Modern hard drives spin very fast! The part that holds the data is a disk called the platter. Assume an aluminum platter that is 3.5in in diameter, 1 mm thick, and spinning at 5400 rpm . Treat the platter as a solid disk - don't worry about the hole in the middle.
What is the angular momentum of the platter in kg - m^2/s when it is spinning at full speed?
To calculate the angular momentum of the platter, we need to use the formula for angular momentum:
Angular Momentum (L) = Moment of Inertia (I) * Angular Velocity (ω)
First, let's calculate the moment of inertia (I) for the platter. The moment of inertia for a solid disk is given by the formula:
I = (1/4) * m * r^2
Where m is the mass of the disk, and r is the radius of the disk.
Given:
Radius (r) = (1/2) * diameter = (1/2) * 3.5in = 1.75in
Radius (r) = 1.75in * 0.0254m/in (converting inches to meters) = 0.04445m
Thickness (h) = 1mm = 0.001m
Density of Aluminum (ρ) = 2700 kg/m^3 (approximate value)
The mass (m) of the disk can be calculated using the formula:
m = (ρ * h * A)
Where A is the cross-sectional area.
The cross-sectional area (A) of the disk can be calculated using the formula:
A = π * (r^2 - (r - h)^2)
Calculating:
A = π * (0.044452 - (0.04445 - 0.001)^2) = 0.00307 m^2
Now, calculating the mass (m):
m = (2700 kg/m^3) * (0.001m) * (0.00307 m^2) = 8.301 kg
Now, let's calculate the angular momentum:
Angular Velocity (ω) = 5400 rpm = (2π/60) * 5400 rad/s = 565.49 rad/s
Angular Momentum (L) = I * ω
L = ((1/4) * m * r^2) * ω
L = ((1/4) * 8.301 kg * (0.04445 m)^2) * 565.49 rad/s
Calculating:
L ≈ 0.2428 kg * m^2 / s
Therefore, the angular momentum of the platter when it is spinning at full speed is approximately 0.2428 kg * m^2 / s.