# Pulley with mass - what is happening in the rope that causes tension forces to differ?

Let's say that we have a block on a surface attached over a pulley with mass to another mass hanging over the edge. We have two different tension forces, and I see that if we didn't, there would be no net torque on the pulley and therefore would not accelerate. My question is specifically what is happening in the rope along the pulley to cause it to have these differing tensions? I think friction must play a role so the rope does not slip over. My teacher said that tension force of the mass on the table would be what incorporates the friction, but I'm not quite sure I understand.

• First off you will have one tension...which is a number not a force. This can be discovered by simply adding more mass to each opposing masses of a material of known weight (say lead) until the rope breaks. Of the question is "how does a pulley act like a lever or shim?" now you're on to something. A pulley does this by first being attached to something solid like an overhanging piece of wood and second by distributing any force acting upon it (the "pull" or the "lift") evenly. The act of pulling creates acceleration which moves the mass upward. Should the act of pulling cease the (the mass – Doctor Zhivago Nov 12 '16 at 5:27
• – user36790 Nov 12 '16 at 5:28
• Over the edge) the opposing mass will still want to return to rest (fall) so a mass of exactly equal weight would have to be applied to create some type of (highly unstable) equilibrium. Far better to simply tie off the "lifting" side of the rope and thus allow the rope tension to hold said mass in place. But again the "tension" is not a force but the breaking point of said rope...meaning the friction created by the fibers of said rope that cohere the rope into a strand, a chord and then a string. – Doctor Zhivago Nov 12 '16 at 5:35

## 1 Answer

Correct : The difference in tension is due to static friction between the rope and the pulley.

It is as though static friction ties an extra mass $M_2$ (not equal to the mass of the pulley) to the rope between the two blocks, $M_1$ on the surface and $M_3$ hanging. The tension $T_2$ in the 2nd section of rope (between $M_2$ and $M_3$) then has to accelerate both $M_2$ and $M_1$, whereas the tension $T_1$ in the 1st section of rope only has to accelerate $M_1$ :

\begin{align}T_2 &=(M_2+M_1)a\\ T_1&=M_1a.\end{align}
So, $T_2 \gt T_1$.

• That's awesome! Thanks! So is static friction working to oppose motion, so it's working against the motion of the rope? – rb612 Nov 12 '16 at 5:30
• Also, does that mean the difference in tension is a slow change? In other words it's not an abrupt change from tension 1 to 2? – rb612 Nov 12 '16 at 5:31
• Yes, friction opposes the tendency toward relative motion. Friction is approximately constant around the pulley, and tension in the rope changes gradually (exponentially, in fact) between making and breaking contact. See Capstan Equation. – sammy gerbil Nov 12 '16 at 14:43