Tarzan spies a 34.0 kg chimpanzee in severe danger, so he swings to the rescue.

Question

Tarzan spies a 34.0 kg chimpanzee in severe danger, so he swings to the rescue. He adjusts his strong, but very light, vine so that he will first come to rest 4.50 s after beginning his swing, at which time his vine makes a 12.0 angle with the vertical.

How long is Tarzan's vine, assuming that he swings at the bottom end of it?

What is the frequency of Tarzan's swing?

What is the amplitude (in degrees) of Tarzan's swing?

Just as he passes through the lowest point in his swing, Tarzan nabs the chimp from the ground and sweeps him out of the jaws of danger. If Tarzan's mass is 65.0 kg , find the frequency of the swing with Tarzan holding onto the grateful chimp.

Find the amplitude (in degrees) of the swing with Tarzan holding onto the grateful chimp.

Explanation / Answer

Let the length of vine be L

Time to come to rest for first time = 4.5 s

Therefore the time period of motion will be 2*4.5 = 9 s

Now we know, time period = 2 pi sqrt(L/g)

=> 9 = 2*pi*sqrt(L/9.81)

=> L = 20.13 m

So length of tarzan's vine = 20.13 m

Frequency = 1/time period = 1/9 Hz = 0.1111 Hz

Amplitude of Tarzan's swing = 12 degree

The time period and frequency is independent of the mass, so the frequency of tarzan holding the chimp will remain same as before. So the frequency = 1/9 Hz = 0.1111 Hz

In the process of tarzan picking up the chimp, using conservation of energy.

65 * g * (L - L cos 12) = (65 + 34) * g * (L - L cos theta)

=> theta = 9.717 degree.

So amplitude (in degrees) of the swing with Tarzan holding onto the grateful chimp = 9.717 degree

[Note: Though the above method gives you the desired answer, the correct method should find the velocity just after picking the chimp by conservation of momentum and then use conservation of energy to measure the angular amplitude as I show below

0.5 v2 = 9.81 * 20.13 * (1 - cos 12)

=> v= 2.9378 m/s

By conservation of angular momentum just before and after picking the chimp

65 * 2.9378 * L = (65 + 34) * v' * L

=> v' = 1.92885 m/s

By conservation of energy

0.5 v'2 = 9.81 * 20.13 * (1 - cos theta)

=> 0.5 1.928852 = 9.81 * 20.13 * (1 - cos theta)

=> theta = 7.87 degree

So the answer should actually be 7.87 degree. But I have shown the calculation of 9.717 degree as that is showing you the answer your textbook says]

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