Normal Force Calculator finds the force a flat surface, slope or elevator floor exerts on an object, with or without a push or pull at any angle, in N, lbf or kgf, with full steps.
Pick the surface first. Angles are measured from the surface, not from the vertical.
Gravity and Result Unit
Step-by-Step Working
- Known values: m = 100 kg, g = 9.80665 m/s², α = 30°
- Weight: W = m × g = 100 × 9.80665 = 980.665 N
- Part of the weight pressing into the slope: W × cos(α) = 980.665 × 0.866025 = 849.281 N
- Formula: FN = m × g × cos(α)
- Substitute: 100 kg × 9.80665 m/s² × cos(30°)
- Result: FN = 849.281 N
Normal Force at Other Slope Angles
| Slope angle | Normal force (N) | Weight along slope (N) |
|---|---|---|
| 0° | 980.7 | 0 |
| 15° | 947.2 | 253.8 |
| 30° | 849.3 | 490.3 |
| 45° | 693.4 | 693.4 |
| 60° | 490.3 | 849.3 |
| 75° | 253.8 | 947.2 |
Normal Force Calculator for Five Surface Setups
Normal force is the push a surface gives back to an object resting on it. It always acts at a right angle to the surface, which is what “normal” means in geometry. A table pushing up on a book and a ramp pushing out on a crate are both normal forces.
The Normal Force Calculator starts with the Surface and Forces menu. It covers a flat surface, an inclined surface, a flat surface with a force pushing down, a flat surface with a force pulling up, and an accelerating elevator floor. Each choice shows only the inputs that case needs.
Mass can be entered in kilograms, grams, pounds or slugs. Gravity sits under Gravity and Result Unit and starts at standard gravity, 9.80665 m/s². Change it there to work a problem on the Moon or Mars, or to match a textbook that rounds g to 9.8 or 9.81.
The main result reads like 849.3 N Perpendicular to the Slope, which is the answer for a 100 kg mass on a 30° incline. The Result Unit option switches that figure to kilonewtons, pounds-force or kilograms-force.
Normal Force Formula for Each Case
Every normal force calculation starts from the weight, m × g. What changes from case to case is how much of that weight presses straight into the surface, and whether another force adds to it or takes some away.
| Situation | Formula | 100 kg example |
|---|---|---|
| Flat surface | FN = m × g | 980.7 N |
| Inclined surface at 30° | FN = m × g × cos(α) | 849.3 N |
| 250 N pushing down at 45° | FN = m × g + F × sin(x) | 1,157 N |
| 250 N pulling up at 45° | FN = m × g − F × sin(x) | 803.9 N |
| Elevator accelerating up at 2 m/s² | FN = m × (g + a) | 1,181 N |
The flat and incline forms match those in OpenStax University Physics, which writes them as N = mg and N = mg cos θ. The last three follow from Newton’s second law applied along the direction perpendicular to the surface.
How to Calculate Normal Force on a Flat Surface
On a level surface with no other vertical force, the normal force equals the weight. This is the vertical normal force formula that most homework problems begin with, and the Flat surface setting applies it.
$$F_{N} = m g$$
A 0.6 kg book on a desk gives 0.6 × 9.80665 = 5.884 N. A 60 kg person standing on a floor gives 588.4 N, which is 132.3 lbf. The surface pushes up exactly as hard as gravity pulls down, so the net vertical force is zero.
How to Calculate Normal Force on an Incline
On a slope, only part of the weight presses into the surface. That part is the weight multiplied by the cosine of the incline angle, measured from the horizontal. The Normal Force Calculator loads this case first.
$$F_{N} = m g \cos(\alpha)$$
For 100 kg on a 30° slope the weight is 980.665 N and cos(30°) is 0.866025. The normal force is 849.3 N, which is 13.4% less than the weight. OpenStax works the same method for a 60 kg skier on a 25° slope and gets about 533 N.
The rest of the weight, m × g × sin(α), points down the slope. The Inclined Plane Calculator follows that part through to the sliding acceleration, travel time and final speed on a ramp.
Normal Force Formula with Applied Force at an Angle
A push or pull at an angle changes the load on the surface. Only its vertical part matters, and that part is F × sin(x) when x is measured from the surface. Two formulas cover the two directions.
$$F_{N} = m g + F \sin(x)$$
$$F_{N} = m g – F \sin(x)$$
The plus sign is for a force pushing down and the minus sign is for a force pulling up. Pushing 250 N down at 45° on a 100 kg box adds 176.8 N, so the normal force is 1,157 N. Pulling the same 250 N upward takes 176.8 N away and leaves 803.9 N.
