# Natural Period of Vibration for Mass with Torque

1. Jun 2, 2010

1. The problem statement, all variables and given/known data

A body of arbitrary shape has a mass m, mass center at G with a distance of D and a radius of gyration about G of K. It is hanging, and pinned at the top. If it is displaced a slight amount of angle P from it's equilibrium position and released, determine the natural period of vibration.

2. Relevant equations

3. The attempt at a solution

Sum of Torques = Ia. Hence, I = K2m

The torque applied by gravity: -mgDSinP = K2ma

Hence, mgDSinP + K2ma = 0
By the small angle approximation..

mgDP + K2mP'' = 0

Is this correct so far? I am lost at what to do next,

Cheers,

Last edited: Jun 2, 2010
2. Jun 2, 2010

### rhoparkour

I just have to ask, you have not defined K so I'm not sure your derivation is correct.

Assuming it is correct, you just got your answer. HINT: Compare your final differential equation with the harmonic oscillator differential equation.

i.e.Compare
$\ddot{x} + \omega^{2}x = 0$

with yours:

$\ddot{P} + \frac{K^{2}}{g D} P = 0$
the rest is just identification.

Tell me if this was helpful, good luck.

Last edited: Jun 2, 2010
3. Jun 3, 2010

Hello,

Thanks for that! I am a little confused about how the K2 coefficient moved from the angular acceleration to the angle, though? ^^

4. Jun 3, 2010

### rhoparkour

Oh my bad, just a typo. Here's the good one:

$P + \frac{K^{2}}{g D} \ddot{P} = 0$

Just got the P miced up with the P'' for a second.

5. Jun 3, 2010