# Question b03f7

Feb 28, 2016

For the SHM approximation to be more accurate.

#### Explanation:

I think you meant $\sin \theta \approx \theta$ instead of $m g \sin \theta \approx \sin \theta$.

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

$\frac{\text{d"^2 theta}{"d} {t}^{2}}{+} \frac{g}{l} \sin \theta = 0$

which is not exactly SHM. Only when $\theta \to 0$, does $\sin \theta \to \theta$, 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.