Finding Angular Frequency for Simple Harmonic Motion - Mass and Rod System

[sjmacewan]

OK, the answer for this problem seems a bit high to me, so I'm going to ask if it all seems alright.

You have a 1m rod of no mass, fixed so that it may rotate about it's center. At the top of the rod is a mass m1 (0.5kg), and the bottom is a mass m2 (1.0kg). Find the angular frequency assuming a small amplitude of oscillations.

I just stared at it for about an hour, trying to find a way to get it into the form

$$\ddot{\psi} + \omega_0 ^2 \psi = 0$$

Eventually i went at it with torques, using the $$\tau = r x F$$ formula.

So...

$$\tau = rm_1g \sin (180- \theta) - rm_2g \sin (\theta)$$

and using small angle approximation, both sins go to theta.

$$\tau = rg (m_2 - m_1) \theta$$
$$\tau = I \ddot{\theta} = -rg (m_2 - m_1) \theta$$
$$I = \sum m r^2 = (m_1 + m_2) r^2$$

$$\ddot{\theta} = \frac{-rg (m_2 - m_1)}{r^2 (m_1+m_2)}$$
$$\ddot{\theta} + \frac{rg (m_2 - m_1)}{r^2 (m_1+m_2)} = 0$$

Which is in the right form. Meaning that the frequency is:
$$\omega_0 = \sqrt{\frac{g(m_2-m_1)}{r(m_1+m_2)}}$$

Now, that may be ALL wrong, but it's what first came to mind. I end up with 2.6Hz for an answer which seems a bit off...

Thanks for any redirections/confirmations/help you can give!

(if the tex is messy I'm working on it still!)

 

[OlderDan]

You could check your answer using the standard physical pendulum equation

http://hyperphysics.phy-astr.gsu.edu/hbase/pendp.html

You already have an expression for I, and the mass and g are easy. You just need to compute the distance from support to CM. I think you will like it. I did not check your computation, but the answer looks good.

 

[sjmacewan]

excellant! what a helpful script! The answer does end up being the same, so I'm going to assume my theory-work was good. It's not EXACTLY how we;ve been doing those questions (been using energies and such) but I found this one much easier to follow.

Thanks a lot! (still open for comments though)

 

[OlderDan]

Thanks a lot! (still open for comments though)

Well, if you look at the first part of the link, you wil find your

$$\ddot{\psi} + \omega_0 ^2 \psi = 0$$

Just replace their variable with yours