Question

To get a flat, uniform cylindrical satellite spinning at the correct rate, engineers fire four tangential rockets, as shown in Fig. 8–50. 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 32 rpm in 5.0 min, starting from rest?

FIGURE 8-50

Problem 45

Short answer

The required force by each rocket is 31 N.

Step-by-step solution

Concepts

Torque is the product of the moment of inertia and the square of the distance. For this problem, find the total moment of inertia and the angular acceleration to find the net required torque.

Given data

The mass of the satellite is \(M = 3600\;{\rm{kg}}\).

The radius of the satellite is \(R = 4.0\;{\rm{m}}\).

The mass of each rocket is \(m = 250\;{\rm{kg}}\).

The initial angular velocity of the satellite is \({\omega _1} = 0\).

The final angular velocity is \({\omega _2} = 32\;{\rm{rpm}} = \frac{{32 \times 2\pi }}{{60}}\;{\rm{rad/s}}\).

The satellite takes \(t = 5.0\;\min = 5.0 \times 60\;{\rm{s}}\) to reach the final angular velocity.

You can assume that the satellite is a solid cylinder.

Let F be the required force by each rocket to get the final angular velocity.

You can assume the rocket as point masses at the edge of the satellite.

Calculation 

The total moment of inertia of the satellite and rocket system is

\(\begin{align}I &= \frac{1}{2}M{R^2} + \left\{ {4 \times \left( {m{R^2}} \right)} \right\}\\ &= \left( {\frac{M}{2} + 4m} \right){R^2}\end{align}\)

The angular acceleration of the satellite is \(\alpha = \frac{{{\omega _2} - {\omega _1}}}{t}\).\(\)

\(\begin{align}\tau &= I\alpha \\\tau &= \left( {\frac{M}{2} + 4m} \right){R^2} \times \frac{{{\omega _2} - {\omega _1}}}{t}\end{align}\).

Now, the torque due to the four rockets at the edge of the satellite is \(\tau = 4FR\).

Comparing the values of the torque,

\(\begin{align}4FR &= \left( {\frac{M}{2} + 4m} \right){R^2} \times \frac{{{\omega _2} - {\omega _1}}}{t}\\F &= \frac{R}{4}\left( {\frac{M}{2} + 4m} \right) \times \frac{{{\omega _2} - {\omega _1}}}{t}\\ &= \left( {\frac{{4.0\;{\rm{m}}}}{4}} \right) \times \left\{ {\frac{{3600\;{\rm{kg}}}}{2} + \left( {4 \times 250\;{\rm{kg}}} \right)} \right\} \times \frac{{\frac{{32 \times 2\pi }}{{60}}\;{\rm{rad/s}} - 0}}{{5.0 \times 60\;{\rm{s}}}}\\ &= 31.26\;{\rm{N}} \approx 31\;{\rm{N}}\end{align}\).

Hence, the required force by each rocket is 31 N.