A capstan is a rotating drum or cylinder over which a rope or cord slides in order to provide a great amplification of the rope’s tension while keeping both ends free (Fig.). Since the added tension in the rope is due to friction, the capstan generates thermal energy. (a) If the difference in tension between the two ends of the rope is 520.0 N and the capstan has a diameter of 10.0 cm and turns once in 0.900 s, find the rate at which thermal energy is generated. Why does the number of turns not matter? (b) If the capstan is made of iron and has mass 6.00 kg, at what rate does its temperature rise? Assume that the temperature in the capstan is uniform and that all the thermal energy generated flows into it. Figure:

Solution 94P Step 1 of 4: (a) If the difference in tension between the two ends of the rope is 520.0 N and the capstan has a diameter of 10.0 cm and turns once in 0.900 s, find the rate at which thermal energy is generated. Why does the number of turns not matter Given data, Force, F=T= 520 N Diameter, D = 10 cm= 0.1 m Time period, t=0.9 s To find, Rate of thermal energy, P= Step 2 of 4: That is, W P = t Using W = FD(2) FD(2) P = t Substituting F= 520 N,D = 0.1 m, t=0.9 s P = (520 N)(0.1)(2(3.14)) 0.9 P=363 W Therefore, the rate at which thermal energy is generated is 363 W.