Solving a Spool Problem: Length of Unwound Rope & Distance of CM Movement

[e(ho0n3]

Here is another one of these spool problems: A large spool of rope stands on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it. The spool rolls behind the person without slipping. (a) What length of rope unwinds from the spool? (b) How far does the spool's center of mass move?

I really haven't a clue on what to do here. I know there is a torque causing the spool to move, but that doesn't help me much. All I can say for sure is that the answer to (a) must be less than L. Any tips?

 

[Gokul43201]

Forget about torques and forces. If you think about it carefully, it's not hard to see that the distance traveled by the spool CoM is equal to the length of rope unwound.

Picture how the last point of contact (between the spool and rope) is above the center and so moves at the same rate that the spool does.

 

[e(ho0n3]

Forget about torques and forces. If you think about it carefully, it's not hard to see that the distance traveled by the spool CoM is equal to the length of rope unwound.

Picture how the last point of contact (between the spool and rope) is above the center and so moves at the same rate that the spool does.

I couldn't picture this in my head at all so I drew a picture. I see what you mean now. I can't believe it was so easy. So the answer to both (a) and (b) is L/2. So much for wanting to become a physicist.

 

[Gokul43201]

I couldn't picture this in my head at all so I drew a picture. I see what you mean now. I can't believe it was so easy. So the answer to both (a) and (b) is L/2.

Correct.

So much for wanting to become a physicist.

Everyone trips up now and then...no big deal ! Give yourself a break.

PS : I drew a picture too. In fact, I'd started drawing before I finished reading. No harm in that.