Question

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 ?

Answer 1

Because large amplitudes (beyond `10^@`) 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 ~~ theta` (the ‘small angle rule’)

This is less true as `theta` increases.

Answer 2

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_"net" = m*g*sintheta`,
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_"net" " and " sintheta`

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

`F_"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

`sintheta = theta`.

When this approximation is valid that initial formula can be written

`F_"net" = m*g*theta`

So in that form, we see that `F_"net" " and " theta` are proportional as required.

So, when amplitude is kept small (allowing use of the `sintheta = 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".