Question

Question #b03f7

Answer 1

For the SHM approximation to be more accurate.

Explanation:

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.