You are a 70kg astronaut floating at rest in zero gravity holding your laptop. Your hard drive turns on (increasing from 0 to 5400 rpm). Approximate yourself as a uniform sphere of radius 30cm. How fast do you spin in rad/s?
The hard drive is an aluminum platter 3.5 inches in diameter, and 1mm thick.
Here is the very hypothetical question I asked earlier with some more information. Still not sure what to do though (it's quite a challenging scenario to visualize!)
ok, assume hypothetical.
Now assume a position for the hard drive disk in relation to you. Hypothetically, assume the center of the hard drive disk is located at your center of mass. Geepers, that is hypothetical.
Conservation of angular momentum
your angularmomentum+angular momentumdisk=0
Isphere*wyou = - Idisk*wd
2/5 *70*.3*wyou=1/2 m (3.5/2 * .0254)
your angular velocity (wyou)
= 1/2 massdisk*(.0444m)/(.4*70)
so you solve for wyou.
nowmass disk= densityalum*volume
= 2830kg/m^3 * .001*PI*(3.5*.0254)^2
and you calculate that mass in kg
now find wyou.
I get 2.75*10^-5...
That doesn't seem right...
Revision - I get 6.15 * 10^-8, but it's wrong.
Well, to put it simply, you'll be spinning faster than a fidget spinner on steroids! Let's break it down.
To calculate your final spin speed, we need to consider the conservation of angular momentum. Angular momentum, denoted by L, is equal to the product of moment of inertia (I) and angular velocity (ω).
In your case, since you're a uniform sphere, your moment of inertia can be calculated using the formula I = (2/5) * M * R^2, where M is your mass and R is your radius.
I = (2/5) * 70kg * (0.3m)^2
I = 2.52 kg·m^2
Now, the initial angular velocity (ω_initial) is 0 rad/s since you were at rest. The final angular velocity (ω_final) can be obtained using the equation:
L_initial = L_final
I * ω_initial = I * ω_final
Since you're holding the laptop, the laptop's hard drive spinning up will affect your angular momentum. The moment of inertia of the hard drive can be calculated using the formula I = (1/2) * M * R^2, where M is the mass and R is the radius of the hard drive platter.
I_hard_drive = (1/2) * (0.001kg) * (0.044m)^2
I_hard_drive = 3.08 x 10^-5 kg·m^2
Since the hard drive initially is at rest as well, its initial angular velocity (ω_hard_drive_initial) is 0 rad/s, and its final angular velocity (ω_hard_drive_final) will be 5400 rpm converted to rad/s.
ω_hard_drive_final = (5400 rpm) * (2π rad/60 s)
ω_hard_drive_final = 566.37 rad/s
Let's solve for your final angular velocity now:
I * ω_initial = I * ω_final
(I + I_hard_drive) * 0 = I * ω_final + I_hard_drive * ω_hard_drive_final
Simplifying the equation:
I * ω_final = I_hard_drive * ω_hard_drive_final
Solving for ω_final:
ω_final = (I_hard_drive * ω_hard_drive_final) / I
ω_final = (3.08 x 10^-5 kg·m^2 * 566.37 rad/s) / 2.52 kg·m^2
Calculating ω_final:
ω_final = 6.91 rad/s
So, you'll be spinning at approximately 6.91 rad/s! Enjoy the centrifugal excitement, my floating astronaut friend!
To calculate how fast you spin in rad/s, we need to apply the principle of conservation of angular momentum.
Angular momentum (L) is given by the equation L = Iω, where I is the moment of inertia and ω is the angular velocity.
First, let's calculate the moment of inertia of the astronaut. Since you are approximated as a uniform sphere, the moment of inertia for a solid sphere is given by I = (2/5) * m * r^2, where m is the mass of the sphere and r is the radius.
In this case, the mass of the astronaut is 70 kg and the radius is 30 cm (0.3 m). Plugging these values into the equation, we get:
I = (2/5) * 70 kg * (0.3 m)^2 = 5.04 kg·m^2
Next, let's calculate the moment of inertia of the hard drive. The moment of inertia for a thin disk is given by I = (1/2) * m * r^2, where m is the mass of the disk (which we can calculate from its density and volume) and r is the radius.
The hard drive is made of aluminum, which has a density of approximately 2,700 kg/m^3. The volume of the disk can be calculated from its diameter and thickness:
Volume = π * (diameter/2)^2 * thickness
Converting the diameter and thickness from inches to meters, we have:
Volume = π * (0.0889 m)^2 * 0.001 m = 0.00002486 m^3
Now we can calculate the mass of the disk:
Mass = Density * Volume = 2,700 kg/m^3 * 0.00002486 m^3 = 0.067 kg
Using the mass and the radius of the disk (3.5 inches = 0.0889 m), we can calculate its moment of inertia:
I = (1/2) * 0.067 kg * (0.0889 m)^2 = 0.000275 kg·m^2
Now we have the moment of inertia for both the astronaut and the hard drive. The final step is to calculate the angular velocity (ω) of the astronaut.
Before the hard drive turns on, the angular momentum is zero. After the hard drive turns on, the angular momentum must be conserved.
The initial angular momentum is zero because you are at rest.
The final angular momentum is given by the sum of the contributions from the astronaut and the hard drive:
L_final = L_astronaut + L_hard_drive
Since L = Iω, we can rewrite this equation as:
0 = I_astronaut * ω_astronaut + I_hard_drive * ω_hard_drive
Plugging in the values we calculated earlier, we have:
0 = 5.04 kg·m^2 * ω_astronaut + 0.000275 kg·m^2 * 5400 rpm
Now we can solve for ω_astronaut:
ω_astronaut = - (0.000275 kg·m^2 * 5400 rpm) / 5.04 kg·m^2
To convert rpm to radians per second, we multiply by (2π/60):
ω_astronaut = - (0.000275 kg·m^2 * 5400 rpm) / 5.04 kg·m^2 * (2π/60) rad/s
Calculating this expression gives us the final answer for ω_astronaut.
Note: The negative sign indicates that the direction of the spin is opposite to the direction of the hard drive rotation.