# Tensions of cable over a pulley

• arpon
In summary, the conversation discusses the concept of tension in a frictionless ideal pulley system. The participants question whether the tensions T_1, T_2, and T_3 are equal and why. The moderator explains that tensions are not forces, but closely related to them. The participants then discuss the direction of tensions and the use of free body diagrams. Eventually, they conclude that T_1 = T_2 = T_3 and provide a diagram to support their reasoning. A word of caution is given regarding the definition of "frictionless" in this context.
arpon
[moderator's note: moved to homework forum so no template]

This is a frictionless ideal pulley.
Are $T_1, T_2, T_3$ equal? If so, why? Please explain in details?

Last edited by a moderator:

Philip Wood said:
Probably, I have mistaken the direction of $T_1$ and $T_2$. They should be downward.
Then I straightened the cable.

So, I assume, $T_1 + T_3 - T_2 = 0$ .But 'why', I am not sure.

Right. Now we have something to go on.

Tensions aren't really forces, though they're closely related. Suppose A and B pull each other in a tug of war, with a rope which at a particular time has a tension of 200 N. That means that the rope pulls A with a force of 200 N, and pulls B with a force of 200 N. These forces are equal and opposite. So in your diagram in post 1, the arrows don't show tensions, but forces. Specifically the two left hand arrows show the forces that the rope exerts on the bottom pulley and the right hand arrow shows the force that the rope exerts on the top pulley. This could cause confusion. It's often best to draw separate diagrams for separate parts of the system showing the forces acting – so-called free body diagrams.

Now your question. The tension in the rope doesn't have a direction (in the same sense that a force has a direction). But it can move freely over the pulleys, and you are right to think that it can be regarded as straight from the point of view of comparing the tensions in the various sections of it. What do you conclude about these tensions? Note that the arrows you put on the straightened rope in post 3 should be removed, as we're dealing with tension, not force. Your equation is also wrong.

Philip Wood said:
Right. Now we have something to go on.

Tensions aren't really forces, though they're closely related. Suppose A and B pull each other in a tug of war, with a rope which at a particular time has a tension of 200 N. That means that the rope pulls A with a force of 200 N, and pulls B with a force of 200 N. These forces are equal and opposite. So in your diagram in post 1, the arrows don't show tensions, but forces. Specifically the two left hand arrows show the forces that the rope exerts on the bottom pulley and the right hand arrow shows the force that the rope exerts on the top pulley. This could cause confusion. It's often best to draw separate diagrams for separate parts of the system showing the forces acting – so-called free body diagrams.

Now your question. The tension in the rope doesn't have a direction (in the same sense that a force has a direction). But it can move freely over the pulleys, and you are right to think that it can be regarded as straight from the point of view of comparing the tensions in the various sections of it. What do you conclude about these tensions? Note that the arrows you put on the straightened rope in post 3 should be removed, as we're dealing with tension, not force. Your equation is also wrong.

So, now I think $T_1 = T_2 = T_3$. I explained the reason in the picture. Please check it whether I am right or Wrong.

You are right. The tension is the same (call it T) throughout the rope if the pulleys are frictionless. The rope therefore exerts an upward force of 2T on the left hand pulley, a downward force of 2T on the right hand pulley (which is counteracted by an equal and opposite force in the rope holding it to the ceiling) and an upward force of T on mass B.

arpon
Philip Wood said:
The tension is the same (call it T) throughout the rope if the pulleys are frictionless.
Just a word of caution. In the context of pulleys, 'frictionless' usually refers to the axle, not the contact with the rope. That being so, it is not sufficient (when the system is accelerating) for the pulleys to be frictionless. For the tension to be the same both sides, the pulleys would also need to be massless.

## 1. What is the tension in a cable over a pulley?

The tension in a cable over a pulley is the force applied by the cable to the pulley. This force is equal to the weight of the object being lifted, assuming there is no friction or other external forces acting on the system.

## 2. How is tension affected by the angle of the cable over a pulley?

The tension in a cable over a pulley is affected by the angle of the cable. As the angle increases, the tension in the cable also increases. This is because the weight of the object is distributed over a larger area, resulting in a larger force required to lift it.

## 3. What happens to the tension in a cable over a pulley if the weight of the object changes?

If the weight of the object changes, the tension in the cable over a pulley will also change. The tension will increase if the weight of the object increases, and decrease if the weight of the object decreases.

## 4. How does the size of the pulley affect the tension in a cable over a pulley?

The size of the pulley does not directly affect the tension in a cable over a pulley. However, a larger pulley may require a longer cable, which can increase the weight and therefore the tension in the cable.

## 5. Is the tension in the cable over a pulley always equal to the weight of the object being lifted?

In an ideal system with no friction or other external forces, the tension in the cable over a pulley will always be equal to the weight of the object being lifted. However, in real-world situations, there may be other factors that affect the tension in the cable.

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