Question #b03f7

1 Answer
Feb 28, 2016

For the SHM approximation to be more accurate.


I think you meant #sintheta ~~ theta# instead of #mgsintheta~~sintheta#.

The simple harmonic motion (SHM) is only an approximation for swinging pendulums, unlike the case for spring mass system where it is exactly SHM.

The differential equation for SHM is

#frac{"d"^2x}{"d"t^2} + omega^2x = "constant"#

For pendulums, the equation of motion is

#frac{"d"^2 theta}{"d"t^2} + g/l sintheta = 0#

which is not exactly SHM. Only when #theta -> 0#, does #sin theta -> theta#, do we begin to observe SHM-like behaviors, such as having a period independent of the amplitude.

For beginners, we do not concern ourselves with frictional losses, although you do have a point that increasing the amplitude increases the rate of energy loss from the system in real life.