How Far Is the Dog from the Shore After Walking on the Boat?

[anotherghost]

Homework Statement

All right, so: dog on a boat. It's the snoopy problem. There's a boat floating on some water, and there's a dog on the end - he starts at X away from the shore. Then, he walks a length L across the boat towards the shore. How far is he away from the shore at the end?

Homework Equations

m1r1 + m2r2 = (m1 + m2)rcm

The Attempt at a Solution

As far as I can get is to set the center of mass as X at the beginning.

Mboat * Xi + Mdog * Xi = (Mboat + Mdog)Xi

and then the dog walks L towards the shore and I am not sure how to set this equation up.

Mboat * (Xi + L) + Mdog * (Xi - L) = (Mboat + Mdog) Xi

See, I know the center of mass doesn't move, because the system is isolated and the only forces are internal. However, I don't know how to make the positions relative - this equation would work if the problem literally stated he walks a length L towards the shore (his new position obviously would be Xi - L) but it says he walks length L down the boat, which means the boat moves too and he doesn't quite make the distance L from an external observer's point of view.

 

[rl.bhat]

In the given problem, the center of mass of boat and dog are not at Xi. Only CM of dog is at Xi. Let d be the CM of boat.
Now CM of (boat + dog) will be -------(1)
As you have said CM of the system remains the same. So when dog walks a length L towards the shore, CM of boat must move away from the shore. Let this be x. Now new position of the dog is [(X - L) + x] and new CM of boat wiil be (d + x). Find the CM of ( dog + boat) in this position and equate it to eq(1) and find x.