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Question
A body is performing S.H.M. Then its ______.
- average total energy per cycle is equal to its maximum kinetic energy.
- average kinetic energy per cycle is equal to half of its maximum kinetic energy.
- mean velocity over a complete cycle is equal to `2/π` times of its π maximum velocity.
- root mean square velocity is times of its maximum velocity `1/sqrt(2)`.
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Solution
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"`
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