Question

a wire hangs from a dark tower so that the upper end is not visible. how can we determine the length of the wire?

Answer 1

Answer:

by solving it as follows.

Explanation:

Let this rope have a radius r. Then, pull the rope backward by a horizontal measured distance x. By the Pythagorean theorem, the vertical distance that you gained is (r- (sqrt(r^2 - x^2))). Let this vertical distance be h.

Suppose you had a speedometer that measured your exact speed at the instant you hit the dark tower. We know from the Work-Energy Theorem that the speed v = sqrt (2gh). We also know the value of h is (r- (sqrt(r^2 - x^2))), which implies the speed measured on the speedometer would be sqrt(2g(r- (sqrt(r^2 - x^2)))). This equation has only 1 unknown (r) and has a maximum degree of 2, meaning we can solve it using elementary methods.

This method requires a speedometer. Here is another creative method which can be implemented using only a measuring tape or ruler.

Measure the distance between your standing reach (hands up all the way) and the bottom of the rope. Cut that off and note that down as y.

Measure your standing reach using the ruler, and use the dark tower as your reference frame. Denote this as dy.

Move the rope out slowly in small increments until the height of the rope matches your standing reach exactly. Once again, the Pythagorean theorem helps us solve for r. First off, note that we cut off part of the rope. Let r’ be the length after cutting, where r’ = r-y. Then, note that r’ is the hypotenuse of the right triangle we created. x, the distance which we moved out from the tower is a leg. The other leg is r’ - dy. Using lots of substituting, we can eventually solve for r’, leading us to solve for r.