Question

A satellite in the shape of a sphere of mass 20,000 kg and radius 5.0 m is spinning about an axis through its center of mass. It has a rotation rate of 8.0 rev/s. Two antennas deploy in the plane of rotation extending from the center of mass of the satellite. Each antenna can be approximated as a rod has mass 200.0 kg and length 7.0 m. What is the new rotation rate of the satellite?

a) 7.7 rev/s
b) 8.0 rev/s
c) 8.3 rev/s
d) 8.6 rev/s

Answer 1

To find the new rotation rate of the satellite, we need to consider the principle of conservation of angular momentum. The initial angular momentum of the satellite is equal to the final angular momentum. The moment of inertia of the satellite with antennas can be calculated using the formula for the moment of inertia of a rod rotating about its center of mass.

Step-by-step explanation:

To find the new rotation rate of the satellite, we need to consider the principle of conservation of angular momentum. The initial angular momentum of the satellite is equal to the final angular momentum. The initial angular momentum is given by the formula:

Li = I1ω1

Where Li is the initial angular momentum, I1 is the moment of inertia of the satellite, and ω1 is the initial rotation rate. Since the antennas are deployed in the plane of rotation and have mass, they will increase the moment of inertia of the satellite. The moment of inertia of the satellite with antennas can be calculated as:

I2 = I1 + Iant

Where I2 is the moment of inertia of the satellite with antennas and Iant is the moment of inertia of the antennas. The moment of inertia of a rod rotating about its center of mass is given by:

Iant = (1/3)mL2

Where m is the mass of the antennas and L is the length of the antennas.

Substituting the given values, we can calculate the new moment of inertia:

I2 = I1 + (1/3)mL2

Finally, the new rotation rate of the satellite is given by:

ω2 = Li/I2

Substituting the given values, we can solve for ω2 to find the new rotation rate of the satellite.