a physical pendulum consists of a uniform solid disk (of radius R = 42.0 cm) supported in a vertical plane by a pivot located a distance d = 13.0 cm from the center of the disk. The disk is displaced by a small angle and released. What is the period of the resulting simple harmonic motion?

T = 2π√(I/mgr)

where I is the moment of inertia of the disk, m is the mass of the disk, g is the acceleration due to gravity, and r is the distance from the pivot to the center of mass of the disk.

I = (1/2)mR^2

m = (ρπR^2)

where ρ is the density of the disk.

r = d + R

T = 2π√((1/2)mR^2/(ρπR^2)gr)

T = 2π√((1/2)(d + R)/(ρπg))

T = 2π√((d + R)/(2ρπg))

T = 2π√((13.0 cm + 42.0 cm)/(2(1000 kg/m^3)π(9.8 m/s^2)))

T = 2.45 s

To find the period of the resulting simple harmonic motion for a physical pendulum, we can use the formula:

T = 2π * √(I / (m * g * d))

Where:
T = Period of the pendulum
π = Pi (approximately 3.14159)
I = Moment of inertia of the pendulum about the pivot point
m = Mass of the pendulum
g = Acceleration due to gravity
d = Distance from the pivot point to the center of mass of the pendulum

In this case, since we have a uniform solid disk, the moment of inertia can be calculated as:

I = (1/2) * m * R^2

Where:
R = Radius of the disk

Now, let's plug in the values given in the problem statement:

R = 42.0 cm = 0.42 m
d = 13.0 cm = 0.13 m
m: The mass of the disk is not given in the problem statement, so we cannot calculate it directly. If you have the mass, please provide it so we can continue with the calculation.
g = 9.8 m/s^2 (standard acceleration due to gravity)

Please provide the mass of the disk so we can continue the calculation.

To find the period of the resulting simple harmonic motion of a physical pendulum, we can use the formula:

T = 2π * sqrt(I / (m * g * d))

Where:
T is the period of the motion,
π is a constant approximately equal to 3.14159,
sqrt() represents the square root,
I is the moment of inertia of the disk,
m is the mass of the disk,
g is the acceleration due to gravity, and
d is the distance between the pivot and the center of mass of the disk.

We are given the radius of the disk, R = 42.0 cm. The moment of inertia of a solid disk is given by the equation:

I = (1/2) * m * R^2

Since the disk is uniform, the mass is distributed uniformly. The total mass of the disk can be calculated using the formula:

m = π * R^2 * ρ

Where:
ρ is the density of the disk.

Now we can calculate the period of the pendulum:

1. Calculate the moment of inertia I:
I = (1/2) * m * R^2

2. Calculate the mass m:
m = π * R^2 * ρ

3. Calculate the period T:
T = 2π * sqrt(I / (m * g * d))

By substituting the given values for R = 42.0 cm and d = 13.0 cm, and assuming a known value for ρ and g, we can calculate the period T of the pendulum.