# Three forces in equilibrium. Uniform rod

• afrocod
In summary, the problem involves a uniform rod resting in equilibrium with one end in contact with a smooth vertical wall and the other end in contact with a smooth inclined plane. Using the fact that the sum of the three forces must equal zero, it can be determined that the reactions at the ends of the rod are equal to W and √2W. The key to solving this problem lies in understanding that a frictionless surface means the contact force is always perpendicular to the surface.
afrocod

## Homework Statement

A uniform rod AB of weight W rests in equilibrium with the end A in contact with a smooth vertical wall and the end B in contact with a smooth plane inclined at 45 to the wall. Find the reactions at A and B in terms of W.

## Homework Equations

The sum of the three forces equal zero.

## The Attempt at a Solution

Well having a look at the answer is W and √2W, it's obviously a square and the rod must be perpendicular to the inclined plane.

So how could I know that it was perpendicular to the plane. What indicated that or how can I prove that assumption to myself?

afrocod said:
Well having a look at the answer is W and √2W, it's obviously a square and the rod must be perpendicular to the inclined plane.

So how could I know that it was perpendicular to the plane. What indicated that or how can I prove that assumption to myself?
What do you mean 'its a square'? And why would you say the rod must be perpendicular to the inclined plane? Please expand on your thinking.

Well I just meant half the square. The right triangle with unit length and hypotenuse square root of two.

So one reaction is equal in magnitude and direction to W and the other is inclined at 45 and thus the hypotenuse with (square root of 2)W.

I drew a lot of diagrams to help me and the the only way this makes sense to me is if rod AB is perpendicular to the inclined plane.

I know nobody has answered because I haven't shown any working out but all I've done is draw diagrams. I can't make any headway into the problem. All I can do is think about why those answers would make sense.

Thanks for replying by the way.

Also I'm just doing this for my own enjoyment. Nobody is doing my homework for me, if you help me out.

afrocod said:
So one reaction is equal in magnitude and direction to W and the other is inclined at 45 and thus the hypotenuse with (square root of 2)W.
Yes, equal in magnitude, but not equal in direction. I think you meant to say perpendicular in direction.

afrocod said:
I drew a lot of diagrams to help me and the the only way this makes sense to me is if rod AB is perpendicular to the inclined plane.
I guess this is where you're going wrong. The wall and the plane are frictionless, so this straightaway tells us something about the direction of any contact force with them.

afrocod said:
Thanks for replying by the way.

Also I'm just doing this for my own enjoyment. Nobody is doing my homework for me, if you help me out.

Ah, no worries. and I get enjoyment from doing physics questions also. which is why I first came to this site. I also like helping people out on questions. But usually I see if I can do the question myself first, since then it is easier to explain.

BruceW said:
I guess this is where you're going wrong. The wall and the plane are frictionless, so this straightaway tells us something about the direction of any contact force with them.

Yes, this is what I was thinking. That there is some piece of intuition given those particular facts that I don't know. I'll take a stab but I really haven't a clue.

Does it tell me that when the rod is in equilibrium on a frictionless surface that the angles between the rod and the vertical is equal to the rod and the inclined plane?

Oh wait, wait hang on. Does it mean that the rod is lying flat on the incline because there was no friction to keep it up?

If it does, I cannot believe that was the answer. I've been looking at this thing for hours. I feel like an absolute moron.

That doesn't seem to work either because W would be √2 times the other length's and not one equal to W and the other √2W.

Yes, it's neither of those things. It might be a 'piece of intuition given those facts' which you are not aware of. It is to do with the definition of contact forces and friction. In fact, it really doesn't have anything to do with the rod, it is just to do with the contact force which a surface will provide.

I'm going to try to give hints without actually saying it outright. You may have used this principle before, but maybe forgotten about it. OK, so say we have a surface, and its normal points in a certain direction. If the surface is frictionless, then what direction must the contact force be in?

Perpendicular to the surface.

yeep. that's it. No friction means the contact force is perpendicular to the surface.

Isn't it always perpendicular to the surface though?

usually the contact force is defined to include frictional forces, so it is not always perpendicular to the surface. (since frictional forces are parallel to the surface).

Ah, I see the solution now. It all seems so simple in retrospect.

Thanks for all the help.

mm, no worries

## 1. What are the three forces in equilibrium?

The three forces in equilibrium refer to the forces that act on an object and balance each other out, resulting in a state of rest or constant motion. These forces are known as the weight, normal force, and friction force.

## 2. How do these forces maintain equilibrium?

In order for an object to remain in equilibrium, the three forces must have equal magnitude and opposite direction. This ensures that there is no net force acting on the object, resulting in a state of balance.

## 3. What is a uniform rod?

A uniform rod is a rigid object that has a constant mass and cross-sectional area throughout its length. This means that the distribution of mass and weight is the same at any point along the rod.

## 4. How does a uniform rod relate to the three forces in equilibrium?

When a uniform rod is in equilibrium, the three forces act on different points of the rod. The weight acts on the center of mass, the normal force acts at the point of contact with a surface, and the friction force acts at the point where the rod is in contact with another object.

## 5. What factors can affect the equilibrium of a uniform rod?

The equilibrium of a uniform rod can be affected by the angle at which the rod is placed, the distribution of weight along the rod, and the presence of external forces such as wind or other objects pushing or pulling on the rod.

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