Physics · Forces & momentum

Normal force calculator

Find the surface reaction on a horizontal or inclined plane, including a signed vertical load on a horizontal surface. The normal force cannot pull a body toward the surface.

Created Oct 3, 2026v1.0

By Thibaut Grzelak, Mechanical Analysis Engineer

Forces & momentum

Inclined surface and block: N is perpendicular to the surface, mg points down. Arrow lengths are schematic.

Quantity to calculate

Mass

Positive mass of the body.

Gravity

Positive local gravitational acceleration.

Incline angle

Inclination from horizontal: 0 ≤ θ < 90°.

Vertical force ↓ +

Positive downward; negative upward. Contact can be lost.

Results

Normal force

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Zero means no compressive contact force in this unilateral-contact model.

Relation and numerical substitutionN = m × g

—

No acceleration normal to the plane while contact is maintained; rigid unilateral contact and gravity uniform over the body.

01Formulas and symbols

Formulas used

NN = m × gZero means no compressive contact force in this unilateral-contact model.

NN = m × g × cos(θ)Zero means no compressive contact force in this unilateral-contact model.

NN = max(0, m × g + F_v)Zero means no compressive contact force in this unilateral-contact model.

02Assumptions and limits

Scope of validity

No acceleration normal to the plane while contact is maintained; rigid unilateral contact and gravity uniform over the body.

03Validation example

Reference numerical case

10 kg × 9.80665 m/s² = 98.0665 N horizontally. At 30°, N = 98.0665 × cos(30°) = 84.928… N.

04References

Technical references

University Physics 1, §6.1 Newton’s laws — OpenStax

FAQ

Can N be negative?

No. This unilateral surface can push but cannot pull. At loss of contact the displayed reaction is zero.

Does this solve sliding with friction?

No. Use the inclined-plane or friction tools for tangential forces; this page isolates the surface reaction.

05Model and conventions

Find the surface reaction on a horizontal or inclined plane, including a signed vertical load on a horizontal surface. The normal force cannot pull a body toward the surface.

06Limits of this model

The incline mode only calculates N, not friction or motion down the slope. When an upward force exceeds weight, N becomes zero and this equilibrium model no longer predicts the trajectory.

07Common mistakes

Measure θ from horizontal. A positive vertical force pushes down; a negative one pulls up. Do not use mg as the normal force on an incline.