How to derive a formula for simple harmonic motion?

[Matej Pertot]

Homework Statement

You need to derive a formula for undamped pendulum simple harmonic motion;
1. Starting from the middle point
2. Starting from the extreme point

Homework Equations

The solutions are;
1. s = s0 sin(2 pi f t)
2. s = s0 cos(2 pi f t)

The Attempt at a Solution

I can derive the first answer -- you have to imagine that it is similar to circular motion. But I have difficulties with the second question; I arrive at solution 1-cos(...) because - doesn't it knly move for the difference between radius (full amplitude) and cosine?

 

[stevendaryl]

Homework Statement

You need to derive a formula for undamped pendulum simple harmonic motion;
1. Starting from the middle point
2. Starting from the extreme point

Homework Equations

The solutions are;
1. s = s0 sin(2 pi f t)
2. s = s0 cos(2 pi f t)

The Attempt at a Solution

I can derive the first answer -- you have to imagine that it is similar to circular motion. But I have difficulties with the second question; I arrive at solution 1-cos(...) because - doesn't it knly move for the difference between radius (full amplitude) and cosine?

Your relevant equations are not completely general. The equation $s = s_0 sin(2 \pi f t)$ describes motion that starts at $s = 0$ at time $t=0$. The more general solution is:

$s = A cos(2 \pi f t) + B sin(2 \pi f t)$

where $A$ and $B$ are constants that have to be chosen based on the value of $s$ at time $t=0$