# Density and simple harmonic motion

## Homework Statement

A mass of density d floats in a liquid of density d_L. The mass is then pushed down a distance x and let go. Use Newton's Second Law to demonstrate that the mass will undergo simple harmonic motion. Recall that the SHM equation is d^2x/dt^2 + w^2*x = 0. Assume there is no friction. Find w in terms of whatever variables needed.

## The Attempt at a Solution

I know that Newton's 2nd law is sum F=ma, and Torque = I*omega. I don't see how I can relate this to simple harmonic motion, which involves things moving back and forth in the same pattern. The answer key says that w=SQRT(D_l * g/(D*H)). However, I don't know what I am missing to solve this problem. I don't know where to start.

I think a free body diagram would be a good place to start. Then I would use Newton's second law. Torque, huh?

When the object is at rest, I have mg pulling down and buoyant force pushing up. They are equal in magnitude. The net torque is also zero.

When the object is pushed down I have f pushing down, mg pulling down, and buoyant force pushing up. This extra f is enough to push it down. My net torque is

T = IW

However, why would I use this? Isn't torque normally used when things are rotated?

T = F x R

What R in this case? Mg and buoyant are both pushing from the center in the free body diagram so I don't think there is an R. So f is the only force that contributes to the torque am I correct?

Torque shouldn't come into play. You've listed some forces, now put them into equations.

For the object at rest

I got

B - mg = 0

When it's pushed down

B - mg -f = -ma

Since B =mg

f = ma

I already know this though so how does it help prove that it's in simple harmonic motion with a repeating pattern?

Never mind I got it now. Thanks a lot for your help.