# Pendulum on the end of a rotating disk

• Smouk
In summary, the conversation discussed the problem of calculating the angle Φ of a pendulum that is shifted due to the rotation of a disk with radius R and angular velocity ω. The conversation outlined the steps to solve the problem, including the use of the centripetal force and gravitational force, and concluded with the correct solution for Φ as a function of ω.
Smouk

## Homework Statement

There's a pendulum with mass m and longitude L strapped to a disk with radius R that rotates with an angular velocity ω. Calculate the angle that the pendulum is shifted (Φ) depending on ω.

You are given m, L, R, ω, g. Calculate Φ depending on ω.

Little drawing (with my amazing Windows Paint skills):

## Homework Equations

Centripetal Force = mω^2r //Will call it CF from now on
Gravitational Force = mg //Will call it GF from now on
The tension is the resultant from both gravitational (opposite direction) and centripetal forces.

## The Attempt at a Solution

Using what we've seen on the 2nd point.
CF = sin(Φ) * L
sin(Φ) = CF/L
Φ = arcsin(mω^2r/L)

Now we have Φ depending on ω but we still have to know what is r exactly so we proceed:
d = sin(Φ) * L
R, r and d form a right triangle so we apply Pythagora´s theorem:
r = sqrt(R^2 + d^2) = sqrt(R^2 + sin^2(Φ) * L^2)

That way we end up with:
Φ = arcsin(mω^2*sqrt(R^2+sin^2(Φ) * L^2) / L)

After doing some calculations you end up with:
sin^2(Φ) - sin^2(Φ) = (mRω^2/L)^2

Which basiclly is:
0 = (mRω^2/L)^2

I just think I'm not understanding this problem or I'm doing something wrong somewhere, I'm not asking for anyone to solve it but just to tell me what I'm doing wrong so I can figure it out.

Thanks to everyone!

For starters
If by CF you mean "centripetal force"
1. the equation CF = sin(Φ) * L is incorrect.
2. the centripetal force points towards the center of the cicle that the mass describes, not tangent.
3. what is the difference between r and R in you drawing? How is r defined?

On edit: Draw a free body diagram in which the vertical axis of rotation and the string are in the plane of the screen. It will be easier for you to see (and for us to explain) what is going on.

kuruman said:
For starters
If by CF you mean "centripetal force"
1. the equation CF = sin(Φ) * L is incorrect.
2. the centripetal force points towards the center of the cicle that the mass describes, not tangent.
3. what is the difference between r and R in you drawing? How is r defined?

On edit: Draw a free body diagram in which the vertical axis of rotation and the string are in the plane of the screen. It will be easier for you to see (and for us to explain) what is going on.
Yeah I just didn't really understand the exercise. I even drew the centripetal force wrong as the pendulum doen't swing like that but outwards, I just ended up figuring it out.

Decomposed both the gravitational force and the inertia (don't really want to say centrifugal force :P) into vectors that are in the same direction of the pendulum's rod and that are also perpendicular to that. We now use those two perpendicular vectors and say they are equal and now we can get from there the angle Φ depending on a given ω.

R was the radius of the disk and r the distance of the mass to the rotational axis (I'm not sure if that's how you say it).

Thank you for trying to help me!

Smouk said:
R was the radius of the disk and r the distance of the mass to the rotational axis (I'm not sure if that's how you say it).
That's a good way to say it. It looks like you were able to finish the problem. Is that right?

kuruman said:
That's a good way to say it. It looks like you were able to finish the problem. Is that right?

Yes!

## What is a Pendulum on the end of a rotating disk?

A pendulum on the end of a rotating disk is a physical system where a pendulum is attached to the edge of a rotating disk. The disk can rotate at a constant speed or accelerate, causing the pendulum to experience various forces and display interesting behaviors.

## What is the purpose of studying a Pendulum on the end of a rotating disk?

Studying a pendulum on the end of a rotating disk can help us understand the principles of rotational motion, forces, and energy. It can also be used to demonstrate and analyze complex physical concepts, such as centripetal force and Coriolis force.

## How does the rotation of the disk affect the pendulum?

The rotation of the disk affects the pendulum in several ways. It can cause the pendulum to swing at different angles and speeds, and it can also introduce additional forces, such as the Coriolis force, that can alter the motion of the pendulum.

## What factors influence the motion of the Pendulum on the end of a rotating disk?

The motion of the pendulum on the end of a rotating disk is influenced by several factors, including the rotational speed of the disk, the length of the pendulum, and the mass of the pendulum. Other factors, such as the angle of the pendulum and the presence of external forces, can also affect its motion.

## What are some real-life applications of a Pendulum on the end of a rotating disk?

A pendulum on the end of a rotating disk has various real-life applications, such as in mechanical clocks, amusement park rides, and gyroscopic stabilization systems. It is also used in scientific experiments and demonstrations to study rotational motion and its effects on objects.

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