What this calculator does
Normal force is the push a surface exerts back on an object resting on it, perpendicular to that surface, and it is what stops objects falling straight through solid ground. On a flat, horizontal surface it simply balances the object's full weight (N = mg), but on a slope only part of the weight presses directly into the surface, so the normal force there is smaller than the object's full weight.
The normal force is one of the more misunderstood forces in introductory physics, mainly because people assume it always equals weight. It only does so on a flat surface with nothing else pushing or pulling. On an incline, the surface only has to resist the component of weight pressing straight into it (mg·cos θ), while the remaining component acts along the slope and is instead resisted by friction or, absent that, causes the object to slide.
The formula
On a flat surface, normal force equals the full weight of the object: N = mg, where g is standard gravity, 9.80665 m/s². On an incline tilted at angle θ from horizontal, only the component of weight perpendicular to the surface counts: N = mg·cos(θ). Setting the incline angle to 0° reduces this exactly to the flat-surface case.
| Term | Meaning |
|---|---|
| N | Normal force, the force the surface exerts perpendicular to itself, in newtons. |
| m | The mass of the object, in kilograms. |
| g | Standard gravity, 9.80665 m/s², a fixed physical constant. |
| θ | The incline angle, measured from horizontal; 0° is a flat surface. |
The inputs explained
| Field | What to enter |
|---|---|
| Mass (kg) | The mass of the object resting on the surface. |
| Incline angle (0 for a flat surface) (°) | The angle of the surface from horizontal. Leave at 0 for a flat, level surface. |
When to use it
Working out friction available on a slope
Friction force is proportional to normal force (friction = μ × N), so finding normal force on an incline is usually the first step before working out how much friction is resisting motion on that slope.
Checking why heavier objects do not always feel proportionally heavier on a ramp
On an incline, only the mg·cos(θ) portion of weight presses into the surface; the steeper the ramp, the smaller that fraction becomes, which is part of why a loaded trolley feels easier to hold back on a gentle ramp than a steep one, force-wise.
Introductory mechanics problems
Normal force is a standard first step in incline, friction and connected-object problems in physics coursework, usually worked out before friction, acceleration or tension are tackled.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How normal force falls as an incline gets steeper
A fixed 10 kg mass, on surfaces tilted at a range of angles from flat to steep.
Questions
What is the formula for normal force?
On a flat, horizontal surface, N = mg (mass times gravity). On an incline, N = mg·cos(θ), where θ is the angle of the surface from horizontal. The incline formula reduces to the flat-surface formula exactly when θ is 0°.
Is normal force always equal to weight?
Only on a flat, horizontal surface with no other vertical forces acting. On an incline, normal force is less than full weight because only part of the weight presses directly into the surface. Extra vertical forces, such as someone pushing down on the object, would also change normal force away from a simple mg.
Why does normal force decrease on a steeper incline?
Weight always acts straight down, but on a slope only the component of that weight perpendicular to the surface contributes to normal force; the steeper the slope, the more of the weight acts along the surface instead of into it, so the perpendicular (normal) component, and hence normal force, shrinks.
How is normal force related to friction?
Friction force is calculated as the coefficient of friction multiplied by normal force (F = μN), so a smaller normal force on a steep incline directly means less friction is available to resist sliding, which is one reason steep slopes are more likely to cause an object to slide.
For the full picture of forces and acceleration on a slope with friction, see the incline with friction calculator.