Try it on paper first. A hint, the key idea and the worked solution are here when you need them.
A power cable hangs between two poles of equal height, m apart. A surveyor takes the origin on the ground directly below the lowest point of the cable. The cable then follows the curve for , with and in metres. A cable that hangs under its own weight takes this shape, called a catenary. The cable has a mass of kg for each metre of its length. The supplier wants the length of the cable between the poles, to order the right amount. The line engineer wants the sag, which is how far the lowest point hangs below the top of a pole. The loader wants the mass of the cable. Find the length, the sag and the mass. Give the answers to two decimal places. A straight line between the tops of the poles would be m, so the cable must be longer than that.
The length of the curve from to is . It comes from adding the tiny hypotenuses . Look for a perfect square under the root.
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. The slope. .
Step 2. A perfect square. , so .
Step 3. The length. m. This is more than m, as expected.
Step 4. The sag. m.
Step 5. The mass. kg.
Answer: length m, sag m, mass kg.
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