Question

A body is performing SHM. Select incorrect statement for the body

• A. average total energy per cycle is equal to its maximum kinetic energy $($ using PE $=$ 0 at mean position$)$
• B. average kinetic energy per cycle is equal to half of its maximum kinetic energy (using PE $=$ 0 of mean position)
• C. root mean square velocity is $\frac{1}{\sqrt{2}}$ times of its maximum velocity
• D. mean velocity over a complete cycle is equal to $\left(2/\pi \right)$ times of its maximum velocity

Answer 1

Answer: (D)

mean velocity over a complete cycle is equal to $\left(2/\pi \right)$ times of its maximum velocity

For SHM, the total energy of the body remains constant and kinetic energy of the body is maximum at mean position. Its maximum value is equal to the total oscillation energy of the body. Hence, average total energy per cycle is equal to its maximum kinetic energy of the body. So the option A is correct.
If the amplitude of oscillation is $a$ and angular frequency is $\omega$, then maximum velocity of the body $=\omega a$ and it varies according to sine law. Therefore , the r.m.s velocity of the body over complete cycle will be $\left(1/\sqrt{2}\right)\omega a$ . Hence , option C is correct.
Average kinetic energy $=\frac{1}{2}m{v}_{r.m.s}^2=\frac{1}{2}.\left(\frac{1}{2}m{\omega }^2{a}^2\right)$ which is half of maximum kinetic energy. Hence B is correct.
As after one complete cycle the body will come back to its initial position, so its displacement over a complete cycle will be zero. Thus, the mean velocity over complete cycle is always is equal to zero. Hence, the option D is wrong.