What Is Normal Force?
Normal force is the perpendicular contact force that a surface exerts on an object resting on it. It prevents the object from falling through the surface. The word "normal" here means perpendicular — the force acts at a right angle to the surface, not along it.
Normal force is a direct consequence of Newton's Third Law of Motion: if an object exerts a force on a surface (its weight), the surface exerts an equal and opposite force on the object. The symbol for normal force is FN or simply N, and its SI unit is the newton (N).
Normal Force Formulas
1. Flat Surface (No External Force)
When an object rests on a horizontal surface with no additional forces applied:
FN = m × g
where m is the mass of the object and g = 9.80665 m/s² (standard gravitational acceleration).
In this case, the normal force equals the object's weight, W = mg.
2. Inclined Surface
On a tilted surface at angle α from the horizontal, only the component of gravity perpendicular to the surface must be balanced:
FN = m × g × cos(α)
As the angle increases, the normal force decreases. At α = 90° the surface is vertical and FN = 0.
3. External Downward Force
If an external force F pushes the object into the surface at angle x above the horizontal:
FN = m × g + F × sin(x)
The downward component of the external force adds to the gravitational force, increasing the normal force.
4. External Upward Force
If an external force F lifts the object at angle x above the horizontal:
FN = m × g − F × sin(x)
The upward component reduces the apparent weight pressing on the surface, decreasing the normal force. If the upward force equals gravity, FN = 0 and the object hovers.
Worked Example
A box has a mass of 100 kg. You push on it at a 45° angle with 250 N of force directed downward into the surface.
- Weight: W = 100 × 9.807 = 980.7 N
- Downward component of push: 250 × sin(45°) = 250 × 0.7071 ≈ 176.8 N
- Normal force: FN = 980.7 + 176.8 = 1157.5 N
The ground pushes back with 1,157.5 N. Because you pushed downward, you actually made it harder to move the box — the increased normal force raises friction. To save effort, push horizontally (x = 0°).
Metric vs. Imperial Units
In the metric system, mass is measured in kilograms (kg) and force in newtons (N). The gravitational acceleration is g = 9.80665 m/s².
In the US customary (imperial) system, mass is commonly given in pounds-mass (lb) or slugs, and force in pound-force (lbf). On Earth's surface, an object with a mass of 1 lb weighs 1 lbf. The relationship is 1 lbf = 4.44822 N.
This calculator accepts mass in kg, grams, metric tonnes, lbs, ounces, or slugs, and external forces in N, kN, lbf, or kip — converting everything internally to SI for the calculation and then displaying results in all major units.
FAQs
Is normal force always equal to weight?
Only on a flat, horizontal surface with no external forces. On an inclined surface or with additional forces applied, the normal force differs from the object's weight.
Can normal force be zero or negative?
Normal force can be zero (e.g., free-fall, or when a strong enough upward force is applied). It cannot be negative because a surface can only push, not pull. If the formula gives a negative result, the object has left the surface.
What is the relationship between normal force and friction?
Friction force = μ × FN, where μ is the coefficient of friction. A higher normal force means more friction, which is why pushing down on an object while trying to slide it makes it harder to move.
Does normal force act on inclined planes?
Yes. On an inclined plane, the normal force is perpendicular to the inclined surface, not vertical. It equals m × g × cos(α), where α is the angle of inclination.
What is the unit of normal force?
In the SI system the unit is the newton (N). In the US customary system it is the pound-force (lbf). 1 lbf ≈ 4.448 N.