A uniform cable of weight ?w? hangs vertically downward, supported by an upward force of magnitude ?w? at its top end. What is the tension in the cable (a) at its top end; (b) at its bottom end; (c) at its middle? Your answer to each part must include a free-body diagram. (?Hint:? For each question choose the body to analyze to be a section of the cable or a pomt along the cable.) (d) Graph the tension in the rope versus the distance from its top end.
Solution to 53P Step 1 Tension is a pulling force that is existent in a non extensible string/cable. Here as the question suggests, an upward pulling force W is supporting a string on which a gravitational force exists. Thus the string is supported at the top and free at the bottom. (a) Here there is an upward pulling force, W. Thus with this force, the top end will pull the bottom end of the string. Thus the tension at the top end of the string is, T=W (b) Here, there is no direct contact force pulling the string in bottom of the string, thus the tension in the string at the bottom end of the string is 0N. T=0N (c) At the middle, the force with which the string section at the middle is pulled upwards is F=W/2 Since there is a pulling force, there would be a tension in the string in the middle section, The middle of the string is in equilibrium, thus equating upward and downward forces, T+W/2=W T=W/2 Thus tension at the middle section of the string is T=W/2