A satellite of mass 1000 kg is tethered to a space station by a 1000-meter-long cable. Calculate the tension in the cable and the satellite's angular velocity when it is released.
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To solve this problem, we can use the principle of conservation of angular momentum.
The equation for angular momentum is given by:
angular momentum = moment of inertia * angular velocity
Since the satellite is initially tethered to the space station, its moment of inertia is calculated with respect to its pivot point, which is at the space station.
The moment of inertia for a satellite rotating about its center of mass is given by the equation:
moment of inertia = M * r^2
Where M is the mass of the satellite and r is the distance between the center of mass of the satellite and the pivot point (in this case, the space station).
Given:
Mass of satellite, M = 1000 kg
Length of cable, r = 1000 m
Using these values, we can calculate the moment of inertia:
moment of inertia = 1000 kg * (1000 m)^2
moment of inertia = 1,000,000,000 kg m^2
Next, we will calculate the angular momentum. Since angular momentum is conserved, it remains constant before and after the satellite is released. Therefore, the angular momentum before release is equal to the angular momentum after release.
The initial angular momentum is given by:
angular momentum_initial = moment of inertia_initial * angular velocity_initial
The final angular momentum is given by:
angular momentum_final = moment of inertia_final * angular velocity_final
Since the cable is released, the final moment of inertia is given by:
moment of inertia_final = M * r^2
Since angular momentum is conserved, we can equate the initial and final angular momenta:
moment of inertia_initial * angular velocity_initial = moment of inertia_final * angular velocity_final
Substituting the given and calculated values:
1,000,000,000 kg m^2 * angular velocity_initial = 1000 kg * (1000 m)^2 * angular velocity_final
Simplifying the equation:
angular velocity_final = (1,000,000,000 kg m^2 * angular velocity_initial) / (1000 kg * (1000 m)^2)
Canceling the units:
angular velocity_final = angular velocity_initial / 1000
Given that the satellite and space station system is initially not rotating (angular velocity_initial = 0), we can solve for the final angular velocity:
angular velocity_final = 0 / 1000
angular velocity_final = 0 rad/s
Therefore, the satellite's final angular velocity when it is released is 0 rad/s.
Now, to calculate the tension in the cable, we need to consider the gravitational force acting on the satellite. The tension in the cable must be equal to the gravitational force for the satellite to remain in equilibrium.
The tension in the cable can be calculated using the equation:
tension = gravitational force
The gravitational force is given by:
gravitational force = mass * gravity
Where the mass is the mass of the satellite and gravity is the acceleration due to gravity.
Given:
Mass of satellite, M = 1000 kg
Acceleration due to gravity, g = 9.8 m/s^2
Using these values, we can calculate the tension in the cable:
tension = 1000 kg * 9.8 m/s^2
tension = 9800 N
Therefore, the tension in the cable is 9800 N.
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