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I'm confused about the direction of the normal contact force acting on a body supported by another body. I have read the following :

If a body is being supported by a surface, the normal force acting on the body is always perpendicular to the surface, irrespective of the tilt of the body. https://physics.stackexchange.com/a/366717/326319

and I'm trying to apply it to different scenarios. enter image description here Assume the surfaces are frictionless.

Case 1 : Circular motion of a particle inside a circle in a vertical plane

Reaction force is always directed towards center of circle. This is because of circle properties : line joining particle and center is perpendicular to tangent at particle.

Case 2 : A rod AB rests in equilibrium inside a hemisphere in a vertical plane. Part of the rod is outside the hemisphere.

At B, the reaction force acts towards center. However, reaction force at A is not towards center of circle. Why is this so? The surface supporting the rod is vertical so shouldn't the normal contact force be horizontal ? How are points A and B different?

Case 3 : A rod AB rests in equilibrium inside a sphere in a vertical plane. Whole rod is inside sphere.

Based on case 2, am I right to think that the reaction forces at both A and B act towards center?

Case 4 : A horizontal rod AB rests in equilibrium inside a sphere in a vertical plane. Whole rod is inside sphere.

enter image description here

The reaction forces at A and B are also towards center right?

Update

After doing some research, I found this :

The normal is at right angles to the contact plane enter image description here

  • If either surface is differentiable at the point of contact it has a tangent plane there
  • If both surfaces have tangent at the contact point then they are necessarily the same plane, or the bodies would intersect.
  • If only one has a tangent plane (as in the picture) , use that.
  • If neither has a tangent plane there, the direction of the normal is indeterminate.

Source : https://www.physicsforums.com/insights/frequently-made-errors-mechanics-friction/#toggle-id-1

I also read this discussion.

So I guess, the normal contact force is always perpendicular to the rod if the rod rests on an edge, except at the tip of the rod. My assumptions for the previous cases seem to be correct.

I also concluded the following for the normal contact force acting on a rod in equilibrium against a wall :

enter image description here

There's friction force as well, but I have excluded it.

But what if the edge of the ladder and the edge of the wall intersect at A as shown below? enter image description here

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  • $\begingroup$ It depends on how the materials deform; this is a case where the simple high school notion of “normal force” breaks down. $\endgroup$
    – knzhou
    May 26, 2022 at 19:27
  • $\begingroup$ what if there is no mention of deformation ? The only thing known is that the rod is uniform and that the sphere/hemisphere is fixed. I'm trying to understand the force diagram for a similar physics question. In particular, for case 2, I don't understand why the reaction force at A is not towards C. $\endgroup$
    – creme332
    May 27, 2022 at 4:19
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    $\begingroup$ In case 2, the implicit assumption seems to be that the rod is hard (so it does not deform, explaining the force at point A being perpendicular to the rod) and its end is slightly rounded (so that at point B, the two surfaces are tangent, and it’s just a regular normal force). But in general this kind of question is under specified and the question writers are making a mistake if they don’t add more information. $\endgroup$
    – knzhou
    May 27, 2022 at 5:34

2 Answers 2

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The “normal” force is defined as the component of the contact force from one object on another which is perpendicular to the surface. In the first three of the situations you present, there is also a component of the contact force (friction) which is parallel to the surface. This assumes that at least one of the surfaces can be considered locally “flat” at the point of contact.

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  • $\begingroup$ In case 2, the rod lies on the rim of the hemisphere. Is the rim of the hemisphere considered to be locally flat? $\endgroup$
    – creme332
    May 27, 2022 at 4:05
  • $\begingroup$ Also, how do you distinguish between points A and B for case 2 ? How are the surfaces different? $\endgroup$
    – creme332
    May 27, 2022 at 4:21
  • $\begingroup$ In case two, if the angle was such that the horizontal components of the two (properly sketched) normal forces were equal, then one could sustain a static system without friction. At point B, the sketched force is the reaction from the normal force which the rod exerts on the sphere. At both points, as long as one curved surface has a non-zero radius then a very small area of contact approaches flatness. $\endgroup$
    – R.W. Bird
    May 27, 2022 at 13:54
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Geometrically speaking, the direction normal to a surface doesn't always exist. When a surface has a sharp angle, it has no defined normal on that angle.

In your case 2, the normal direction is undefined on $A$. That doesn't mean that the contact force doesn't exist: you can for example assume that the system is at rest, and you'll deduce the contact force on $A$ from the rest.

Same on case 5. The best you can do is to develop the contact force on the horizontal and vertical directions, but without knowing beforehand its overall direction. So if the answer in this book assumes that its normal to the rod without any proof, then the author was careless.

Also, as pointed out on another comment above, if there's friction, the contact force is no longer normal (its tangential component is the friction force).

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  • $\begingroup$ Is there a way to deduce the direction of the normal contact force for the different cases without calculations? In an exam, time is limited and I have to draw the force diagram correctly on the first try. For time being, I am simply memorizing the different cases. $\endgroup$
    – creme332
    May 27, 2022 at 7:40
  • $\begingroup$ You could write Newton's third law to deduce the contact force corner$\to$rod from its opposite rod$\to$corner. The latter is easily defined, since the rod has no angle on the contact point. $\endgroup$
    – Miyase
    May 27, 2022 at 8:06
  • $\begingroup$ I don't get it. There are 2 possibilities for the direction of the normal contact force at A in case 2 : either it is normal to rod or normal to surface on which rod is resting. How does Newton's law distinguish between these two cases? On another note, what would happen when the edge of rod coincides with corner (see the new diagram I've added to my question) $\endgroup$
    – creme332
    May 27, 2022 at 8:21
  • $\begingroup$ In case 2, the surface has no normal at $A$ because of the angle. So you can't find the correct orientation of the contact force just by looking at the surface. You have to switch to the rod and use Newton's third law. $\endgroup$
    – Miyase
    May 27, 2022 at 8:25

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