# Mechanical energy of a pendulum- I just can't get it

1. Nov 8, 2007

### BlueSkyy

Mechanical energy of a pendulum- I just can't get it!!

1. The problem statement, all variables and given/known data

A pendulum of length 140 cm swings with an amplitude of 1.9 cm. Its mechanical energy is 5.7 mJ. What is the mechanical energy of the same pendulum when it swings with an amplitude of 3.7 cm?

2. Relevant equations

ME = PE + KE
PE = 1/2 k x^2
PE = mgh
KE = 1/2 m v^2

and I also know that at the top of the arc (amplitude) all of the energy is PE

3. The attempt at a solution

I keep trying to solve this and always end up with 11.1 mJ - this answer is wrong! I don't know what else to do.

2. Nov 8, 2007

### Shooting Star

Why do you need to find the KE? Can you find the height difference between the extremities of the pendulum in the two cases?

3. Nov 8, 2007

### Retsam

I think we can just use the equation :

PE = 0.5kx^2

Where k is just a constant, so first we must find k from our initial data :

PE(2)/(x^2) = k
5.7(2)/(1.9^2) = k = 3.158

Now just use this value of k to solve the second part :

PE = 0.5(3.158)(3.7^2)
PE = 21.62 mJ

ME = PE at the max amplitude point, so ME = 21.62 mJ

I think that is right.

4. Nov 8, 2007

### BlueSkyy

Thank you so much Retsam! That worked :D
I didn't know I could use PE = 1/2 k x^2 for pendulums; now I know!
Thank you!

5. Nov 8, 2007

### Shooting Star

For small amplitudes, you can use PE = kx^2/2. That means, you are essentially treating the simple pendulum as an SHM, in the regime where x is approximately equal to sin x. Geometrically, you would have got the same result, but this is the fastesr way.