In the Normal Force Calculator, enter the size of the force as a positive number and pick push or pull from the first menu. A force at 0° runs along the surface and leaves the normal force at m × g. A force at 90° acts straight down or straight up.
Normal Force in an Accelerating Elevator
An elevator floor has to hold the weight and also supply the net force that speeds the rider up or slows the rider down. Upward acceleration is entered as a positive number and downward as a negative one.
$$F_{N} = m (g + a)$$
A 100 kg load in an elevator accelerating upward at 2 m/s² feels 1,181 N from the floor, or 1.204 g. Accelerating downward at 2 m/s² drops it to 780.7 N. The Newton’s Second Law Calculator handles the net force m × a on its own.
| Acceleration (m/s²) | Normal force on 100 kg (N) | g-force felt (g) |
|---|---|---|
| −9.807 | 0 | 0 |
| −4.903 | 490.3 | 0.5 |
| 0 | 980.7 | 1 |
| 2 | 1,181 | 1.204 |
| 4.903 | 1,471 | 1.5 |
| 9.807 | 1,961 | 2 |
| 19.61 | 2,942 | 3 |
These are the rows the Normal Force Calculator lists under Normal Force at Other Accelerations for the 2 m/s² example. At −9.807 m/s² the floor falls away as fast as the load does. The tool then reports 0 N with the note Free Fall: Weightless.
Is Normal Force the Same as Weight?
No. Weight is the pull of gravity on an object, and normal force is the push of a surface on it. The two are equal in size only on a level surface with no other vertical force and no vertical acceleration.
The second result card shows the gap. It gives the weight, the percentage change from weight, and a headline such as 0.866 g Felt by the Object. That figure is the normal force divided by the object’s weight under standard gravity.
The two forces are also not an action-reaction pair, because both act on the same object. The reaction to the table’s push on a book is the book’s push on the table. The Newton’s Third Law Calculator works with pairs like that one.
How Is Normal Force Measured?
A bathroom scale measures normal force directly. The spring or load cell inside reads how hard the platform pushes up on you, then displays that force as kilograms or pounds. That is why a scale reads high in an elevator that is speeding up.
The first result card of the Normal Force Calculator gives the same idea in numbers. For the 30° slope it shows 190.9 lbf, 86.6 kgf and 0.8493 kN. The 86.6 kgf figure is what a scale lying on the slope under the 100 kg mass would display.
Can Normal Force Be Negative or Zero?
A surface can push but it cannot pull, so the normal force never drops below zero. If the formula gives a negative number, contact has been lost. The tool reports 0 N with the note Object Lifts Off the Surface.
Pulling up on the 100 kg box at 45° lifts it once the force reaches 1,387 N. At 30° it takes 1,961 N, because a shallow pull has a smaller vertical part. The fourth card of the Normal Force Calculator lists the pull that lifts the object at the angle you entered.
Friction and Sliding Next to the Normal Force
Most people run a Normal Force Calculator as a step toward friction. The largest friction a surface can supply is the friction coefficient times the normal force, μ × FN. The third result card answers the next question, which is whether the object stays put.
On the 30° slope it reads 4.903 m/s² Down the Slope, the acceleration with no friction at all. It also lists a friction μ of 0.5774 to stay put, which is tan(30°), and shows that 50% of the weight acts along the slope.
With an applied force, the same card shows the part of the force that runs along the surface. For the 250 N push at 45° that is 1.768 m/s² along the surface and a μ of 0.1527 to hold the box still. The Friction Calculator turns μ and a normal force into a friction force.
The fourth card on a slope compares other angles. At half the angle, 15°, the normal force is 947.2 N, and at double the angle, 60°, it is 490.3 N. The change per extra degree is −8.558 N at 30°, and it grows as the slope steepens.
| Slope angle | Normal force on 100 kg (N) | Weight along slope (N) |
|---|---|---|
| 0° | 980.7 | 0 |
| 15° | 947.2 | 253.8 |
| 30° | 849.3 | 490.3 |
| 45° | 693.4 | 693.4 |
| 60° | 490.3 | 849.3 |
| 75° | 253.8 | 947.2 |
A car on a banked curve is a related case. There the normal force tilts inward and supplies part of the turning force, so it can be larger than the weight. The Angle of Banking Calculator covers that setup.
