# Displacement x of simple harmonic oscillation

## Homework Statement

The amplitude of simple harmonic oscillation is A = 10 cm. Find a displacement x when K = 1/6 U. Here K is a kinetic energy and U is a potential energy.

## Homework Equations

$$KE = \frac{1}{2} k A^2$$
$$U = \frac{1}{2} k x^2$$

## The Attempt at a Solution

I'm not sure the correct method of doing this problem but here is what I have attempted based on an example from my notes:

$$KE = \frac{1}{6} U$$
$$\frac{1}{2} k A^2 = \frac{1}{6} \frac{1}{2} k x^2$$
$$A^2 = \frac{1}{6} x^2$$
$$(0.1)^2 = \frac{1}{6} x^2$$
$$x \approx 0.24 m$$

## Answers and Replies

vela
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Your formula for kinetic energy isn't correct.

You also need another equation (or law) regarding energy.

This may be it, I think:

$$K = \frac{1}{6} U$$
$$U = \frac{1}{2} kx^2$$
$$KE = U + K = U + \frac{1}{6} U = \frac{7}{6} U = \frac {7}{6} \frac {1}{2} kx^2$$
$$KE = \frac {1}{2} kA^2$$
$$\frac{1}{2} kA^2 = \frac {7}{6} \frac {1}{2} kx^2$$
$$A^2 = \frac{7}{6}x^2$$
$$(0.1)^2 = \frac {7}{6}x^2$$
$$x \approx 0.0926 m$$

vela
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Looks good.

By the way, what does KE stand for? In my first post, I mistakenly thought you were referring to the kinetic energy as KE is a common abbreviation for it.

I was using KE as the kinetic energy of the spring... wasn't sure if energy in a spring should be referred to as kinetic energy or just energy.

Should it just be E for energy?

vela
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You're using the approximation that the spring is massless, so it has no kinetic energy. It only has potential energy. Your KE is the total energy of the system.