Question

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 with respect to the origin and the kinetic energy of all the particles with respect to the center of mass as 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 true
• B. Both Statement I and Statement II are false
• C. Statement I is false but Statement II is true
• D. Statement I is true but Statement II is false

Hint:

The total kinetic energy of a system can always be decomposed into the kinetic energy of the center of mass motion and the kinetic energy of motion relative to the center of mass.

Answer 1

The Correct Option is A

Step 1: Analyze Statement I.
For a system consisting of many particles, the total kinetic energy of the system is defined as the sum of the individual kinetic energies of all the particles. Mathematically, \[ K_{\text{total}} = \sum \frac{1}{2} m_i v_i^2. \] Hence, Statement I is true.

Step 2: Analyze Statement II.
The kinetic energy of a system of particles can be split into two parts: \[ K_{\text{total}} = \frac{1}{2} M V_{\text{CM}}^2 + \sum \frac{1}{2} m_i v_i'^2, \] where \( M \) is the total mass, \( V_{\text{CM}} \) is the velocity of the center of mass with respect to the origin, and \( v_i' \) is the velocity of the \( i \)-th particle with respect to the center of mass. This is a standard result in mechanics. Hence, Statement II is also true.

Step 3: Final conclusion.
Both Statement I and Statement II are true.

Final Answer: \[ \boxed{\text{Both Statement I and Statement II are true}} \]