A standard CD has a diameter of 12.0 cm, and a hole that is centered on the axis of symmetry and has a diameter of 1.50 cm. The CD’s thickness is 1.2 mm, and it's mass is 17 g. If a (very thin) pin extending through the hole of the CD suspends it in a vertical orientation, the CD may oscillate about an axis parallel to the pin, rocking back and forth. How long does it take to complete 8 complete oscillations?

To find the time it takes to complete 8 complete oscillations, we need to determine the period of oscillation for the CD. The period, T, is the time it takes for the CD to complete one full oscillation.

We can use the formula for the period of a physical pendulum to calculate T:

T = 2π * √(I / (m * g * d))

Where:
- T is the period of oscillation
- π is a mathematical constant approximately equal to 3.14159
- I is the moment of inertia of the CD
- m is the mass of the CD
- g is the acceleration due to gravity
- d is the distance from the center of mass of the CD to the axis of rotation

To find the moment of inertia, I, we can use the formula:

I = 0.5 * m * R^2 + m * d^2

Where:
- I is the moment of inertia
- m is the mass of the CD
- R is the radius of the CD (diameter divided by 2)
- d is the distance from the center of mass of the CD to the axis of rotation

Given:
- Diameter of the CD, D = 12.0 cm
- Diameter of the hole, d_hole = 1.50 cm
- Radius of the CD, R = D / 2 = 12.0 cm / 2 = 6.0 cm = 0.06 m
- Thickness of the CD, h = 1.2 mm = 0.0012 m
- Mass of the CD, m = 17 g = 0.017 kg
- Acceleration due to gravity, g ≈ 9.8 m/s^2

Let's calculate the moment of inertia, I:

I = 0.5 * 0.017 kg * (0.06 m)^2 + 0.017 kg * (0.06 m - 0.0075 m)^2

I = 0.01242 kg*m^2

Now, we can calculate the period of oscillation, T:

T = 2π * √(0.01242 kg*m^2 / (0.017 kg * 9.8 m/s^2 * 0.0075 m))

T ≈ 1.299 seconds

Since 8 complete oscillations are required, we multiply the period by 8:

Total Time = T * 8

Total Time ≈ 1.299 seconds * 8

Total Time ≈ 10.392 seconds

Therefore, it takes approximately 10.392 seconds to complete 8 complete oscillations.