You are a 70 kg astronaut floating at rest in zero gravity holding your laptop. Your hard drive turns on (increasing from 0 to 5400. How fast do you spin in rad/s ? Approximate yourself as a uniform sphere of radius 30 cm - I know you've always wanted to do that.
To determine the rotational speed, we will use the principle of conservation of angular momentum. The angular momentum is given by the equation:
L = Iω
Where:
L is the angular momentum,
I is the moment of inertia, and
ω is the angular velocity.
The moment of inertia for a uniform sphere is given by:
I = (2/5) * m * r^2
Where:
m is the mass of the sphere, and
r is the radius of the sphere.
Given:
Mass of the astronaut (m) = 70 kg
Radius of the sphere (r) = 30 cm = 0.3 m
Using these values, we can calculate the moment of inertia (I) as follows:
I = (2/5) * 70 kg * (0.3 m)^2
I = 1.68 kg⋅m^2
Now, we can calculate the initial angular momentum (L_initial) when the astronaut is at rest:
L_initial = I * ω_initial
Since the astronaut is at rest, the initial angular velocity (ω_initial) is 0 rad/s. Hence, the initial angular momentum (L_initial) is also 0 kg⋅m^2/s.
The final angular momentum (L_final) with the turned-on hard drive can be calculated as:
L_final = I * ω_final
We are given:
Change in hard drive's angular momentum = 5400 kg⋅m^2/s
Hence, the final angular momentum is also 5400 kg⋅m^2/s.
Using the conservation of angular momentum, we can equate the initial and final angular momenta:
L_initial = L_final
0 kg⋅m^2/s = 5400 kg⋅m^2/s
Solving this equation, we find that ω_final (angular velocity) is undefined or infinite. This means that the astronaut would experience an infinite rotational speed when the hard drive turns on.
However, note that in reality, other factors such as friction and external torques would act on the system, preventing the astronaut from rotating infinitely fast.