Example 1.3 of Chapter 1 proposed that mined material on the moon might be projected off the moon's surface by a rotating boom that slings the material into space. Assume the boom rotates in a horizontal plane with constant angular velocity \(\omega\), and let \(r\), the distance of the payload from the rotation axis at one end of the boom, be the single generalized coordinate.

(a) Find the Lagrangian for a bucket of material of mass \(m\) that moves along the boom.

(b) Find its equation of motion.

(c) Solve it for \(r(t)\), subject to the initial conditions \(r=r_{0}\) and \(\dot{r}=0\) at \(t=0\).

(d) If the boom has length \(R\), find the radial and tangential components of the bucket's velocity, and its total speed, as it emerges from the end of the boom.

(e) Find the power input \(P=d E / d t\) into a bucket of mass \(m\) as a function of time. Is the power input larger at the beginning or end of the bucket's journey along the boom?

(f) Find the torque exerted by the boom on the bucket, as a function of the position \(r\) of the bucket on the boom. There would be an equal but opposite torque back on the boom, caused by the bucket, which might break the boom. At what part of the bucket's journey would this torque most likely break the boom? 

(g) If \(R=100\) meters and \(r_{0}=1\) meter, what must be the rotational period of the boom so that buckets will reach the moon's escape speed as they fly off the boom?

Data from Example 1.3

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Someday we may want to construct spacecraft or space colonies not on the earth or the moon but in space itself, using mined metals and other materials lifted off the moon. The moon has the advantage of a much smaller escape velocity than that of the earth, and no atmosphere to retard motion. Instead of using expensive rockets and fuel, could it be possible to achieve the escape velocity from the airless moon by slinging containers of material from its surface using a rapidly rotating boom? A sturdy boom of length & might swing around in a horizontal plane on the moon's surface about a central vertical axis at constant angular velocity w. A payload container starting near the rotation axis of the boom might then slide with increasing speed out along the length of the boom and then project outward at a very high velocity when it leaves the end of the boom. payload A boom with payload on the moon's surface, rotating in a horizontal plane.