# NCERT Class 11 Physics Solutions: Chapter 7 – System of Particles and Rotation Motion-Part 15

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**Question 7.33:**

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

(a) Show

Where is the momentum of the particle (of mass ) and Note is the velocity of the particle relative to the centre of mass.

Also, prove using the definition of the centre of mass

(b) Show

Where is the total kinetic energy of the system of particles, is the total kinetic energy of the system when the particle velocities are taken with respect to the centre of mass and 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 Section.

(c)

Where is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass. Remember rest of the notation is the standard notation used in the chapter. Noteand 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 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.)

**Answer:**

**Explanation:**

Take a system of moving particles.

Mass of the particle

Velocity of the particle

Hence, momentum of the particle,

Velocity of the centre of mass

The velocity of the particle with respect to the centre of mass of the system is given as:

Multiplying mi throughout equation, we get:

Where,

Momentum of the particle with respect to the centre of mass of the system

We have the relation:

Taking the summation of momentum of all the particles with respect to the centre of mass of the system, we get:

Where,

Position vector of i^th partical with respect to the centre of mass

As per the definition of the centre of mass, we have:

**Explanation:**

We have the relation for velocity of the particle as:

Taking the dot product of equation with itself, we get:

Here, for the centre of the mass of the system of particles,

Where,

Total kinetic energy of the system of particles

Total kinetic energy of the system of particles with respect to the centre of mass

Kinetic energy of the translation of the system as a whole

**:**

**Explanation:**

Position vector of the particle with respect to origin

Position vector of the particle with respect to the centre of mass

Position vector of the centre of mass with respect to the origin

It is given that:

We have from part

Taking the cross product of this relation by we get:

Where,

**:**

**Explanation:**

We have the relation:

Where, is the position vector with respect to the centre of mass of the system of particles

We have the relation:

Where, is the rate of change of velocity of the particles with respect of the centre of the system

Therefore, according to Newton’s third low of motion, we can write:

External force acting on the particles

i.e. , External torque acting on system as a whole