Normal Force Unit and kg-Force Conversions
The SI unit of normal force is the newton, the same as for any other force. One newton is the force that accelerates 1 kg at 1 m/s². It equals about 0.2248 lbf, or the weight of roughly 102 grams.
A newton is not equal to a kilogram, since one is a force and the other is a mass. The link between them is gravity. To calculate kg-force, multiply the mass in kilograms by 9.80665, which gives newtons. One kilogram-force is exactly 9.80665 N in the NIST Guide to the SI.
| Mass on a flat surface | Normal force (N) | Normal force (lbf) |
|---|---|---|
| 1 kg | 9.807 | 2.205 |
| 60 kg | 588.4 | 132.3 |
| 100 kg | 980.7 | 220.5 |
| 1 metric ton (1,000 kg) | 9,807 | 2,205 |
Going the other way, divide newtons by 9.80665. A force of 12 N is the weight of 1.224 kg. One metric ton is 1,000 kg, while a US short ton is 907.2 kg. The kg to Newtons Calculator does these conversions in both directions.
g-Force and the Normal Force
A g-force is a normal force in disguise. It compares the push of a seat or floor on your body with your ordinary weight. Standing still you feel 1 g, which is 9.80665 N for every kilogram of body mass.
At 5 g the seat pushes five times harder than usual. For a 70 kg person that is 350 kgf, or 3,432 N. The elevator setting of the Normal Force Calculator reproduces any g level, because an upward acceleration of 4 g on top of gravity gives 5 g in total.
| g-force | Push on a 70 kg person (kgf) | In newtons | In pounds-force |
|---|---|---|---|
| 1 g | 70 | 686.5 | 154.3 |
| 5 g | 350 | 3,432 | 771.6 |
| 9 g | 630 | 6,178 | 1,389 |
| 10 g | 700 | 6,865 | 1,543 |
| 20 g | 1,400 | 13,729 | 3,086 |
| 50 g | 3,500 | 34,323 | 7,716 |
Can a Human Survive 10 g or More?
It depends on how long the force lasts and which way it points. The FAA’s pilot safety brochure on acceleration says military fighter jets may reach 9 G. Head-to-foot G drains blood from the head, and vision narrows, grays out and blacks out before the pilot loses consciousness.
Very short bursts are a different matter. On December 10, 1954, Col. John Stapp rode a rocket sled to 632 mph and stopped in 1.4 seconds. The US Air Force account puts the load at over 40 times the pull of gravity. He came away with bruising, blisters and temporary blindness.
So there is no single g-force that kills a person. A few g held for minutes can be more dangerous than 20 g or 50 g lasting a few thousandths of a second in a crash. The Car Crash Calculator estimates the g-force and impact force on a belted or unbelted occupant.
What Is Negative g-Force?
Negative g is a force that pushes from foot to head. It happens at the top of a sharp roller coaster drop or when an aircraft pushes over into a dive. The FAA brochure says the body tolerates it even less well than positive G. Blood is forced toward the head, which causes a “red out”.
Normal Force Among the Types of Forces
School physics usually lists seven types of forces. Six are contact forces, which are applied force, normal force, friction, tension, air resistance and spring force. The seventh is gravity, which acts at a distance. Lists such as The Physics Classroom’s types of forces add electrical and magnetic forces as well.
All of them share one formula for force, which is F = m × a for the net force on an object. The Force Calculator works from that formula. A second everyday example of a force, after the normal force, is the stretch of a spring, which the Hooke’s Law Calculator covers.
What Are 5 Examples of Normal Force?
A floor pushing up on your feet is the most familiar one. A table holding up a book, a chair seat supporting you and a ramp pressing out on a parked car are three more. A wall pushing back on your hand when you lean on it is a fifth, and there the normal force is horizontal.
Which Force Is the Weakest and Which Is the Strongest?
Among the four fundamental forces, gravity is the weakest and the strong nuclear force is the strongest. OpenStax College Physics rates gravity at about 10−38 of the strong force. The normal force is electromagnetic in origin. It comes from atoms in the two surfaces repelling each other.
Is Gravity a Force?
In Newtonian physics, yes. Gravity is the force that gives an object its weight, and it is the m × g term in every formula the Normal Force Calculator uses. General relativity describes gravity as curved spacetime instead, but the force picture is accurate for everyday problems.
The Gravitational Force Calculator finds that pull between any two masses, such as a planet and a person standing on it. Every other dynamics tool on the site is listed under Dynamics and Statics Calculators.