Question

A body is performing S.H.M. Then its ______.

1. average total energy per cycle is equal to its maximum kinetic energy.
2. average kinetic energy per cycle is equal to half of its maximum kinetic energy.
3. mean velocity over a complete cycle is equal to `2/π` times of its π maximum velocity. 
4. root mean square velocity is times of its maximum velocity `1/sqrt(2)`.

Answer 1

a, b and d

Explanation:

In the case of S.H.M, the average total energy per cycle

= Maximum kinetic energy (K₀)

= Maximum potential energy (U₀)

Average KE per cycle = `(0 + K_0)/2 = K_0/2`

Let us write the equation for the SHM x = a sin ωt.

Clearly, it is a periodic motion as it involves a sine function.

Let us find velocity of the particle, `v = (dx)/(dt) = d/(dt) (a sin ωt) = aω cos ωt`

Mean velocity over a complete cycle,

`v_"mean" = (int_0^(2pi) ωa cos θd θ)/(2pi)`

= `(ωa[sin θ]_0^(2pi))/(2pi)`

= 0

So, `v_"mean" ≠ 2/pi v_"max"`

Root mean square speed,

`v_(rms) = sqrt((v_"min"^2 + v_"max"^2)/2`

= `sqrt((0 + v_"max"^2)/2`

`v_(rms) = 1/sqrt(2) v_"max"`