Solve Buoyant Force/Simple Harmonic Motion Homework

[DrunkApple]

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]

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.

 

[DrunkApple]

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

 

[Doc Al]

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.)

 

[asteriatic]

Sure! Let's break down the problem step by step:

1. First, we need to determine the buoyant force acting on the balloon. This can be found using Archimedes' principle, which states that the buoyant force is equal to the weight of the fluid displaced by the object. In this case, the fluid is air and the object is the balloon.

Buoyant force (Fb) = ρVg

Where ρ is the density of the fluid (in this case, air), V is the volume of the displaced fluid, and g is the acceleration due to gravity.

We are given the density of air (ρ = 1.29 kg/m^3) and the acceleration due to gravity (g = 9.8 m/s^2), but we still need to find the volume of air displaced by the balloon.

2. To find the volume of air displaced, we can use the fact that the balloon is filled with helium, which has a density of 0.18 kg/m^3. This means that the balloon has a volume of 1 kg of helium / 0.18 kg/m^3 = 5.56 m^3.

3. Now that we have the volume of air displaced, we can plug this into our equation for the buoyant force:

Fb = ρVg = (1.29 kg/m^3)(5.56 m^3)(9.8 m/s^2) = 71.03 N

4. Next, we need to find the restoring force (k) of the spring. This can be done using Hooke's Law, which states that the restoring force is directly proportional to the displacement from equilibrium.

F = -kx

Where F is the restoring force, k is the spring constant, and x is the displacement from equilibrium.

In this case, the spring is actually the tension in the string, so we can use the same equation:

F = -kx = -mg

Where m is the mass of the balloon and g is the acceleration due to gravity. We can rearrange this equation to solve for k:

k = mg/x

We are given the mass of the balloon (m = 1 kg) and the length of the string (x = 1.51 m), so we can plug these values in to find the spring constant:

k = (1 kg)(9.8 m/s^2)/(