Solving for the Centre of Mass in an Astronaut Space Walk

[JenL]

Homework Statement

a) Two astronauts, A of total mass 90kg and B of total mass 110kg are tied together by a light rope 15m long during a space walk. A starts to reel in the rope at her end. After 8.5s she has reduced the length of the rope by 5.5m. What distance does the centre of mass move in this time? Explain your answer.

Homework Equations

x=md/(M+m)

The Attempt at a Solution

a) I thought that because A was pulling the rope the centre of mass would move towards A? As the astronauts are in outer space this is an isolated system but would the pulling of the rope by A be counted as an external force? Or is it an internal force?

 

[gneill]

a) I thought that because A was pulling the rope the centre of mass would move towards A? As the astronauts are in outer space this is an isolated system but would the pulling of the rope by A be counted as an external force? Or is it an internal force?

It is definitely an internal force, originating inside the system. While A was pulling on the rope, the rope was pulling back :)

 

[JenL]

So if the rope pulls back does that mean it is not moving? Sorry I still do not fully get this

 

[Nathanael]

So if the rope pulls back does that mean it is not moving?

Indeed. The only way to change the motion of the center of mass of a system is for some external force to act on the system.

 

[BvU]

So if the rope pulls back does that mean it is not moving? Sorry I still do not fully get this

The way to see this is by considering Newton three: action = - reaction.

The center of mass has some coordinate for which (in a simple case)
(m₁+ m₂) x = m₁ x₁+ m₂ x₂
so (m₁+ m₂) v = m₁ v₁+ m₂ v₂
and (m₁+ m₂) a = m₁ a₁+ m₂ a₂
and with Newton two (F = ma), F₂₁ the force by m₂ on m₁ and F₁₂ the force by m₁ on m₂:
(m₁+ m₂) a = F₂₁ + F₁₂ which, according to Newton three is zero !

In short: internal forces cancel for the equations of motion of the center of mass. Cute, isn't it ?

 

[Nirrjhhar]

So what is the answer ?

 

[gneill]

So what is the answer ?

Hi Nirrjhhar, Welcome to PF.

Sorry, but we don't hand out homework results here. You have to show your effort at a solution, then we can point out where its right or wrong, offer advice and other help so that you can solve the problem yourself. See the forum guidelines for more information.