Simple Pendulum Motion & Time Period Formula with SHM Approximation

[shadowrunner]

Homework Statement

The motion of a simple pendulum is only approximately simple harmonic. How small should the amplitude should be for the approximation to hold good?. Obtain the general expression for the time period of a simple pendulum. How much does the actual time period differ from the approximate time period when the amplitude is 15 degree?

Homework Equations

The Attempt at a Solution

On obtaining the period of oscillation using SHM, we approximate sin(a)=a in radians. I was thinking of keeping it as sin(a). Any help with the first question? The amplitude one.

 

[gabbagabbahey]

The approximation $\sin\theta\approx\theta$ is derived via Taylor series expansion. To determine how accurate it is, you need to look at the Taylor series remainder.

 

[shadowrunner]

$$\ T=2pi \sqrt{L/g}(1\theta^2/16+11\theta^4/3072+173\theta^6/737280+22931\theta^8/1321205760+...)$$

That was the equation I got for time period. But I can't approximate the smallest value of amplitude.

 

[gabbagabbahey]

$$\ T=2pi \sqrt{L/g}(1\theta^2/16+11\theta^4/3072+173\theta^6/737280+22931\theta^8/1321205760+...)$$

Surely you mean

$$T=2\pi\sqrt{\frac{L}{g}}\left(1+\frac{1}{16}\theta_0^2+\frac{11}{3072}\theta_0^4+\frac{173}{737280}\theta_0^6+\ldots\right)$$

where $\theta_0$ is the initial amplitude (angular displacement) of the pendulum...right?

That was the equation I got for time period. But I can't approximate the smallest value of amplitude.

Read the question again. You aren't asked to approximate the smallest value of amplitude.

 

[shadowrunner]

"How small should the amplitude should be for the approximation to hold good?"
It is given in the question. I don't know how to do it.

 

[gabbagabbahey]

The approximation they are referring to is

$$T=2\pi\sqrt{\frac{L}{g}}\left(1+\frac{1}{16}\theta _0^2+\frac{11}{3072}\theta_0^4+\frac{173}{737280}\theta_0^6+\ldots\right)\approx2\pi\sqrt{\frac{L}{g}}$$

...how large can you make $\theta_0$ before this is no longer a good approximation?