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?

sorry, I need a picture.

there isn't a picture

ok, well if I have it right the pin is at the edge of the hole, 0.75 cm above the center of gravity of the CD

for a small angle A the restoring moment about that pin is
-m g (.0075)sin A = -.0075 m g A
= -.0075*.017*9.81 A = -.00125 A

the moment of inertia of the CD about its center is
(1/2) m r^2 - moment of inertia of hole
= (1/2) .017(.06^2 + .0075^2)
= .0085 (.00366) = 3.11*10^-7
use parallel axis to move that up to the pin
I = 3.11*10^-7 + .017(.0075)^2
= 3.11*10^-7 + 9.56*10^-7
= 12.67*10^-6

moment = I d^2 A/dt^2
-.00125 A = 12.67*10^-6 d^2A/dt^2
if A = sin 2 pi t/T
then d^2A/dt^2 = - (2 pi/T)^2 A
so
.00125 = 12.67*10^-6 (2 pi/T)^2

(2 pi/T)^2 = 9.86*10^1 = 98.6

2 pi/T = 9.93
T = .633 seconds
times 8 = 5 seconds

better get a CD and try it :)

To find how long it takes for the CD to complete 8 complete oscillations, we need to calculate the period of one oscillation and then multiply it by 8.

The period of an oscillation can be found using the formula:

T = 2π√(I/mg)

Where:
T = Period of one oscillation
π = Pi, approximately 3.14159
I = Moment of inertia
m = Mass of the object
g = Acceleration due to gravity

Let's calculate the different values step by step:

1. Determine the moment of inertia (I):
The moment of inertia of a thin disc rotating about its axis of symmetry can be given by:

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

Where:
m = Mass of the CD
r = Radius of the CD

The radius can be calculated by dividing the diameter by 2:

r = 12.0 cm / 2 = 6.0 cm = 0.06 m

Now substitute the values:

I = (1/2) * 0.017 kg * (0.06 m)^2
I = 0.000036 kg * m^2

2. Calculate the period of one oscillation (T):
Now we can plug in the values we have into the formula:

T = 2π√(I/mg)
T = 2π√(0.000036 kg * m^2 / (0.017 kg * 9.8 m/s^2))
T = 2π√(0.0021176 s^2)
T = 2π * 0.04603 s
T ≈ 0.2895 s

3. Multiply the period by 8:
To find the time for 8 complete oscillations, simply multiply the period (T) by 8:

Time = T * 8
Time ≈ 0.2895 s * 8
Time ≈ 2.316 s

Therefore, it takes approximately 2.316 seconds for the CD to complete 8 oscillations.