( a \). Their centre of mass has an acceleration

1. Zero

2. \( \frac{a}{2} \)

3. \( a \)

4. \( 2a \)

View Answer

The acceleration of the centre of mass is given by \( a_{\text{cm}} = \frac{m_1 a_1 + m_2 a_2}{m_1 + m_2} \). Since \( m_1 = m_2 = m \), \( a_1 = 0 \), and \( a_2 = a \), we get \( a_{\text{cm}} = \frac{m(0) + m(a)}{2m} = \frac{a}{2} \).

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Question 9: easy

Read the statements marked as assertion (A) and reason (R) and choose the correct option.

Assertion (A): If no external force acts on a system, the velocity of the centre of mass remains constant.

Reason (R): If there is no external force on system, then momentum of system is conserved.

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. (A) is false but (R) is true

View Answer

If the net external force on a system is zero, its linear momentum is conserved (\(vec{P} = M\vec{v}_{\text{cm}} = \text{constant}\)). Consequently, the velocity of the centre of mass \(\vec{v}_{\text{cm}}\) remains constant.

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Question 10: easy

Assertion (A): Two particles undergo rectilinear motion along different straight lines. Then the centre of mass of system of given two particles also always moves along a straight line.

Reason (R): If direction of net momentum of a system of particles (having non-zero net momentum) is fixed, the centre of mass of given system moves along a straight line.

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

Both (A) and (R) are true. The velocity of the center of mass is given by \(V_{CM} = \frac{P_{total}}{M_{total}}\). If particles move rectilinearly, their velocities are constant, making \(P_{total}\) constant in direction. A fixed direction of \(V_{CM}\) means rectilinear motion. Hence, (R) correctly explains (A).

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