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Calculation of Center of Mass

Practice Questions:

Question 1: easy

Center of mass is closer to

1. Heavier Mass

2. Closer Mass

3. Mid Point

4. Any one of the above

View Answer

Center of Mass is the weighted mean of masses so it is closer to heavier mass.

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Question 2: easy

Centre of mass is the point which divides the line joining the masses in

1. ratio of masses

2. reverse ratio of masses

3. 1:1

4. 1:2

View Answer

Centre of mass is the point which divides the line joining the masses in reverse ratio of masses.

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Question 3: easy

Masses 2 kg and 5kg are located at origin and (7,0) respectively. Centre of mass of the arrangement is located at

1. (0,5)

2. (5,0)

3. (0,0)

4. (7,0)

View Answer

Centre of mass is the point which divides the line joining the masses in reverse ratio of masses.

Here ratio of mass is 2:5 so distance from 2kg mass located at origin will be 5/7 of 7 cm so

\[ x_{cm}= \frac{2(0)+5(7)}{5+2}= 5 \]

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Question 4: easy

Given below are two statements:

Statement I: Centre of mass of any object always coincide with centre of gravity.

Statement II: Centre of gravity is the point where total gravitational torque on the body is zero.

In the light of the above statements, choose the most appropriate answer from the options given below.

1. Both statements I and II are correct

2. Both statements I and II are incorrect

3. Statement I is correct but II is incorrect

4. Statement I is incorrect but II is correct

View Answer

Statement I is false because the center of mass and center of gravity only coincide in a uniform gravitational field. Statement II is true because the center of gravity is defined as the point about which the net gravitational torque is zero.

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Question 5: easy

Assertion (A): Centre of mass of a body in pure rolling on a horizontal surface always moves in a straight line.

Reason (R): Centre of mass of a body must be inside 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

View Answer

Assertion (A) is true; the center of mass of a body in pure rolling on a horizontal surface moves along a straight line (rectilinear motion). Reason (R) is false; the center of mass can be outside the physical boundaries of the body (e.g., for a ring or a hollow sphere).

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Question 6: easy

Assertion (A): In two particle system when viewed from center of mass reference frame, if one particle stops then other one will also stop simultaneously, irrespective of external forces acting on system.

Reason (R): Centre of mass of a system is a point about which total momentum of system is always constant and non-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

View Answer

Assertion (A) is true.

In the center of mass (CM) frame, the total momentum of the system is always zero. If \(p_1 + p_2 = 0\), then \(p_1 = -p_2\). If \(p_1 = 0\), then \(p_2\) must also be zero.

Reason (R) is false; total momentum of the system about the center of mass is always zero, not 'constant and non-zero'.

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Question 7: easy

Assertion (A): A half filled bottle is more stable than a fully filled identical bottle when kept in upright position.

Reason (R): A half filled bottle has lesser mass than a fully filled bottle. (The fluid and bottles are identical).

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

View Answer

Assertion (A) is false because a fully-filled bottle has a lower center of gravity and lacks the destabilizing sloshing effect of a partially-filled bottle.

Reason (R) is true as a half-filled bottle naturally contains less mass

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Question 8: easy

Assertion (A): The centre of mass of a system of two particles is closer to the heavier particle.

Reason (R): Algebraic sum of mass moments about centre of mass 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

View Answer

For a two-particle system, the center of mass \( R_{CM} \) is defined such that the sum of mass moments about it is zero: \( m_1r_1 = m_2r_2 \). If \( m_1 > m_2 \), then \( r_1 < r_2 \), meaning the COM is closer to the heavier particle.

Thus both A and R are true, and R explains A.

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Question 9: easy

The centre of mass of a system of particles depends on

1. Position of the particles

2. Relative distance between the particles

3. Masses of the particles

4. All of these

View Answer

The position of the centre of mass of a system of particles is defined as \( \vec{R}_{cm} = \frac{\sum m_i \vec{r}_i}{\sum m_i} \). It clearly depends on individual masses, their coordinates (positions), and consequently the relative distances between them.

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Question 10: easy

Two particles of masses \(2\text{ kg}\) and \(6\text{ kg}\) located at the point \((1\text{ m}, 1\text{ m}, 1\text{ m})\) and \((2\text{ m}, 2\text{ m}, 1\text{ m})\) respectively. The distance of centre of mass from \(2\text{ kg}\) mass will be

1. \(\frac{\sqrt{2}}{4}\text{ m}\)

2. \(\frac{3\sqrt{2}}{4}\text{ m}\)

3. \(\frac{1}{\sqrt{2}}\text{ m}\)

4. \(\frac{3\sqrt{2}}{8}\text{ m}\)

View Answer

Distance between masses is \(d = \sqrt{(2-1)^2+(2-1)^2+0^2} = \sqrt{2}\text{ m}\)
Distance of center of mass from \(m_1\) is \(r_1 = \frac{m_2 d}{m_1+m_2} = \frac{6\sqrt{2}}{8} = \frac{3\sqrt{2}}{4}\text{ m}\).

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