Tarzan has foolishly gotten himself into another scrape with the animals and mus

Question

Tarzan has foolishly gotten himself into another scrape with the animals and must be rescued once again by Jane. The 60.0 kg Jane starts from rest at a height of 4.50 m in the trees and swings down to the ground using a thin, but very rigid, 32.0 kgvine 8.50 m long. She arrives just in time to snatch the 72.0 kg Tarzan from the jaws of an angry hippopotamus. What is Jane's (and the vine's) angular speed

Part A: just before she grabs Tarzan?

Part B: just after she grabs him?

Part C: How high will Tarzan and Jane go on their first swing after this daring rescue?

Explanation / Answer

given

a )

M = 60 kg

m = 32 kg

L = 8.50 m

H = 4.5 m

mT = 75 Kg

change in potential energy of jane = PEj = M X g X H

change in potential energy of vine = PEv = m X g X ( H / 2 )

Ivine = m X L2/3   

vine's KEv = ( 1 / 2 ) I X w2

= ( 1 / 2 ) m L2 X w2/3  

jane's KEj = ( 1 / 2 ) M X v2

= ( 1 / 2 ) M X ( w X L)2

using energy conservation

KEv + KEj = PEj + PEv  

( 1 / 2 ) m L2 X w2/3   + ( 1 / 2 ) M X ( w X L)2 = M X g X H + m X g X ( H / 2 )

w2 = 2 X g X ( H/ L2 ) X ( M + m / 2 ) / ( M + m / 3 )

w2 = 2 X 9.8 X ( 4.5 / 8.52 ) X ( 60 + 32 / 2 ) / ( 60 + 32 / 3 )

w2 = 92.777 / 70.66

w2 = 1.31

w = 1.14 rad/sec

b )

before

the rod momentum is = I X w

= ( m X L2/3) X w

and jane momentum is = ( M X L2 ) X w

after

the rod momentum is= ( m X L2/3) X w

and jane momentum is = ( ( M + mT ) X L2 X W

( m X L2/3) X w + ( M X L2 ) X w= ( m X L2/3) X w + ( ( M + mT ) X L2 X W

then

by simplyfying

W = w X ( M + m / 3 ) / ( M + mT + m/3 )

W = 1.14 X ( 60 + 32 / 3 ) / ( 60 + 72 + 32 / 3 )

W = 0.6073 rad/sec

c )

the KE of jane and trazan = ( 1 / 2 ) ( M + mT ) ( W X L )2  

for vine = m X L2 X W2/6    

the potential energy is

for jane and trazan

= ( M + mT ) X g X h

for vine = m X g X ( h / 2 )

by equating the above values

( 1 / 2 ) ( M + mT ) ( W X L )2 + m X L2 X W2/6    = ( M + mT ) X g X h + m X g X ( h / 2 )

h = ( 1 / 2 ) X (W2 X L2 / g ) X ( M + mT + m / 3 ) / ( M + mT + m / 2 )

h = ( 1 / 2 ) X (0.602 X 8.52 / 9.8 ) X ( 60 + 72 + 32 / 3 ) / ( 60 + 72 + 32 / 2 )

h = 188.3112 / 148

h = 1.2723 m

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