Pendulum on the end of a rotating disk

[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!

 

[kuruman]

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.

 

[Smouk]

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!

 

[kuruman]

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?

 

[Smouk]

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

Yes!