Normal Force Formula: N = mg cos θ on Slopes, Lifts and Floors

The normal force formula for an object resting on a slope is $N = mg\cos\theta$ , and on a level floor, with θ = 0, it becomes $N = mg$ . A 10 kg box on a 30° ramp is pushed on by the ramp with $10 \times 9.8 \times 0.866 \approx 84.9\ \text{N}$ .

Normal means perpendicular. The normal force is the push a surface gives, at right angles to itself, to stop an object sinking into it. It is not a fixed value: it grows or shrinks to match whatever else is pressing the object into the surface. That is why it matters so much in the laws of motion and why the calculator below assumes g = 9.8 m/s².

Quick facts

Level floor
$N = mg$
Incline
$N = mg\cos\theta$
Lift accelerating up
$N = m(g + a)$
SI unit
newton (N)
Dimensional formula
[M¹ L¹ T⁻²]
Direction
Perpendicular to the contact surface, away from it

Formula for normal force in common situations

Normal force for different set ups
Situation	Normal force	Why
Box on a level floor	$N = mg$	Only weight presses down
Box on an incline at θ	$N = mg\cos\theta$	Only the part of the weight across the slope presses in
Pushed down at angle φ below horizontal	$N = mg + F\sin\phi$	The push adds to the weight
Pulled up at angle φ above horizontal	$N = mg - F\sin\phi$	The pull lifts part of the weight
Person in a lift with upward acceleration a	$N = m(g + a)$	Floor must also provide the acceleration
Car at the top of a hump of radius r	$N = mg - \frac{mv^2}{r}$	Part of the weight supplies the centripetal force

In each row, start from $\sum F = ma$ perpendicular to the surface and solve for N. Memorising the rows is less useful than being able to rebuild them.

Why the normal force on a slope uses cos θ

Split the weight mg into two parts. Along the slope it is $mg\sin\theta$ , the part that tries to slide the object down. Into the slope it is $mg\cos\theta$ . The surface only has to resist the second part, so if nothing else pushes in or out, $N = mg\cos\theta$ .

Check the limits. At θ = 0 the slope is flat and cos 0 = 1, giving N = mg. At θ = 90° the surface is a vertical wall and cos 90° = 0, so a block falling past it feels no normal force. A formula that passes both checks is probably right.

Worked examples on the normal force

Example 1: a person standing still

Find the normal force on a 60 kg student standing on level ground.

$N = mg = 60 \times 9.8 = 588\ \text{N}$

Example 2: box on a ramp

A 10 kg box rests on a ramp inclined at 30°. Find the normal force.

$N = 10 \times 9.8 \times \cos 30^\circ = 98 \times 0.866 = 84.9\ \text{N}$

This is the value the calculator gives with m = 10 and th = 30.

Example 3: scale reading in a lift

The same 60 kg student stands on bathroom scales in a lift accelerating upward at 2.0 m/s². What do the scales read?

Up is positive: $N - mg = ma$ , so $N = m(g + a) = 60 \times (9.8 + 2.0)$ .

$N = 60 \times 11.8 = 708\ \text{N}$

The scales show the normal force, so the student appears heavier. Accelerating downward at 2.0 m/s², they would read 60 × 7.8 = 468 N.

Normal force formula and friction

Friction depends on the normal force through $f = \mu N$ . Anything that changes N changes friction. Pulling a suitcase with the handle tilted up lifts some of its weight, cuts N, and makes it easier to drag. Pressing down on a sanding block raises N and increases the grip. See the friction force formula and the coefficient of friction formula for the numbers.

Normal force is not the reaction to weight

A common belief is that the normal force and the weight form a Newton's third law pair. They do not. The weight is the pull of Earth on the box; its partner is the pull of the box on Earth. The normal force is the push of the floor on the box; its partner is the push of the box on the floor. On a level floor with no acceleration, N happens to equal mg because the net force is zero, but the lift example shows they can differ. The weight formula page explains the difference between real and apparent weight.

If the normal force works out negative, the object has left the surface. Set N = 0 at that point instead.

Mistakes with N to watch for

Using N = mg on a slope or with an angled force.
Using sin θ instead of cos θ for the component into the slope.
Forgetting that a calculator must be in degree mode for cos 30°.
Drawing N vertical on a slope instead of perpendicular to the surface.

Frequently asked questions

What is the formula for normal force?

On a level surface with no other vertical forces, N = mg. On an incline at angle θ, N = mg cos θ. Other cases follow from balancing forces perpendicular to the surface.

Is normal force always equal to weight?

No. It equals the weight only on a level surface when nothing else pushes up or down and there is no vertical acceleration.

In what unit is normal force measured?

In newtons, N, just like weight, tension and friction.

Why is it called the normal force?

In geometry, normal means perpendicular. The force acts at right angles to the surface of contact.