Solving Harmonic Pendulum Homework

[joemama69]

Homework Statement

A pendulum consists of a solid, uniform sphere of mass M and radius
R attached to one end of a thin, uniform rod of mass m and length L.
The pendulum swings freely about the other end of the rod. Find the
period of small oscillations of this pendulum.

Homework Equations

The Attempt at a Solution

T = -mhgsinQ = Ia = Id²Q/dt²

d²Q/dt² +mghsinQ/I = 0

does sinQ = Q for small approx

should i also be able to find the actual I

 

[AEM]

Homework Statement

A pendulum consists of a solid, uniform sphere of mass M and radius
R attached to one end of a thin, uniform rod of mass m and length L.
The pendulum swings freely about the other end of the rod. Find the
period of small oscillations of this pendulum.

It almost seems from the way the problem is worded that you are expected to use

$$\tau = I \alpha$$

where tau is the torque, I is the moment of inertia, and alpha is the angular acceleration. Once you have set up the differential equation that goes with this, you can pick off what you need to answer the question.

Assuming that is the case, the torque is the force (due to gravity) times the distance from the center of mass of the ball to the pivot point, plus the force acting on the center of mass of the rod times the distance between the center of mass of the rod to the pivot point:

$$Mg sin(\theta) (R+L) + mg sin(\theta) (\frac{L}{2})$$


The moment of inertia for the system is the sum of the moment of inertia of the ball about the distant pivot point (use the parallel axis theorem) plus the moment of inertia of the rod about the end point (pivot point). I'll let you figure that out. (One of those two moments of inertia you can look up.)

Then your differential equation is:

$$(I_{total}) \frac{d^2 \theta} {d^2 t} = \tau$$


One of your tasks will be to simplify the terms for I total and the torque into something neater. Since it's a small angle oscillation, replace the sines with just the angle. Simplify the way the equation looks and try to make it match the equation for simple harmonic motion. Check and see what terms in the D.E. for SHM determine the period, and figure out your answer.