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 5.50 m in the trees and swings down to the ground using a thin, but very rigid, 32.0 kg vine 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 speedHow high will Tarzan and Jane go on their first swing after this daring rescue?

Answer 1

Change in gravitational potential energy for Jane = PEj = M*g^H

Change in gravitational potential energy for vine = PEv = m*g*H/2

I = moment of inertia of vine = m*L^2/3

Rotational kientic energy of vine = KEr = (1/2)I*w^2 = (1/2)mL^2*w^2/3

Janes KE = (1/2)M*v^2 = KEj = (1/2)M*(w*L)^2

Kinetic energy after equals the change in the potential energy.

KEr + KEj = PEj + PEv

(1/2)mL^2*w^2/3 + (1/2)M*(w*L)^2 = M*g*H + m*g*H/2

w^2*L^2[m/6 + M/2] = g*H[M + m/2]

w^2*L^2*(M + m/3) = 2*g*H[M + m/2

w^2 = 2*g*(H/L^2)*[M + m/2]/[M + m/3]

M = 60 kg

m = 32 kg

L = 8.5 m

H = 5.5 m

w = 1.2667 rad/s (angular speed just before she grabs Tarzan)

After she grabs Tarzan we must use conservation of angular momentum.

Before:

Momentum of the rod = I*w = [m*L^2/3]*w

Momentum of Jane = [M*L^2]*w ..... Moment of inertia for a point mass is M*L^2

After (use W for the angual velocity after Jane grabs Tarzan):

Momentum of the rod = I*W = [m*L^2/3]*W

Momentum of Jane = [(M + T)*L^2]*W ... where T is the mass of Tarzan

Equate the sums of each.

[m*L^2/3]*w + [M*L^2]*w = [m*L^2/3]*W + [(M + T)*L^2]*W

[m/3]*w + [M]*w = [m/3]*W + [(M + T)]*W

w*[M + m/3] = W*[M + T + m/3]

W = w*[M + m/3]/[M + T + m/3]

W = 0.6275 rad/s (angular speed just after she grabs Tarzan]

To determine how high Jane and Tarzan will go you need to equate the KE after Jane picks up Taran with the potential energy at the height of their swing. You need the KE of the vine as well.

Kinetic energy.

Jane + Tarzan = KEj = (1/2)(M + T)v^2 = (1/2)(M + T)(W*L)^2

Vine = KE2 = (1/2)I*W^2 = m*L^2*W^2/6

Potential energy.

Jane + Tarzan = PE1 = (M + T)*g*h

Vine = PE2 = m*g*(h/2) .... again center of mass height change is 1/2 that of Jane

Equate the sums of these:

(1/2)(M + T)(W*L)^2 + m*L^2*W^2/6 = (M + T)*g*h + m*g*(h/2) .

(1/2)W^2*L^2*{ M + T + m/3] = g*h*[M + T + m/2]

h = (1/2)*(W^2*L^2/g)*[M + T + m/3]/[M + T + m/2]

h = 1.399 m