Artificial Gravity on a Spinning Spaceship - Find the Period

[kchurchi]

Homework Statement

Spinning Space Ship
One way to provide artificial gravity (i.e., a feeling of weight) on long space voyages is to separate a spacecraft into two parts at the ends of a long cable, and set them rotating around each other. A craft has been separated into two parts with a mass of 86000 kg each, at the ends of a cable with their centers of mass 106 m apart, rotating around the center point of the cable with a period of 232.2 seconds.

If the cable is reeled in so that the the centers of the two pieces are now only 74.2 m apart, what will the new period be?

Homework Equations

T = 1/f or f = 1/T
ω = 2*pi*f
L = I*ω
I = Ʃ m*r^2
L(initial) = L(final)

The Attempt at a Solution

First, I noted that this scenario should follow the principle of conservation of angualr momentum. Therefore L(initial) = L(final) should be true. Then, I followed this series of steps...

L = I*ω
L = (Ʃ m*r^2)*ω
L = (2*(m*r^2))*ω
L = (2*(m*r^2))*2*pi*f
L = (2*(m*r^2))*2*pi*1/T

Upon finding this initial angular momentum, I thought I could find the final L by setting this expression equal to L(final) with the new radius to the axis point and with T set as an unknown variable.

Please let me know if this is on the right track? I am a little bit hesitant about it so any feedback would be awesome.

 

[BruceW]

it looks good to me, keep on with it :)

 

[kchurchi]

Thanks Batman! (Haha, referring to your username, Bruce W.). I was wondering if you could help me with the second part of the question. I am stumped.

The two pieces of the spacecraft are returned to their initial configuration with a period of 232.2 seconds. A radiation leak is detected in one of the engines. An escape pod from each section of the spacecraft are ejected as indicated in the figure. Notice that the direction of rotation for the two pieces of the spacecraft is also indicated in the figure.
(One of the pods is rotating into the page and the other is rotating out of the page).

Escape Pods
The escape pods each have a mass of 12000 kg and are ejected with a speed of 77.99 m/s along thedirections indicated in the figure with θ = 32.5 degrees.
What is the new period of the remaining spacecraft after the two escape pods are ejected?

I was initially just trying to use conservation of angular momentum to solve this problem (as I did in the first part). However, I got confused because they provide us with a speed and a direction for the escape pods?

 

[marie24607]

I have this same problem and can't find the new period of the remaining spacecraft s after the escape pods are ejected. Did you ever figure out how to do this?

 

[Nickie]

Your approach is on the right track, however there are a few things to consider. First, the principle of conservation of angular momentum states that the total angular momentum of a system remains constant unless acted upon by an external torque. In this case, the system is not experiencing any external torques, so the initial and final angular momentum should be equal.

Second, the equation for angular momentum is L = I*ω, where I is the moment of inertia and ω is the angular velocity. In this scenario, the moment of inertia is not changing, so we can set the initial and final moments of inertia equal to each other.

Finally, we can use the equation T = 2π/ω to find the period, where ω is the angular velocity. Since the initial and final angular velocities are equal (due to conservation of angular momentum), we can set the initial and final periods equal to each other.

Therefore, the steps to solve this problem would be:

1. Calculate the initial angular momentum using the equation L = I*ω, where I is the moment of inertia and ω is the initial angular velocity.

2. Set the initial and final angular momenta equal to each other and solve for the final angular velocity.

3. Use the equation T = 2π/ω to find the final period, where ω is the final angular velocity.

I hope this helps!