What is the eigenfrequency of a disc with mass 20 kg and no sliding motion?

[dirk_mec1]

Homework Statement

Determine the eigenfrequency of the disc with mass m = 20 kg of fig. 11-38. Assume that the disc doesn't slides.

http://img80.imageshack.us/img80/8408/86061698.jpg

Homework Equations

The correct answer is 1.01.

The Attempt at a Solution

If the angle \theta is measured in radians then the distance traveled downwards is R \cdot \theta, now Newton in the direction of the spring leads to:

m R \ddot{ \theta} + k R \theta = 0

Thus the eigenfrequency is:

\omega = \sqrt{ \frac{kR }{mR} } = \sqrt{60}

2\pi f= \omega \rightarrow f= \frac{\sqrt{60} }{2 \pi} = 1.23 Hz.

 

[vela]

There has to be a torque on the disk if it's going to rotate instead of just slide. That means there's another force on the disk that you need to account for.

 

[dirk_mec1]

But in the picture there's no torque given, so the exercise is wrong?

 

[vela]

The picture doesn't show any of the forces and torques. As with every dynamics problem, you need to begin the analysis of the situation by identifying what forces are involved and go from there.

 

[dirk_mec1]

Well I've got the gravity force and the spring force, but both go through in the center of the disc and

\sum M_{com} =I \ddot{\theta}

is then zero. What am I doing wrong?

 

[vela]

You're still missing a few forces. Think about the disk and the surface it's rolling on.

 

[dirk_mec1]

But there isn't any information about friction forces, right? Then there's the normal force but it's perpendicular to the surface so it doesn't matter, right?

 

[vela]

There's information about friction. The problem states that the disc doesn't slide.

 

[dirk_mec1]

So the friction force is mg*sin(30)?

 

[vela]

No. Just call it F for now. Write down the equations of motion for the disc.

 

[dirk_mec1]

I got it thanks, Vela. I used sum of moments is zero around the contact point of the disc and Newton along the direction of the disc, eliminated F and transformed the eigenfequency to Hertz this results in 1.006 Hz = 1.01 Hz.

 

[vela]

The sum of the moments about the contact point isn't 0 if the disc accelerates.