[Physics] When do we assume that a rope has different tensions in different places

forcesfree-body-diagramnewtonian-mechanicsstring

Working a problem with 2 pulleys and masses in class this week the professor at one point assumed that the tension on the two sides of a pulley would be different. Every other time we've done a similar problem, we just assumed that the tension would be the same on both sides. And I didn't understand his explanation of why we had to do it that way this time.

My question is when can we assume the tension on both sides of a rope are the same and when do we need to assume different tensions?

Best Answer

You cannot assume equal tension throughout if the pulley is not "massless" (assuming the rope does not slip over the pulley.

Then adding a heavy load in one end will be carried by the string all the way up (Newton's 3rd law states that for all crosssections of the rope on this side, the forces must be equal).

But if the pulley has inertia by having mass, then it "helps" by holding up the mass - as if someone grapped the rope and held it back. The tension on the other side is therefore lower, since the pulley "helps" so this side does not carry the whole weight alone.

Also, if the rope is not rigid (if it is an elastic rubber band e.g.), you will feel different tension in the rope when adding a load in one end over a pully as long as stretching happens.