Chapter 6.9, Problem 74ES

Suppose that a hollow tube rotates with a constant angular velocity of $\omega \text{ rad/s}$ about a horizontal axis at one end of the tube, as shown in the accompanying figure. Assume that an object is free to slide without friction in the tube while the tube is rotating. Let $r$ be the distance from the object to the pivot point at time $t>0$ , and assume that the object is at rest and $r=0$ when $t=0$ . It can be shown that if the tube is horizontal at time $t=0$ and rotating as shown in the figure, then

$r=\frac{g}{2{\omega }^2}\left[\sinh \left(\omega t\right)-\sin \left(\omega t\right)\right]$

during the period that the object is in the tube. Assume that $t$ is in seconds and $r$ is in meters, and use $g=9.8{\text{ m/s}}^2$ and $\omega =2\text{ rad/s}$ .

(a) Graph $r$ versus $t$ for $0\le t\le 1$ .

(b) Assuming that the tube has a length of $1\text{ m}$ , approximately how long does it take for the object to reach the end of the tube?

(c) Use the result of part (b) to approximate $dr/dt$ at the instant that the object reaches the end of the tube.