The **normal force equation** provides a means to calculate the force exerted on an object in different situations. In the case of an object resting on a flat surface, the normal force (F_{N}) is equal to the weight of the object (mg). This relationship is expressed by the equation F_{N} = mg. When dealing with an object on an inclined surface, the equation for the normal force becomes F_{N} = mg × cos (α), where α represents the angle of the inclined surface.

In scenarios involving external forces, the normal force equation can be modified accordingly. When a downward external force is applied to an object, the equation becomes F_{N} = mg + F sin(θ), where F denotes the magnitude of the external force and θ is the angle between the horizontal surface and the direction of the external force. Conversely, if an upward external force is acting on the object, the equation becomes F_{N} = mg – F sin(θ). It is important to note that α and θ are measured in degrees, while the unit for the external force, F, is newton (N). Additionally, m and g represent the mass and acceleration due to gravity, respectively. By utilizing these normal force equations, one can accurately calculate the normal force exerted on an object in various scenarios.

## Practice problems

### Problem #1

Determine the normal force acting on a 2 kg flower pot placed on a desk. The gravitational acceleration is given as g = 9.81 m/s^{2}.

**Solution**

Given data:

- Normal force acting on a flower pot, F
_{N}= ? - Mass of a flower pot, m = 2 kg
- Gravitational acceleration, g = 9.81 m/s
^{2}

Applying the formula, when an object is resting on the flat surface:

- F
_{N}= mg - F
_{N}= 2 × 9.81 - F
_{N}= 19.62 N

Therefore, the normal force acting on a flower pot is **19.62 N**.

### Problem #2

Calculate the normal force acting on a 3 kg book placed on an inclined surface inclined at a 45° angle.

**Solution**

Given data:

- Normal force acting on a book, F
_{N}= ? - Mass of a book, m = 3 kg
- Angle of an inclined surface, α = 45°

Applying the formula, when an object is placed on an inclined surface:

- F
_{N}= mg × cos (α) - F
_{N}= 3 × 9.81 × cos (45°) - F
_{N}= 29.43 × 0.7071 - F
_{N}= 20.8099 N

Therefore, the normal force acting on a book is **20.8099 N**.

### Problem #3

A wooden crate with a mass of 20 kg is resting on the ground. An external downward force of 8 N is applied to the crate at a 30° angle. Calculate the normal force acting on the wooden crate.

**Solution**

Given data:

- Mass of a wooden crate, m = 20 kg
- External downward force applied on a wooden crate, F = 8 N
- Angle between horizontal surface and external downward force, θ = 30°
- Normal force acting on a wooden crate, F
_{N}= ?

Applying the formula, when an external downward force is acting on an object:

- F
_{N}= mg + F sin (θ) - F
_{N}= [20 × 9.81] + [8 × sin (30°)] - F
_{N}= 196.2 + [8 × 0.5] - F
_{N}= 196.2 + 4 - F
_{N}= 200.2 N

Therefore, the normal force acting on a wooden crate is **200.2 N**.

### Problem #4

Find the normal force acting on a 4 kg block when an external upward force of 10 N is applied at a 45° angle.

**Solution**

Given data:

- Normal force acting on a block, F
_{N}= ? - Mass of a block, m = 4 kg
- External upward force applied on a block, F = 10 N
- Angle between horizontal surface and external upward force, θ = 45°

Applying the formula, when an external upward force is acting on an object:

- F
_{N}= mg – F sin (θ) - F
_{N}= [4 × 9.81] – [10 × sin (45°)] - F
_{N}= 39.24 – [10 × 0.8509] - F
_{N}= 39.24 – 8.509 - F
_{N}= 30.731 N

Therefore, the normal force acting on a block is **30.731 N**.

## Related

## More topics

- Force equation
**Normal force equation**- Net force formula
- Applied force formula
- Magnetic force equation
- Centripetal force equation
- Centrifugal force equation
- Spring force equation
- Tension force formula
- Electric force equation