# Normal force equation

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 (FN) is equal to the weight of the object (mg). This relationship is expressed by the equation FN = mg. When dealing with an object on an inclined surface, the equation for the normal force becomes FN = 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 FN = 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 FN = 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.

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## 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/s2.

Solution

Given data:

• Normal force acting on a flower pot, FN = ?
• Mass of a flower pot, m = 2 kg
• Gravitational acceleration, g = 9.81 m/s2

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

• FN = mg
• FN = 2 × 9.81
• FN = 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, FN = ?
• 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:

• FN = mg × cos (α)
• FN = 3 × 9.81 × cos (45°)
• FN = 29.43 × 0.7071
• FN = 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, FN = ?

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

• FN = mg + F sin (θ)
• FN = [20 × 9.81] + [8 × sin (30°)]
• FN = 196.2 + [8 × 0.5]
• FN = 196.2 + 4
• FN = 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, FN = ?
• 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:

• FN = mg – F sin (θ)
• FN = [4 × 9.81] – [10 × sin (45°)]
• FN = 39.24 – [10 × 0.8509]
• FN = 39.24 – 8.509
• FN = 30.731 N

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