# As time period doesnot depend on amplitude so then why it is said that in measurement of time period amplitude must be kept small and also that amplitude must be zero ?

Nov 23, 2017

Because large amplitudes (beyond ${10}^{\circ}$) invalidate the assumptions in the equation used.

#### Explanation:

If you look at the derivation it includes the line that the restoring force is proportional to $\sin \theta \approx \theta$ (the ‘small angle rule’)

This is less true as $\theta$ increases.

Nov 24, 2017

The Explanation describes why that is true for a simple pendulum.

#### Explanation:

You did not say what was oscillating. I will answer for the case where it is a simple pendulum that is oscillating. If it is a pendulum, amplitude must be small because the "time period does not depend on amplitude" rule applies to pendulums only if it is exhibiting simple harmonic motion.

Simple harmonic motion of a physical system requires that the force restoring the object (bob) to the equilibrium position must be proportional to the displacement from the equilibrium position.

Go to the site
http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html
Scroll down to the section with a heading of "Period of Simple Pendulum". The first formula in that section is
${F}_{\text{net}} = m \cdot g \cdot \sin \theta$,
The angle theta (in radians) is the displacement. This equation comes from valid application of trigonometry.

In that formula, m and g are constants. Therefore this formula says that

${F}_{\text{net" " and }} \sin \theta$

are proportional. But the rule for simple harmonic motion says that

${F}_{\text{net" " and }} \theta$

must be proportional. You know that sine is not a linear relation to the angle. It plots a sine wave. But, when the value of $\theta$ is small, it is a very good approximation to say that

$\sin \theta = \theta$.

When this approximation is valid that initial formula can be written

${F}_{\text{net}} = m \cdot g \cdot \theta$

So in that form, we see that ${F}_{\text{net" " and }} \theta$ are proportional as required.

So, when amplitude is kept small (allowing use of the $\sin \theta = \theta$ approximation), time period is independent of amplitude.

I hope this helps,
Steve

P.S. I do not know why you may have seen "also that amplitude must be zero".