Normal Force
The normal force is the contact force exerted by a surface on an object, acting perpendicular (normal) to the surface at the point of contact. It is a reaction force that prevents objects from passing through solid surfaces and adjusts in magnitude to balance components of other forces. On a flat horizontal surface, the normal force on a stationary object equals its weight; on an inclined surface, it equals the component of weight perpendicular to the slope.
Key Formula
N = m × g × cos(θ)
LaTeX: N = mg\cos\theta
| Symbol | Meaning | Unit |
|---|---|---|
| N | Normal force | Newton (N) |
| m | Mass of the object | kilogram (kg) |
| g | Acceleration due to gravity (9.8 m/s² on Earth) | m/s² |
| θ | Angle of the surface from horizontal | degrees (°) or radians (rad) |
Worked Example
Problem
A 20 kg box rests on a ramp inclined at 30° to the horizontal. What is the normal force acting on the box? Use g = 9.8 m/s².
Solution
N = m × g × cos(θ) = 20 × 9.8 × cos(30°) = 196 × 0.866 = 169.7 N.
Answer
The normal force on the box is approximately 169.7 N, perpendicular to the ramp surface.
Normal Force for a 10 kg Object at Different Surface Angles
| Surface Angle θ (°) | cos(θ) | Normal Force N (N) | Scenario |
|---|---|---|---|
| 0 | 1.000 | 98.0 | Flat horizontal floor |
| 15 | 0.966 | 94.7 | Gentle ramp |
| 30 | 0.866 | 84.9 | Moderate incline |
| 45 | 0.707 | 69.3 | Steep ramp |
| 60 | 0.500 | 49.0 | Very steep slope |
| 90 | 0.000 | 0 | Vertical wall (no support) |
Interactive Tools
PhET Forces and Motion Basics
Visualise normal force and its interaction with weight on flat surfaces
Open ToolDesmos Graphing Calculator
Plot N = mg cos(θ) to see how normal force changes with incline angle
Open ToolKhan Academy — Normal Force
Lesson on normal force with worked examples on flat and inclined surfaces
Open Tool
Wikimedia Commons, CC BY-SA
Related Terms
Weight
Weight is the gravitational force exerted on an object due to a gravitational field, typically Earth's. It is a vector quantity directed toward the centre of the gravitational body and varies depending on the local gravitational acceleration. A person who weighs 686 N on Earth would weigh only about 114 N on the Moon, because the Moon's gravitational acceleration is approximately one-sixth that of Earth.
Friction
Friction is a contact force that opposes the relative motion or tendency of motion between two surfaces in contact. It arises from microscopic interactions between surface irregularities and is directly proportional to the normal force through the coefficient of friction. Friction is essential in everyday life — it allows us to walk, cars to brake, and objects to remain stationary on inclined surfaces — but it also causes energy loss as heat in machines.
Newton's Third Law
Newton's Third Law of Motion states that for every action force, there is an equal and opposite reaction force. These two forces always act on different objects simultaneously, forming an action-reaction pair. The law explains how rockets propel themselves through space, how we walk by pushing the ground backward, and why swimming is possible by pushing water backward to move forward.
'Normal' comes from Latin 'normalis' meaning made according to a carpenter's square — referring to a right angle. In geometry and physics, 'normal' means perpendicular to a surface. The concept became central to Newtonian mechanics in the analysis of contact forces.