Need to answer these questions to be able to do a lab, thanks! :)

Question

1)A physical pendulum is made of a solid disk of radius R hung by a screw that is free to rotate. Show that the period of oscillation is T=2pi*sqrt(3R/2g)

 

2) Consider a physical pendulum made up of two point masses (each mass M separated by a distance d) on a light rod allowed to rotate about the screw (L is the distance from the midpoint between the masses to the screw). 

 

a) Write down an expression for the moment of inertia of this pendulum. You can assume the centre of mass is the point midway between to two masses. 

 

b) Write down an expression for the period of oscillation of this pendulum. 

Answer 1

1) The moment of inertia for a solid disk suspended by a screw on its edge is given by: (3/2)mR²

 

This can be derived in many ways, but the easiest is to see the following: The moment of inertia of a disk with the axis of rotation through its center is (1/2)mR², this should already have been taught to you (if not, see any of ten thousand online resources that derive this). Here, the disk is being rotated about an axis at the disk's edge. This, understandably, changes the moment of inertia. The exact change is actually just mR²! This is because the distance from the center of mass to the new axis of rotation is R; the parallel axis theorem states that you can find the moment of inertia about a parallel axis from the equation:

 

I = I_cm + Md²,                                (1)

 

where I_cm is the moment of inertia for the axis of rotation that passes through the center of mass and "d" is the distance from the center of mass that the parallel axis is located. So, from (1), we have

 

I = (1/2)mR² + mR² = (3/2)mR²                        (2)

 

The equation for finding the period of oscillation for a physical pendulum is (after many simplifications, including the small angle approximation):

 

T = 2π*sqrt(I/(mgL))                       (3)

 

where "L" is the distance from the axis of rotation to the center of mass.

 

It should be more than doable for you to answer the rest of number one now! Just plug the answer from (2) into (3), replacing "L" with a suitable variable... and voila! The answer should appear!

 

 

2a) I will assume that the screw is a parallel axis and is a distance L from the center of mass (as stated) and work from there. If it isn't a parallel axis, this probably won't be all that helpful.

 

The moment of inertia for a rotational axis about the center of mass is easy enough to find in this case:

 

I_cm = m(d/2)² + m(d/2)² = (1/2)md²            (4)

 

This is found easily enough by just using the most basic moment of inertia equation. Now we use the parallel axis theorem and (1) to find the total moment of inertia:

 

I = I_cm + ML² = (1/2)md² + 2mL² = (1/2)m(d² + 4L²)         (5)

 

 

2b) Period of oscillation is found using (3). Plug (5) into (3) and you'll have your answer!

 

I hope this helps! Let me know if you need anything else to help clarify these concepts. For reference, I will list the concepts that these questions are testing on:

 

Moments of Inertia

Physical Pendulums

Parallel Axis Theorem

 

I recommend using Hyperphysics for some nice information on these topics!