# What is reaction force

1. Jul 24, 2014

### 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 $v^2\,=\,rgh\cos x$ .

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