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?

Question

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?

Answer 1

To find the time it takes to complete 8 complete oscillations, we can use the formula:

T = 2π√(I / (mgd))

Where:
T is the period of oscillation,
π is a mathematical constant (approximately 3.14),
I is the moment of inertia of the CD,
m is the mass of the CD,
g is the acceleration due to gravity, and
d is the distance from the axis of rotation to the center of mass.

First, we need to calculate the moment of inertia (I) of the CD. The moment of inertia of a disc about its diameter is given by the formula:

I = (1/4) * m * (r1^2 + r2^2)

Where:
m is the mass of the CD,
r1 is the outer radius of the CD (half of the diameter),
r2 is the inner radius of the CD (half of the hole diameter).

Given that the diameter of the CD is 12.0 cm, the outer radius is 6.0 cm, and the hole diameter is 1.50 cm. Therefore, the inner radius is 0.75 cm.

Now, let's calculate the moment of inertia (I) using the formula:

I = (1/4) * m * (r1^2 + r2^2)
I = (1/4) * 17g * ((6.0cm)^2 + (0.75cm)^2)

Next, we need to convert the units of mass and radius to SI units.

12.0 cm = 0.12 m
1.5 cm = 0.015 m

Now, we can calculate the moment of inertia (I) in SI units:

I = (1/4) * 0.017 kg * ((0.06 m)^2 + (0.0075 m)^2)

Once we have the value of I, we can substitute it into the formula for the period of oscillation (T):

T = 2π√(I / (mgd))

Given that the thickness of the CD is 1.2 mm, we can convert it to meters:

1.2 mm = 0.0012 m

Now, we substitute the values into the formula:

T = 2π√(I / (mgd))
T = 2π√((1/4) * 0.017 kg * ((0.06 m)^2 + (0.0075 m)^2) / ((0.017 kg) * (9.8 m/s^2) * (0.0012 m)))

Finally, we can calculate the value of T:

T = 2π√((1/4) * ((0.06 m)^2 + (0.0075 m)^2) / (9.8 m/s^2 * 0.0012 m))

With the value of T, we can multiply it by 8 to find the time taken for 8 complete oscillations.