Statement-1: Kinetic energy of system = \( \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 + \dots + \frac{1}{2} m_n v_n^2 \) Statement-2: Kinetic energy of system = Kinetic energy of center of mass + kinetic energy with respect to center of mass.

Question

A wave disturbance in a medium is described by \[ y(x,t) = 0.02 \cos\left(50 \pi t + \frac{\pi}{2}\right) \cos(10 \pi x) \] where \( x \) and \( y \) are in meters and \( t \) is in seconds. Which statement(s) is/are correct?

Answer 1

To determine the correctness of the statements given, let's analyze each one individually:

Statement 1: Kinetic energy of a system = \(\frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 + \dots + \frac{1}{2} m_n v_n^2 \)

This statement describes the total kinetic energy of a system consisting of multiple particles. The kinetic energy of each particle is given by \(\frac{1}{2} mv^2\), where \(m\) is the mass of the particle and \(v\) is its velocity. Therefore, the total kinetic energy of the system is the sum of the kinetic energies of all the particles, which is expressed as:

\(\frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 + \dots + \frac{1}{2} m_n v_n^2 \)

This is a correct and standard expression in classical mechanics.

Statement 2: Kinetic energy of a system = Kinetic energy of center of mass + kinetic energy with respect to center of mass.

According to the principle of the center of mass in mechanics, the total kinetic energy of a system can be decomposed into two parts: the kinetic energy associated with the motion of the center of mass and the kinetic energy of the particles relative to the center of mass. This can be mathematically expressed as:

Total Kinetic Energy = \(\frac{1}{2} M V_{cm}^2 + \sum \frac{1}{2} m_i v_{i,rel}^2 \)

where \(M\) is the total mass of the system, \(V_{cm}\) is the velocity of the center of mass, and \(v_{i,rel}\) is the velocity of each particle with respect to the center of mass. This decomposition shows that the total kinetic energy includes the motion about the center of mass as well as the internal motion relative to it.

Therefore, Statement 2 is also true.

Conclusion:

Both Statement 1 and Statement 2 are correctly derived from the principles of classical mechanics. Therefore, the correct answer is:

Statement I is true; Statement II is true.

Answer 2