What are Reaction Forces and how do they relate to Newton's third law of motion?

[Greg Bernhardt]

Definition/Summary

Between two bodies in contact, there is a pair of Reaction Forces.

They are always equal but opposite (this is precisely Newton's third law), and so it is usual (though slightly misleading) to refer to "the Reaction Force", in the singular.

The component of the Reaction Force perpendicular (normal) to the surface, in the outward direction, is known as the Normal Force.

The component of the Reaction Force parallel to the surface is known as the Friction Force.

If the Normal Force becomes zero, then the two bodies (literally) lose contact.

If the surface of contact is curved, then, in addition to the Reaction Force, there may be a Reaction Torque (a rotational force).

Equations

$\mathbf{R}_{AB}\,=\,- \mathbf{R}_{BA}$

Extended explanation

The Reaction Force is basically the "missing force" needed to balance all the other forces and the acceleration.

Where more than two bodies are in contact, the best way of calculating all the reaction Forces is usually to deal with each block separately, starting at one end, and calculating the Reaction Forces one at a time until reaching the other end.

For example, if n blocks of equal mass m and with vertical sides are being pushed with (obviously) equal acceleration on a flat horizontal table by a force F applied perpendicularly to the side of the nth block, then the Reaction Force between the kth and (k+1)th block is kF/n horizontally.

Similarly, if n blocks of equal mass m rest on each other, or hang vertically from each other under gravity, then then the Reaction Force between the kth and (k+1)th block is kmg vertically.

The Reaction Force is principally used to determine whether a body will slide or topple or lose contact.

Sliding:

The relative acceleration between the surfaces in the normal direction is (obviously!) zero, whether the surfaces are sliding or not.

So the Normal Force can be calculated without reference to sliding or to the Friction Force.

Then, if the force needed to prevent sliding is greater than the Normal Force times the coefficient of static friction, the surfaces will slide.

Toppling:

Toppling means rotating about an edge of the area of contact.

If the line of the Reaction Force passes through the area of contact (or through any chord of that area, for a non-convex area), then there will be no toppling.

For example, if a uniform cylinder of mass m and radius r is at rest on a flat horizontal table, the Reaction Force is mg vertically upward through the centre of the bottom face, so as to balance the only other force (the weight of the cylinder).

However, if a force is applied to a point at height z on the side of the cylinder, with an upward component N, and with a radial horizontal component H small enough not to overcome the friction with the table, so that the cylinder remains at rest, the Reaction Force is a Friction Force of -H and a Normal Force of mg - N upward.

The position of the point of application of that Reaction Force is at a distance x beyond the centre of the bottom face, which is most easily found by taking Moments about the point of application: $mgx = Hz + N(x+r)$ so $x\,=\,\frac{Hz+Nr}{mg-N}$

If N is sufficiently large that $x\,>\,\frac{Hz+Nr}{mg-N}$ then the cylinder will topple.

Losing contact:

A positive Normal Force between two bodies keeps them in contact.

A zero (or negative) Normal Force will not keep them in contact.

To find out whether and when contact will be lost (for example, when a rollercoaster descends a steep slope, or tries to loop-the-loop), calculate the Normal Force, and find when it becomes zero.

For example, for planar motion with speed v at an angle x below the horizontal along a path with radius of curvature r (r for a flat surface, of course, is infinite) and subject to gravity g, the Normal Force will be $mgh\cos x - mv^2/r$, and contact will be lost when and if $v^2\,=\,rgh\cos x$ .

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[kuruman]

Good answer, Greg. Now this post can be removed from your 0 replies list.

 

[haruspex]

If the surface of contact is curved, then, in addition to the Reaction Force, there may be a Reaction Torque

In addition? Surely any torque arises from the forces?
I would write instead "If the contact is over an area then the reaction forces may be distributed over that area, but there will be a single equivalent net force with a line of action such that it exerts the same net torque".

The Reaction Force is basically the "missing force" needed to balance all the other forces and the acceleration.

Not sure that's helpful. The starting point must be any info re how the bodies may move in relation to each other. If we assume they move as one, we can deduce what that motion is then see what the reaction force must be by considering one body in isolation. We can then check whether the assumption was correct: does the normal component go negative; does the tangential component exceed max static friction?
Things get tricky here for friction over a curved surface contact.

I usually define the normal force between rigid bodies as the force pair of minimum magnitude that prevents interpenetration.

the force needed to prevent sliding

A bit ambiguous. "The force that would be needed to prevent sliding in the absence of friction ".

 

[sophiecentaur]

The component of the Reaction Force perpendicular (normal) to the surface, in the outward direction, is known as the Normal Force.

The component of the Reaction Force parallel to the surface is known as the Friction Force.

Those statements should be read in the context of this very simplified / idealised scenario. It's important not to get hung up on the arbitrary names used. Resolving into Normal and Tangential forces is only for convenience but it may not always fit the situation. The answer to "Why do it that way?" is just because it works best - mostly.

 

[jbriggs444]

I'd remove almost every occurrence of the word "Reaction" and replace it with "contact". The notion of a "reaction force" (as opposed to what other kind of force?) has no relevance to the rest of the discussion.

The distinction between an "interaction" force and an "inertial" or "fictitious" force might be usefully made, but that is not the subject matter of the OP here.

 

[sophiecentaur]

I'd remove almost every occurrence of the word "Reaction" and replace it with "contact". The notion of a "reaction force" (as opposed to what other kind of force?) has no relevance to the rest of the discussion.

The distinction between an "interaction" force and an "inertial" or "fictitious" force might be usefully made, but that is not the subject matter of the OP here.

I sympathise with your frustration but I have to wonder whether it may be a bit too late to change terminology. Forces are forces, whatever you call them and they should be marked on a diagram with suitable keys or labels. N3 always applies so people can take it from there. The Maths will yield the answer

 

[haruspex]

remove almost every occurrence of the word "Reaction" and replace it with "contact".

Good point, I agree completely. According to Newton's terminology, a reaction force arises to oppose an exerted force, which is not the usage here.

 

[sophiecentaur]

a reaction force arises to oppose an exerted force,

Isn't that a bit specific. If you and I both push against each other there is still a third law pair but which would be which?
I think this thread could be going nowhere because very few of the contributions are actually wrong and who is actually bothered by what Sir Isaac was actually referring to? He got it right, didn't he?

 

[jbriggs444]

Isn't that a bit specific. If you and I both push against each other there is still a third law pair but which would be which?
I think this thread could be going nowhere because very few of the contributions are actually wrong and who is actually bothered by what Sir Isaac was actually referring to? He got it right, didn't he?

As we all agree, it is arbitrary. If you have an "action force" then its third law partner will be a "reaction force". If neither is the "action" then it is simply the case that the third law partners are equal and opposite. Indeed, Newton got that part exactly right.

"Actioni contrariam semper et æqualem esse reactionem: sive corporum duorum actiones in se mutuo semper esse æquales et in partes contrarias dirigi."

[To every action there is always opposed an equal reaction; or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.]

 

[haruspex]

the mutual actions of two bodies upon each other are always equal, and directed to contrary parts

Thanks, I had never heard that part. Pity it is not the form generally quoted.