George of the Jungle, with mass m, swings on a light vine hanging from a station

Question

George of the Jungle, with mass m, swings on a light vine hanging from a stationary tree branch. A second vine of equal length hangs from the same point, and a gorilla of larger mass M swings in the opposite direction on it. Both vines are horizontal when the primates start from rest at the same moment. George and the gorilla meet at the lowest point of their swings. Each is afraid that one vine will break, so they grab each other and hang on. They swing upward together, reaching a point where the vines make an angle of 22.5 with the vertical. (a) Find the value of the ratio m/M. 0.1556 Your response differs from the correct answer by more than 10%. Double check your calculations. (b) Try this at home. Tie a small magnet and a steel screw to opposite ends of a string. Hold the center of the string fixed to represent the tree branch and reproduce a model of the motions of George and the gorilla. What changes in your analysis will make it apply to this situation? Assume the magnet is strong so that it noticeably attracts the screw over a distance of a few centimeters. Then the screw will be moving faster just before it sticks to the magnet. Does this make a difference? yes no

Explanation / Answer

a First we have to find the velocity of each of them at lowest pointof swing

As we know here the only force doing wok is gravitationalforce

So kinetic energy at bottom = Change in potentialenergy

   mv2/2 = mgh

   here v = ?2gl (Where l inlength of rope)

So both of them has same velocity at bottom .

Now we have to conserve momentum at bottom

Initial momentum = Mv -mv

Final momentum =(M+m)v'

so v' = (M-m) v /(M+m)

This velocity raises them to height equivalent to 22.50

Total height raised = l cos ?

So the kinetic energy must be equal to the change in potentialenergy

(M+m)g(l cos ?) = (M+m)v'2 / 2

Put the values of v' to get answer

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