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Normal Force on Incline Calculator

Enter mass and slope angle to find the normal force on the incline.

Result

169.86 N
98.07 N
196.13 N

How it is calculated

The weight of the object is split into two parts relative to the slope. The weight is W = m × g, with the mass m in kilograms and gravity g in m/s²; if the gravity box is empty, 9.80665 is used, and you can type another value (for example 1.62 for the Moon). With the slope angle θ in degrees from the horizontal, the normal force perpendicular to the surface is N = m × g × cos θ, and the component pulling the object down the slope is F∥ = m × g × sin θ. All three forces are shown in newtons with two decimals. The angle must lie between 0 and 90 degrees, the mass cannot be negative and gravity must be above zero; otherwise no result is shown. The result assumes a rigid, stationary object on a flat incline with no other applied force. Friction is not applied, so the tool does not say whether the object slides.

Worked example

A 20 kg crate sits on a 30 degree ramp on Earth. Weight: 20 × 9.80665 = 196.13 N. The cosine of 30 degrees is 0.8660, so the normal force is 196.13 × 0.8660 = 169.86 N, shown as 169.86 N. The sine is 0.5, so the force along the slope is 196.13 × 0.5 = 98.07 N. The two components are perpendicular, so they combine back to the weight: the square root of 169.86² + 98.07² is 196.13 N.

Try this example →

Common mistakes

  • Measuring the angle from the vertical instead of from the horizontal. The angle is between the slope and the horizontal; a 30 degree slope gives cos 30 for the normal force, not sin 30.
  • Entering the angle in radians. The field is in degrees, so 0.52 is read as about half a degree and the normal force comes out almost equal to the weight (196.12 N for 20 kg).
  • Entering the weight in newtons, or the mass in grams, in the mass field. The field is kilograms and the tool multiplies by g itself.
  • Assuming the normal force equals the weight. That holds only on a flat surface; on a slope it is smaller by the factor cos θ.
  • Entering an angle above 90 degrees, for example 95, or a negative one. The tool shows no result, because a slope runs from 0 (flat) to 90 (vertical wall) degrees.

About this calculator

On an incline, with or without friction, the normal force is the component of a resting object's weight that acts perpendicular to the slope's surface: N = mg·cos(θ), where m is mass, g is gravitational acceleration, and θ is the incline angle from horizontal. The remaining weight component, mg·sin(θ), acts parallel to the surface (downslope) and is what friction or a holding force must counteract to keep the object in place.

This decomposition is one of the first applications of vector resolution taught in introductory physics and statics courses, used to analyze everything from a block on a ramp to a climber on a rock face. Engineers designing ramps, conveyor systems and retaining structures use the same N = mg·cos(θ) relationship to compute the load a surface must bear and, combined with a friction coefficient, whether an object will stay put or slide.

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Comments (1)

  • Editorial team

    The tool gives forces, not motion. To see whether an object starts sliding, compare the tangent of the slope angle with the friction coefficient of the two surfaces: at 30 degrees the tangent is about 0.577. At 90 degrees the normal force is zero, since the surface is a vertical wall. Which pair of surfaces and which friction coefficient are you working with?

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