Two astronauts (see figure below), each having a mass M , are connected by a rop

Question

Two astronauts (see figure below), each having a mass M, are connected by a rope of length d having negligible mass. They are isolated in space, moving in circles around the point halfway between them at a speed v. (Use any variable or symbol stated above as necessary.)

(a) Calculate the magnitude of the angular momentum of the system by treating the astronauts as particles.

Li =

(b) Calculate the rotational energy of the system.

KEi =

By pulling on the rope, the astronauts shorten the distance between them to d/2.

(c) What is the new angular momentum of the system?
Lf =

(d) What are their new speeds?
vf =

(e) What is the new rotational energy of the system?
KEf =

(f) How much work is done by the astronauts in shortening the rope?
Wnet =

Explanation / Answer

a)

Intial angular momentum

Li=mvr+mvr =2mvr

L=2*m*v*(d/2)=mvd

b.

rotational energy of the system

Ki=(1/2)mV2+(1/2)mV2

Ki=mV2

c.

By Conservation of angular momentum

Li =Lf

Lf=mvd

d.

Vf=2Lf/md =2*mvd/md

Vf=2V

e.

Kf=(1/2)m(2V)2+(1/2)m(2V)2=4mV2

f.

Wnet =Kf-Ki =4mV2-mV2

Wnet =3mV2

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