To get a flat, uniform cylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 10–61. Suppose that the satellite has a mass of 3600 kg and a radius of 4.0 m, and that the rockets each add a mass of 250 kg. What is the steady force required of each rocket if the satellite is to reach 28 rpm in 5.0 min, starting from rest?

Verified step by step guidance

1. Determine the moment of inertia of the satellite. Since the satellite is a flat, uniform cylinder, its moment of inertia about its central axis is given by the formula: $I=\frac{1}{2}M{R}^2$, where $M$ is the total mass of the satellite (including the rockets) and $R$ is its radius.
2. Convert the final angular velocity from rpm to rad/s. The formula for this conversion is: $\omega =\frac{2}{\pi }\times \frac{\mathrm{rpm}}{60}$. Use this to find the angular velocity $\omega$ in rad/s.
3. Calculate the angular acceleration $\alpha$ using the formula: $\alpha =\frac{\omega }{t}$, where $t$ is the time in seconds (convert 5.0 minutes to seconds).
4. Determine the total torque required to achieve the angular acceleration using the formula: $\tau =I\alpha$, where $I$ is the moment of inertia and $\alpha$ is the angular acceleration.
5. Find the force required by each rocket. Since the torque is generated by four tangential rockets, the torque due to one rocket is $\tau =FR$. Rearrange this formula to solve for the force $F$ exerted by each rocket: $F=\frac{\tau }{R}$.

Key Concepts

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, typically expressed in radians per second or revolutions per minute (rpm). In this question, the satellite needs to achieve a specific angular velocity of 28 rpm, which will determine the necessary forces applied by the rockets to reach that speed within a given time frame.

Newton's Second Law of Motion

Newton's Second Law states that the force acting on an object is equal to the mass of that object multiplied by its acceleration (F = ma). In the context of the satellite, the engineers must calculate the net force required to accelerate the satellite to the desired angular velocity, taking into account both the satellite's mass and the additional mass from the rockets.

Torque

Torque is the rotational equivalent of linear force, representing the tendency of a force to rotate an object about an axis. The rockets must generate sufficient torque to overcome the inertia of the satellite and achieve the desired angular velocity. The torque produced by each rocket can be calculated based on the force exerted and the distance from the axis of rotation.