Average kinetic energy for a simple harmonic motion
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Average kinetic energy for a simple harmonic motion

[From: ] [author: ] [Date: 12-04-16] [Hit: ]
. f for frequency and v for velocity.For frequency (f), the equivalent angular velocity (ω) is ..Passing through the equilibrium point the velocity has its max value given by .......
What will be the average kinetic energy of a particle of mass m executing simple harmonic motion with amplitude ‘a’ and frequency ‘ν’ during its motion from the position of equilibrium to the end?
How to derive?

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I'm using symbols .. 'f' for frequency and 'v' for velocity.

For frequency (f), the equivalent angular velocity (ω) is .. ω = 2πf rad/s

Passing through the equilibrium point the velocity has it's max value given by ..
v = ω a .. .. (from v = ±ω√(a² - x²) .. x = 0 at equil.point)
v = 2πf.a

Initial KE = ½ mv² .. .. ½ m(2πfa)² .. .. KE(i) = 2m(πfa)²
Final KE = 0 (assuming that's what you mean by '' .. to the end.'' ie comes to rest due to frictional forces)

Average KE = {KE(i) - KE(f)} / 2 .. .. 2m(πfa)² / 2 .. .. .. ►Av KE = m(πfa)²
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