Fictitious forces on a rigid body

[chris25]

Homework Statement

A pendulum is designed for use on a gravity-free spacecraft. The pendulum consists of a mass at the end of a rod of length l. The pivot at the other end of the rod is forced to move in a circle of radius R with angular frequency ω. Let θ be the angle the rod makes with the radial direction. Show this system behaves exactly like a pendulum of length l in a uniform gravitational field g = ω𝑅^2. That is, show that θ(t) is a solution for one system if and only if it is for the other.

Relevant Equations

F=wR^2

I was confused by how to work this problem in a rotating frame. The solution read that the centrifugal force on the mass should be of magnitude 𝑚𝑤𝑅^2. However, I thought it would be 𝑚𝑤𝐿^2 where L is the distance between the mass and the center of the circle (L = l + R). What am I missing here?

 

[TSny]

I think you need to make some approximations based on assuming that $R \gg l$. I don't like it that they said "behaves exactly like..."

 

[haruspex]

I think you need to make some approximations based on assuming that $R \gg l$. I don't like it that they said "behaves exactly like..."

If that diagram came with the question, it does not intend $R \gg l$.

 

[chris25]

If that diagram came with the question, it does not intend $R \gg l$.

Do you have any idea why the solution is what it is?

 

[TSny]

If that diagram came with the question, it does not intend $R \gg l$.

Yes, thanks. I jumped the gun. It turns out that no approximations need to be made other than neglecting the Coriolis force.
[Edit: Actually, since the Coriolis force would act along the pendulum rod, it would only affect the tension in the rod. It would not affect the equation of motion for $\theta$. So, I think there is an exact correspondence!]

 

[TSny]

Do you have any idea why the solution is what it is?

Have you studied how to get the equation of motion from a Lagrangian?

 

[chris25]

Have you studied how to get the equation of motion from a Lagrangian?

Not yet, I've only studied newtonian mechanics

 

[TSny]

OK. You can just use Newtonian mechanics.

Let $r$ be the instantaneous distance from the center of the circle of radius $R$ to the pendulum bob. How would you express the centrifugal force $F_c$ on the bob in terms of $r$? Use some trigonometry to find the component of $F_c$ along the tangent of the circle of radius $l$ that the bob moves along. Nice simplification occurs.

 

[Chestermiller]

Not yet, I've only studied newtonian mechanics

In Newtonian Mechanics, have you learned about expressing Newton's 2nd law of motion as reckoned from an accelerating frame of reference?

 

[Chestermiller]

Homework Statement:: A pendulum is designed for use on a gravity-free spacecraft. The pendulum consists of a mass at the end of a rod of length l. The pivot at the other end of the rod is forced to move in a circle of radius R with angular frequency ω. Let θ be the angle the rod makes with the radial direction. Show this system behaves exactly like a pendulum of length l in a uniform gravitational field g = ω𝑅^2. That is, show that θ(t) is a solution for one system if and only if it is for the other.
Relevant Equations:: F=wR^2

I was confused by how to work this problem in a rotating frame. The solution read that the centrifugal force on the mass should be of magnitude 𝑚𝑤𝑅^2. However, I thought it would be 𝑚𝑤𝐿^2 where L is the distance between the mass and the center of the circle (L = l + R). What am I missing here?

The centrifugal force on the bob is in the radial direction from the center of rotation of the rod, and is given by $m\omega^2 \bf{L}$. So the centrifugal force is not that produced by a uniform gravitational field. You need to determine the component of this force in the theta direction of the bob motion (see figure). When you complete the analysis, the motion will be the same as if the rod were not rotating and the bob were in a uniform gravitational field of magnitude $g=\omega^2R$.