Electrical power is transmitted across long distances through cables strung from one support to another. The wires are usually made of many smaller wires wrapped into a flexible bundle. For example one similar to the image below has 7 steel wires making a strong inner core, and 26 aluminum wires around it to make a cable 27 mm in diameter (over 1 inch). It has a mass of 1.63 kg/m and can sustain a load of 679,000 N safely without risk of failure. When lines are strung from one support to another, their enormous mass creates a tension in the cable that exerts a gravitational pull on its supports.

Suppose such a cable is strung between poles spaced 1000 meters apart across a rural landscape. The cable is allowed to sag, following a natural curve called a catenary, increasing its length negligibly compared to the distance between the poles, while potentially lowering the cable to nearly contacting the ground at midpoint.

Consider a cable tension of 200,000 N that results in a sag of about 10 meters at the middle of the cable over this 1000 meter span. The sag would add only 1% to the length of the cable.

1.) If a small mass compared to the total cable mass were added to the middle of the cable, how would this affect the tension in the cable?

2.) What is the speed of a wave along this cable that could develop from wind or a vibration in the towers supporting the cable?

3.) What is lowest natural frequency of the cable's vibration created by a standing wave between two supports 1000 meters apart?

4.) What would happen to the tension, the sag, and the resonant frequency of vibration if ice accumulated on the wires?