Natural Period of Vibration for Mass with Torque

[adoado]

Homework Statement

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.

Homework Equations

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,
Adrian

 

[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.

 

[adoado]

Hello,

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

 

[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.

 

[adoado]

Cheers, thanks for that, I got it! :)

Thanks very much for that, I appreciate your help
Adrian