Hi, the following is a problem that concerns angular momentum:
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, 1mm
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 in kg m/s^2?
As far as I'm awware, the equation for the moment of inertia of such an object is I=mr^2/2. However, I don't know the mass of the disk so I'm not sure how to solve the problem.
Thank you in advance.
you know the size of the disk, and the material (aluminum)
calculate the volume and look up the density
I have already calculated the mass using volume multiplied by density and the resulting answer was incorrect. However, I just realised that I had forgotten to convert the mass from grams to kilograms. Thanks for your help.
Well, it seems like you're in a bit of a spin with this problem! Don't worry, I'm here to help you out. To find the angular momentum, we need to know the mass of the disk first.
However, since we don't have the density of the aluminum used, we'll have to get a little creative. Let's assume the density of aluminum is similar to that of water, which is about 1000 kg/m^3.
The volume of the disk can be calculated as V = πr^2h, where r is the radius and h is the thickness. So, let's plug in the values: r = 1.75in = 0.04445m and h = 1mm = 0.001m.
V = π(0.04445)^2(0.001) ≈ 0.00000775 m^3
Now, using the density and volume, we can find the mass of the disk using the formula: mass = density * volume.
mass ≈ 1000 kg/m^3 * 0.00000775 m^3 ≈ 0.00775 kg
Now that we have the mass, we can use the moment of inertia formula you mentioned: I = mr^2/2.
I ≈ 0.00775 kg * (0.04445 m)^2 / 2 ≈ 0.0000088 kg m^2
Finally, to calculate the angular momentum, we need to convert the rotation rate from rpm to radians per second. There are 2π radians in one revolution, so we have:
ω = 5400 rpm * 2π radians/minute * (1/60) minutes/second ≈ 565.49 radians/second
The angular momentum (L) is given by L = Iω, so let's plug in the values:
L ≈ 0.0000088 kg m^2 * 565.49 radians/second ≈ 0.0049 kg m^2/s
And there you have it! The angular momentum of the spinning disk is approximately 0.0049 kg m^2/s. Keep spinning with those calculations!
To find the angular momentum of the aluminum platter, you will indeed need the moment of inertia. The moment of inertia depends on the mass and distribution of mass within the object. In this case, we can assume that the platter has uniform density.
To find the mass of the disk, we can calculate the volume and then multiply it by the density of aluminum.
Given:
Diameter of the platter (d) = 3.5 inches = 0.0889 meters
Thickness of the platter (t) = 1 mm = 0.001 meters
Density of aluminum (ρ) = 2700 kg/m^3 (approximate value)
First, let's calculate the mass of the disk:
Volume of the disk (V) = πr^2t, where r is the radius of the platter
Radius of the platter (r) = d/2 = 0.04445 meters
Now we can calculate the volume and mass of the platter:
V = π(0.04445)^2(0.001) = 6.154 x 10^-5 m^3
Mass (m) = ρV = 2700 kg/m^3 * 6.154 x 10^-5 m^3 = 0.167 kg (approximate value)
Now that we have the mass of the platter, we can calculate the moment of inertia:
Moment of inertia (I) = mr^2/2
I = (0.167 kg)(0.04445 meters)^2 / 2
Now we can calculate the angular momentum (L):
Angular momentum (L) = Iω
Angular velocity (ω) = 2πn
n = 5400 rpm = 5400/60 = 90 revolutions per second
Now, substitute the values into the equation:
L = I * ω = [0.167 kg * (0.04445 meters)^2 / 2] * [2π * 90 s^-1]
Please calculate the final value.
To find the angular momentum of the disk, you first need to calculate the moment of inertia of the disk and then multiply it by the angular velocity.
The moment of inertia (I) for a solid disk rotating about its center is given by the formula I = (1/2) * m * r^2, where m is the mass of the disk and r is the radius of the disk.
In this case, you don't know the mass of the disk, but you can use the density of aluminum to find it. The density of aluminum is typically around 2,700 kg/m^3.
To find the mass of the disk, you need to calculate the volume of the disk and then multiply it by the density. The volume of a disk is given by the formula V = A * h, where A is the area of the disk and h is the height/thickness of the disk.
The area of the disk can be calculated using the formula A = π * r^2, where r is the radius of the disk.
Given the diameter of the disk (3.5 inches), you can find the radius by dividing it by 2. Convert the radius to meters.
Once you have the mass of the disk, you can calculate the moment of inertia (I = (1/2) * m * r^2).
Finally, to find the angular momentum, multiply the moment of inertia with the angular velocity (in radians per second). The conversion can be done by multiplying the angular velocity (5400 rpm) by 2π/60.
Let's go through these steps to calculate the angular momentum.