Given below are two statements : Statement I : For a mechanical system of many particles total kinetic energy is the sum of kinetic energies of all the particles. Statement II : The total kinetic energy can be the sum of kinetic energy of the center of mass w.r.t to the origin and the kinetic energy of all the particles w.r.t. the center of mass as the reference. In the light of the above statements, choose the correct answer from the options given below :

Given below are two statements For a mechanical system of many particles total kinetic energy

Q. Given below are two statements :

Statement I : For a mechanical system of many particles total kinetic energy is the sum of kinetic energies of all the particles.

Statement II : The total kinetic energy can be the sum of kinetic energy of the center of mass w.r.t to the origin and the kinetic energy of all the particles w.r.t. the center of mass as the reference.

In the light of the above statements, choose the correct answer from the options given below :

(A) Both Statement I and Statement II are false

(B) Statement I is false but Statement II is true

(D) Both Statement I and Statement II are true

Correct Answer: Both Statement I and Statement II are true

Explanation

For a system consisting of many particles, the total kinetic energy of the system is obtained by adding the kinetic energies of all individual particles.

Hence, Statement I is correct.

The total kinetic energy of a system of particles can also be written as the sum of two parts:

One part is the kinetic energy associated with the motion of the center of mass of the system with respect to the origin.

The second part is the kinetic energy of all particles measured with respect to the center of mass frame.

Mathematically,

$$ K_{\text{total}} = \frac{1}{2} M V_{\text{CM}}^2 + \sum \frac{1}{2} m_i v_i'^2 $$

where $M$ is the total mass, $V_{\text{CM}}$ is the velocity of the center of mass, and $v_i'$ are velocities of particles relative to the center of mass.

Thus, Statement II is also correct.

Therefore, both the given statements are true.

Explanation

Related JEE Main Questions