# What is the tension in the rope used to pull two blocks with friction?

• capandbells
In summary: So you should get 2A = a_a - a_b, which gives you T = 2m_a*m_b/(m_a + m_b)A + f. And this agrees with the answer provided by the textbook.In summary, the problem involves two blocks, Ma and Mb, connected by a massless rope and pulled by an accelerated pulley. The blocks experience a friction force f between each other due to their relative motion. By setting up equations for the forces in the x and y directions, and incorporating the acceleration of the pulley, we can solve for the tension in the rope, which is given by T = 2m_a*m_b/(m_a + m_b)A + f.
capandbells

## Homework Statement

Mass Ma lies on top of mass Mb, as shown. Assume Mb > Ma. The two blocks are pulled from rest by a massless rope passing over a pulley. The pulley is accelerated at rate A. Block Mb slides on the table without friction, but there is a constant friction force f between Ma and Mb due to their relative motion. Find the tension in the rope.

## The Attempt at a Solution

Newton's second law in the y direction for both masses:
$N_1 - M_a g = 0$
$N_2 - N_1 - M_b g = 0$
In the x-direction:
$T_a - F_a = M_a a_a$
$T_b + F_a = M_b a_b$
where
$F_a = \mu M_a g$

Ok, so I suppose we can assume the tension in the rope is the same at every point since it's massless. So $T_a = T_b = T$. Now, the thing is, friction can't cause the blocks to move, it can only oppose motion. So maybe the friction here has to be static friction and we assume the blocks are not moving relative to each other? Otherwise, one of the blocks would have to be moving opposite the direction in which we apply the force, which seems wrong. I'm really not sure where to go with this problem.

I think you are doing alright . In the question it is given that there is relative motion between the blocks , so the friction between the blocks is kinetic in nature . You just take it as 'f' . No need to introduce coefficient of friction.

Apart from the equations you have formed you need one more relation i.e between the acceleration of pulley and the acceleration of the blocks .

I guess that is the part that I am having the most difficulty with. I don't see any way to incorporate the acceleration applied to the pulley. The only thing I can think of is that accelerating the pulley to the right is the same as subjecting the pulley to a gravitational acceleration to the left and holding it in place. Then we could say that the blocks experience a fictitious "force" to the left from this acceleration. But other than that I am stumped.

Look at the picture I have attached .

Let a,b,c be the coordinates of the lower block ,upper block and the pulley respectively.Express the length of the rope in terms of a,b,c .Then use the fact that rope length remains constant . This will give you the relationship between the acceleration of the three entities .

#### Attachments

• rope.PNG
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Last edited:
Oh... ok. So L = (c - a) + (c - b) and differentiating twice, $2A = a_a + a_b$. Solving for T we get $T = 2 \frac{m_a m_b}{m_a + m_b}A + f$

capandbells said:
Oh... ok. So L = (c - a) + (c - b) and differentiating twice, $2A = a_a + a_b$

Correct .

capandbells said:
Solving for T we get $T = 2 \frac{m_a m_b}{m_a + m_b}A + f$

This doesn't look right .Please recheck.

ranjhahah
capandbells said:
Solving for T we get $T = 2 \frac{m_a m_b}{m_a + m_b}A + f$
I think you got a sign wrong on one of the f terms. Note that in your original equations F was +ve in one and -ve in the other, while T had the same sign in both.

## 1. What is a pulley?

A pulley is a simple machine that uses a wheel with a groove and a rope or chain to change the direction of a force.

## 2. How does a pulley work?

A pulley works by distributing the weight of an object between multiple ropes or chains, making it easier to lift or move the object.

## 3. What are the two blocks in a two blocks and a pulley system?

The two blocks in a two blocks and a pulley system refer to the two objects that are being connected by the pulley, with one block being the object being lifted and the other block being the counterweight.

## 4. What is the purpose of using a pulley in a two blocks and a pulley system?

The purpose of using a pulley in a two blocks and a pulley system is to reduce the amount of force needed to lift or move an object. It also allows for a more controlled and efficient movement of the object.

## 5. How does the number of pulleys affect the mechanical advantage of a two blocks and a pulley system?

The more pulleys there are in a two blocks and a pulley system, the greater the mechanical advantage. This means that it requires less force to lift or move the object, making it easier to do work.

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