Question

To get a flat, uniform cylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in the figure (Figure 1) . Suppose that the satellite has a mass of 4100kg and a radius of 4.9 m, and that the rockets each add a mass of 250 kg.

GIANCOLI.ch08.p33.jpg

What is the steady force required of each rocket if the satellite is to reach 47 rpm in 6.0 min, starting from

rest?

Answer 1

Given data

Angular velocity \(\omega=47 \mathrm{rpm}\) Time \(t=6 \mathrm{~min}\)

Mass of the satellite \(M_{s}=4100 \mathrm{~kg}\)

Mass of the rocket \(M_{r}=250 \mathrm{~kg}\)

Radius of the satellite \(r=4.9 \mathrm{~m}\)

Calculate the angular velocity in \(\mathrm{rad} / \mathrm{s}\) \(\omega=47 \times\left(\frac{\pi}{30}\right)\)

\(\omega=4.92 \mathrm{rad} / \mathrm{s}\)

Calculate the angular acceleration of the satellite \(\omega=\omega_{0}+\alpha t\)

Substitute the value

$$ \begin{aligned} 4.92 &=0+\alpha \times(6 \times 60) \\ \alpha &=\frac{4.92}{360} \\ \alpha &=1.36 \times 10^{-2} \mathrm{rad} / \mathrm{s}^{2} \end{aligned} $$

Calculate the total moment of inertia \(I=I_{s}+I_{r}\)

Here, \(I_{s}\) is moment of inertia of satellite and \(I_{r}\) is moment of inertia of rocket. \(I=\left(\frac{1}{2} M_{s} r^{2}\right)+4\left(M, r^{2}\right)\)

Substitute the value \(I=\left(\frac{1}{2} \times 4100 \times 4.9^{2}\right)+4\left(250 \times 4.9^{2}\right)\)

\(I=49220.5+24010\)

\(I=73230.5 \mathrm{~kg} \cdot \mathrm{m}^{2}\)

Calculate the torque \(\tau=I \alpha\)

Substitute the value \(\tau=73230.5 \times 1.36 \times 10^{-2}\)

\(\tau=995.93 \mathrm{Nm}\)

Hence the torque is \(995.93 \mathrm{Nm}\)

Calculate the force required for each rocket \(\tau=4 F r\)

Substitute the value \(995.93=4 \times F \times 4.9\)

\(F=\frac{995.93}{19.6}\)

\(F=50.81 \mathrm{~N}\)

Hence the force required for each rocket is \(50.81 \mathrm{~N}\)

Answer 2

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