# Buoyant Force/SHM

## Homework Statement

A light balloon filled with helium of density 0.18 kg/$m^{3}$ is tied to a light string of length 1.51 m. The string is tied to the ground, forming an “inverted” simple pendulum as in the figure. The acceleration of gravity is 9.8 m/$s^{2}$. If the balloon is displaced slightly from equilibrium, find the period of the motion. Take the density of air to be 1.29 kg/$m^{3}$. Answer in units of s.

## Homework Equations

T = 2$\pi$$\sqrt{m/k}$
Fb = ρVg

## The Attempt at a Solution

I know Archimede's principle and SHM apply here, but I cannot make the connection. Can anyone take me step by step please?

Doc Al
Mentor
If it was a regular pendulum, how would you solve it? If you can analyze a regular pendulum--a mass on a string--then you'll be able to figure out this problem.

If it was just a normal pendulum, I would just plug in the number no?

Doc Al
Mentor
If it was just a normal pendulum, I would just plug in the number no?
Write the equation of motion for a pendulum (torque = I*alpha) and then the expression for its period (the solution).

Then write a comparable equation for this inverted pendulum and see how it differs and how you'd modify the solution for the normal pendulum to suit this problem. (The same equations have the same solutions.)