if the aplitude oscillation of a pendulam is very small,then time period of oscillation depends on?
Answers
If a pendulum is initially displaced only through a small angle - say, five or six degrees, there will be only an imperceptible difference in its period of oscillation for different amplitudes. So the period does not change appreciably as the amplitude diminishes over time - even until the oscillations die out entirely.
The reason is that if the amplitude of the oscillation is very small compared to the pendulum’s length - or equivalently, if the initial angular amplitude is very small compared to one radian of angle (which is approximately 57°), the pendulum is essentially in what is called simple harmonic motion. That is, the displacement from equilibrium is described by a sinusoidal function with respect to time - much like a mass which oscillates on the end of a spring.
The period of a pendulum is essentially constant for small angular displacements. But increase the initial angle to, say, 30° or 45°, and the initial period will be measurably larger, and then gradually decrease back to its “small angle” value as the oscillations die out.
The mathematics is not particularly difficult to show that the period is essentially independent of the amplitude for small angular displacements. The solution can be found in any introductory level physics book that includes calculus. But the solution for the period of a pendulum is significantly more difficult for larger angles of initial displacement. And that is the reason there is not a simple answer to the question.
But that leads to an interesting thought about measurement precision. If the period of oscillation depends on the angular amplitude for larger oscillation amplitudes, it must also depend on amplitude for even small amplitudes. It just doesn’t change very much for small oscillations - and we claim it is essentially independent of amplitude. That is, we use those qualifying words “essentially” or “approximately” or “nearly” or “very little change” or “not appreciably different” in the description. That just means that if the required precision for knowing the period is not too high, we can get away with saying the period is independent of amplitude as long as the amplitude is small enough.
And, incidentally, if the pendulum is what’s called a “simple” pendulum - for example, a small mass on the end to a long string, and only oscillates through a small angle, the period only depends on the length of the string and the acceleration due to gravity. That is, not only does it not depend on the amplitude, it doesn’t depend on the amount of the mass either.