A 500-kg disk-shaped nonrotating satellite gets hit by a 5-g piece of space debris moving at 1 km/s. If the debris buries itself near the satellite’s rim, how rapidly will the satellite rotate after the hit?
To determine the rotational speed of the satellite after the hit, we need to apply the principle of conservation of angular momentum. Angular momentum is defined as the product of moment of inertia and angular velocity.
The moment of inertia, denoted by I, is a property that depends on the mass and shape of an object. For a disk-shaped satellite, the moment of inertia can be calculated using the equation:
I = (1/2) * m * r^2
Where:
- I is the moment of inertia
- m is the mass of the satellite (given as 500 kg)
- r is the radius of the satellite
The mass of the debris is given as 5 g (0.005 kg), and it buries itself near the satellite's rim. This means that it can be treated as an additional mass added to the satellite at the edge. Thus, the total mass of the satellite after the impact is 500 kg + 0.005 kg = 500.005 kg.
The velocity of the debris is given as 1 km/s. Since it buries itself near the rim, we can assume that the satellite initially has zero angular velocity.
To calculate the final angular velocity, we can set the initial and final angular momenta equal to each other:
(Initial Angular Momentum) = (Final Angular Momentum)
Initially, the angular momentum is zero since the satellite is not rotating:
0 = I_initial * 0
After the impact, the angular momentum is given by:
(Final Angular Momentum) = I_final * ω_final
Where:
- ω_final is the final angular velocity
Substituting the values, we have:
0 = (1/2) * m * r^2 * 0 + (1/2) * (500.005 kg) * r^2 * ω_final
Simplifying the equation:
0 = 0 + (1/2) * (500.005) * r^2 * ω_final
Simplifying further:
0 = 250.0025 * r^2 * ω_final
Since we want to find the final angular velocity, we can solve the equation for ω_final:
ω_final = 0
Therefore, the final angular velocity of the satellite after the impact is zero. This means that the satellite does not rotate.