A physical pendulum consists of a body of mass contained in the xy-plane. The moment of inertia of the object about an axis perpendicular to the plane and passing through the object's center of mass is I_cm .

The object oscillates in the xy-plane about the point S a distance d from the center of mass as shown. What is the period of the pendulum for small angle oscillations where sin(theta )neraly =theta ?

I assume your mass is m and moment of inertia about the center of mass is Ic

Then the total moment of inertia about the point S is I = (Ic + m d^2)

If the xy plane is vertical and this is a physical pendulum hanging from point S then the gravity force down is mg at the center of mass. At angle theta, which I call Th, this has components mg cos theta along the rod and mg son Theta toward the center. The torque about S is therefore m g d sin Th trying to bring the pendulum back to center
Then m g d sin Th = -I alpha
where alpha is the angular acceleration
If Th = x sin 2 pi f t (harmonic motion)
then V =dTh/dt 2 pi f x cos 2 pi f t
and alpha = dv/dt = -(2 pi f)^2 sin 2 pi f t
which is -(2pif)^2 Th
then
m g d sin Th = I ( 2 pi f)^2 T
now for small angles sin Th = Th in radians so
m g d = I (2 pi f)^2
or
m g d = (Ic + m d^2)( 2 pi f)^2
solve for f, the frequency
then
period = 1/f

Thank you very much Damon

To find the period of the pendulum, we can use the equation for the period of a physical pendulum:

T = 2π√(I_cm / (m * g * d))

where:
- T is the period of the pendulum
- I_cm is the moment of inertia of the object about the axis passing through its center of mass
- m is the mass of the object
- g is the acceleration due to gravity
- d is the distance from the center of mass to the point of oscillation (point S in this case)

Since the pendulum is oscillating in the xy-plane and the angle theta is small, we can approximate sin(theta) as theta. Therefore, we can substitute sin(theta) with theta in the equation:

T = 2π√(I_cm / (m * g * d))

This will give us the period of the pendulum for small angle oscillations.

To find the period of the physical pendulum for small angle oscillations, we can use the formula:

T = 2π√(I_cm / (m * g * d))

Where:
T is the period of the pendulum,
π is a constant approximately equal to 3.14,
I_cm is the moment of inertia of the object about an axis perpendicular to the plane and passing through the object's center of mass,
m is the mass of the object,
g is the acceleration due to gravity, and
d is the distance from the center of mass to the point of oscillation.

Now, let's examine how to obtain the values needed to calculate the period:

1. Moment of Inertia (I_cm): The moment of inertia of the object can be obtained based on its geometry, by using the mass and the distribution of mass around the axis of rotation. The formula for the moment of inertia depends on the specific shape of the object. You would need to use the relevant formula for the shape of the object in order to determine I_cm.

2. Mass (m): The mass of the object can be measured using a balance or a scale. It is typically provided in kilograms (kg).

3. Acceleration due to Gravity (g): On Earth, the acceleration due to gravity is approximately 9.8 m/s^2. However, this value may vary depending on the location. If more precision is required, you can check the specific value for your location.

4. Distance (d): The distance from the center of mass to the point of oscillation can be measured using a ruler or a measuring tape. It is typically provided in meters (m).

Once you have obtained the values for I_cm, m, g, and d, you can substitute them into the period formula mentioned earlier:

T = 2π√(I_cm / (m * g * d))

By performing the calculations, you will find the period (T) of the physical pendulum for small angle oscillations.