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?
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
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