Why are the angles in rope equilibrium the same?

In summary, the conversation discusses the equilibrium position of a pulley with a mass attached to a rope between two parallel walls. The solution states that in equilibrium, the inclined angles of the rope must be the same, despite the assumption that the pulley is ideal and has no mass or friction. This is because the most important property of an ideal pulley is that it changes the direction of the tension without changing its magnitude. Therefore, in order to have equilibrium, the horizontal components of the tension on each side of the pulley must be balanced, which can only occur when the angles are equal.
  • #1
bolzano95
89
7

Homework Statement


Between two parallel walls we tighten a rope. One end of a rope is tied a little higher on the wall than the end on the opposite wall. On the rope we put a pulley with mass m which slides to equilibrium position.

Homework Equations

The Attempt at a Solution


I assumed that in equilibrium the angles with which the ends of the rope are connected with the wall are different. But in the solution states that in the equilibrium the inclined angles are the same.
Interesting.
How can this be explained?
I know that forces in x direction cancel each other out, but I wouldn't conclude that the forces in the rope are also the same.
 
Physics news on Phys.org
  • #2
Is the tension on one side of the pulley different from the other side? Presumably this is an ideal pulley.
 
  • #3
Yes, the pulley here is presumed ideal. The problem doesn't give anything except what is written above under problem statement.
 
  • #4
bolzano95 said:
Yes, the pulley here is presumed ideal. The problem doesn't give anything except what is written above under problem statement.
You don't need more to answer the question. What is the most important property of an ideal pulley other than it is assumed massless and there is no friction at the bearings?
 
  • #5
Ok, so in this case I disregard the mass of the pulley and the friction but what I still don't get is why in the equilibrium the inclined angles are the same.
 
  • #6
Because the most important property of an ideal pulley is that it changes the direction of the tension without changing its magnitude. Write down the horizontal components of the tension on each side of the pulley and see what must be necessary in order to have equilibrium.
 
  • #7
Lets suppose the angles and therefore forces are different. I would write horizontal components for equilibrium as:
[tex]F_{1}cos\alpha= F_{2}cos\beta [/tex]

and for vertical:
[tex]F_{g}= F_{1}sin\alpha + F_{2}sin\beta [/tex]

I still don't get it, sorry.
 
  • #8
bolzano95 said:
Lets suppose the angles and therefore forces are different. I would write horizontal components for equilibrium as:
You cannot suppose that the forces are different. This is an ideal pulley that has the property that I explained in post #6. What happens to ##F_1 \cos\alpha=F_2 \cos\beta## when ##F_1=F_2## as is the case here?
 
  • #9
You are right. Then the angles are the same.
 
  • #10
Here is another way to look at it. The magnitudes of the tensions on either side are always equal. As long as the angles are different, the pulley will accelerate because the horizontal tension components are unbalanced. Once the angles become equal, the horizontal components are balanced, the pulley stops accelerating and is in equilibrium.
 

1. Why are the angles in rope equilibrium equal?

The angles in rope equilibrium are equal because of the principle of static equilibrium. This principle states that in order for an object to be in equilibrium, the sum of forces acting on it must be equal to zero. In the case of a rope in equilibrium, the tension forces acting on the rope must be equal and opposite, resulting in equal angles.

2. How does the tension in a rope affect the angles in equilibrium?

The tension in a rope directly affects the angles in equilibrium. As the tension increases, the angles become smaller, and as the tension decreases, the angles become larger. This is because the tension force is one of the forces acting on the rope and must be taken into account when determining the angles.

3. Why is it important for the angles in a rope to be equal in equilibrium?

Having equal angles in a rope in equilibrium ensures that the forces acting on the rope are balanced. If the angles were not equal, it would indicate that the forces are not balanced and the object is not in equilibrium. This could result in the object moving or falling.

4. Can the angles in rope equilibrium ever be unequal?

No, the angles in rope equilibrium must always be equal. This is due to the principle of static equilibrium, which requires that the sum of forces acting on an object must be equal to zero. If the angles were unequal, it would indicate that the forces are not balanced and the object is not in equilibrium.

5. How does the length of the rope affect the angles in equilibrium?

The length of the rope does not directly affect the angles in equilibrium. However, it can indirectly affect the angles by changing the tension in the rope. A longer rope will have a greater tension force, resulting in smaller angles, while a shorter rope will have a smaller tension force, resulting in larger angles.

Similar threads

  • Introductory Physics Homework Help
Replies
12
Views
1K
  • Introductory Physics Homework Help
Replies
13
Views
2K
  • Introductory Physics Homework Help
Replies
6
Views
2K
  • Introductory Physics Homework Help
Replies
8
Views
2K
  • Introductory Physics Homework Help
Replies
8
Views
1K
  • Introductory Physics Homework Help
Replies
5
Views
7K
  • Introductory Physics Homework Help
Replies
1
Views
1K
  • Introductory Physics Homework Help
Replies
16
Views
2K
  • Introductory Physics Homework Help
Replies
4
Views
3K
Replies
18
Views
1K
Back
Top