In another solar system, a 100,000 kg spaceship and a 200,000 kg spaceship
are connected by a motionless 90 m long tunnel. The rockets start their
engines at the same time, and each produces 50,000 N of thrust in opposing
directions. What is the system's angular velocity after 60 seconds?
To determine the system's angular velocity, we need to apply the principle of conservation of angular momentum.
The angular momentum of the system 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.
In this case, we have two spaceships connected by a tunnel, so the total moment of inertia of the system is the sum of the individual moments of inertia of the spaceships. The moment of inertia for a spaceship rotating about an axis passing through its center of mass is given by the equation:
I = m * r^2
where m is the mass of the spaceship and r is the distance between the spaceship's center of mass and the axis of rotation.
Since both spaceships have the same mass and the tunnel axis is equidistant from the center of mass of each spaceship, the moment of inertia for each spaceship is the same. Let's denote it as I_ship.
So, the total moment of inertia of the system is:
I_total = 2 * I_ship
Next, we need to calculate the moment of inertia for one spaceship. Given that the spaceships are connected by a motionless tunnel, we can consider the system as a rigid body. The moment of inertia for a rigid body rotating about an axis through one end perpendicular to its length is given by the equation:
I = 1/3 * m * L^2
where m is the mass of the rigid body and L is its length.
In this case, we have the same mass for each spaceship, so the moment of inertia for one spaceship is:
I_ship = 1/3 * m * L^2
Now, we can calculate the total moment of inertia of the system:
I_total = 2 * I_ship = 2 * (1/3 * m * L^2)
Given the information in the question, the mass of each spaceship is 100,000 kg and the length of the tunnel is 90 m. Thus:
I_total = 2 * (1/3 * 100,000 kg * (90 m)^2)
Next, we need to calculate the angular momentum. The angular momentum of the system is conserved, which means the initial angular momentum is equal to the final angular momentum. Before the rockets start their engines, the system is at rest, so the initial angular momentum is zero.
L_initial = 0
After the rockets start their engines, each produces a thrust of 50,000 N in opposing directions. Since the tunnel is motionless, the spaceships will experience equal and opposite torques, resulting in a net torque of zero, which means the angular momentum is conserved.
So, the final angular momentum is also zero:
L_final = 0
Now, we can solve for the angular velocity:
0 = I_total * ω
Simplifying the equation:
0 = 2 * (1/3 * 100,000 kg * (90 m)^2) * ω
Simplifying further:
0 = 2 * (1/3 * 100,000 kg * 8100 m^2) * ω
0 = (2 * 1/3 * 100,000 kg * 8100 m^2) * ω
0 = (2/3 * 100,000 kg * 8100 m^2) * ω
0 = (2/3 * 100,000 kg * 8100 m^2) * ω
Rearranging the equation to solve for ω:
ω = 0 / (2/3 * 100,000 kg * 8100 m^2)
ω = 0
Therefore, the system's angular velocity after 60 seconds is 0.