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Standard XII

Physics

Center of Mass as an Average Point

The sum of mo...

Question

The sum of moments of masses of all the particles in a system about the centre of mass is always:

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Solution

Moments of mass:
The sum of all the particles in the system's moments about COM is always zero.
When we multiply a particle's mass by its directed distance from this point, we get the particle's moment.
It measures the tendency of the mass to produce a rotation about that point.
Sum of moments $m_{1} r_{1} + m_{2} r_{2} + m_{3} r_{3} = 0$
Here,
$m$ is mass.
$r$ is the distance from the center of mass.

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Similar questions

Q. Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass :
(a) Show

p = p_{i}^{'} + m_{i} V

where

p_{i}

is the momentum of the

i

th particle (of mass

m_{i}

) and

p_{i}^{'} = m_{i} v_{i}^{'}

. Note

v_{i}^{'}

is the velocity of the

i

th particle relative to the centre of mass.
Also, prove using the definition of the center of mass

\sum p_{i}^{'} = 0

(b) Show

K = K^{'} + \frac{1}{2} M V^{2}

where

K

is the total kinetic energy of the system of particles,

K^{'}

is the total kinetic energy of the system when the particle velocities are taken with respect to the centre of mass and

M V^{2} / 2

is the kinetic energy of the translation of the system as a whole (i.e. of the centre of mass motion of the system). The result has been used in Sec. 7.14.
(c) Show

L = L^{'} + R \times M V

where

L^{'} = r_{i}^{'} \times p_{i}^{'}

is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass. Remember

r_{i}^{'} = r_{i} - R

; rest of the notation is the standard notation used in the chapter. Note

L^{'}

and

M R \times V

can be said to be angular momenta, respectively, about and of the centre of mass of the system of particles.
(d) Show

\frac{d L^{'}}{d t} = \sum r_{i}^{'} \times \frac{d p^{'}}{d t}

Further, show that

\frac{d L^{'}}{d t} = τ_{e x t}^{'}

where

τ_{e x t}^{'}

is the sum of all external torques acting on the system about the centre of mass.
(Hint : Use the definition of centre of mass and Newtons Third Law. Assume the internal forces between any two particles act along the line joining the particles.)

Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass:

(a) Show p_i = p’_i+ m_iV

Where p_i is the momentum of the i^th particle (of mass m_i) and p′ _i = m_i v′ _i. Note v′ _i is the velocity of the i^th particle relative to the centre of mass.

Also, prove using the definition of the centre of mass

(b) Show K = K′ + ½MV²

Where K is the total kinetic energy of the system of particles, K′ is the total kinetic energy of the system when the particle velocities are taken with respect to the centre of mass and MV²/2 is the kinetic energy of the translation of the system as a whole (i.e. of the centre of mass motion of the system). The result has been used in Sec. 7.14.

Where is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass. Remember r’_i= r_i– R; rest of the notation is the standard notation used in the chapter. Note L′ and MR × V can be said to be angular momenta, respectively, about and of the centre of mass of the system of particles.

(d) Show

Further, show that

where τ′_ext is the sum of all external torques acting on the system about the centre of mass.

(Hint: Use the definition of centre of mass and Newton’s Third Law. Assume the internal forces between any two particles act along the line joining the particles.)

The sum of moments of masses of all the particles in a system about the centre of mass is always:

Q. The sum of moments of all the particles in a system about the centre of mass is always

Q. Why do the algebraic sum of moment of masses of various particles of a system about its centre of mass is zero?

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