A book resting on a table is an example of a contact force between two surfaces. Specifically, it is a normal force that counteracts the downward force of gravity acting on the book. The normal force arises from electromagnetic interactions between atoms in the book and atoms in the table surface.

## What is a force?

A force is a push or pull on an object. Forces cause objects to accelerate, slow down, remain in place, or change shape. Forces are measured in units of newtons (N) in the SI system.

Some examples of forces include:

- Friction force – a force that resists the motion of two surfaces sliding against each other
- Tension force – a pulling force exerted by a string, rope, cable, etc. when it is stretched
- Drag force – a force exerted by a fluid that resists the motion of an object through it
- Gravitational force – an attractive force between two objects that have mass
- Normal force – a force that counteracts a load pressing on a surface perpendicular to the surface

Forces are vector quantities, having both magnitude and direction. Force vectors are often represented by arrows in diagrams, with the length of the arrow proportional to the force’s magnitude.

## Normal Force

The normal force is a contact force that arises when two solid surfaces come in contact and press against each other. The normal force is always perpendicular or normal to the surface interface.

The normal force exists to counteract orthogonal forces trying to push one object into another object. The classic example is a book resting on a table. The book experiences a downward gravitational force due to its mass. But instead of accelerating down into the table, the book remains still. This is because the table’s surface exerts an upwards normal force on the book, equal in magnitude to the gravitational force but opposite in direction.

### Normal Force Equation

The normal force can be calculated using the following equation:

**FN = m x g x cos(θ)**

Where:

FN = Normal force (N)

m = Mass of the object (kg)

g = Acceleration due to gravity (9.81 m/s2 on Earth)

θ = Angle between the surface and the gravitational force vector

For a book sitting flat on a table, θ = 0 degrees. So the normal force equation simplifies to:

**FN = m x g**

This shows that the normal force equals the weight of the object.

### Normal Force vs Friction Force

The normal force is different from the friction force between two surfaces. While the normal force is perpendicular to the surface contact, friction acts parallel to the surface.

Friction arises from intermolecular attractive forces between microscopic protrusions on the two surfaces. If one surface slides relative to the other, these attractions resist the lateral motion, giving rise to kinetic friction.

Static friction refers to friction forces that prevent initiation of sliding motion. Kinetic friction is weaker than static friction. The maximum static friction is proportional to the normal force, through the coefficient of static friction.

So for a book on a table, both static friction and normal forces are involved. Static friction keeps the book from sliding off the table when given a small horizontal push. The normal force supports the book against gravity.

## Forces on a Book on a Table

The primary forces acting on a book resting on a table are:

- Gravitational force (Fg)
- Normal force from the table (FN)
- Friction force (f)

Here is a diagram showing the force vectors when viewing the book from the side:

The gravitational force acts downwards through the book’s center of mass. Its magnitude equals the mass of the book times the gravitational acceleration:

**Fg = m x g**

Where m is the mass of the book and g is 9.81 m/s2 on Earth.

This force vector always points vertically downwards parallel to the gravitational field of the Earth.

To counteract Fg, the table’s surface exerts a normal force (FN) straight upwards on the book. FN has the same magnitude as Fg but opposite direction:

**FN = m x g**

Finally, static friction (f) acts horizontally to prevent the book from sliding across the table. The maximum value of f depends on FN and the coefficients of static and kinetic friction.

As long as the book remains at rest, the vector sum of Fg, FN, and f is zero. This satisfies the equilibrium condition:

**ΣF = 0**

If a horizontal force is applied exceeding the static friction, the book will accelerate and start sliding. Kinetic friction arises which is typically less than static friction.

## Book on an Incline

If the book rests on an inclined plane instead of a flat horizontal table, the normal force will be less than the full weight of the book.

On an incline, the gravitational force can be decomposed into components that are perpendicular and parallel to the ramp surface:

- Fg sin(θ) – parallel component
- Fg cos(θ) – perpendicular component

Where θ is the incline angle relative to the horizontal.

The perpendicular component determines the normal force:

**FN = Fg cos(θ) = m x g x cos(θ)**

Here is a diagram of the forces when on an incline:

The normal force is now less than the full weight mg. Meanwhile, the parallel gravitational component causes the book to accelerate down the ramp. Static friction fs opposes this parallel motion.

## Normal Force and Pressure

The normal force is directly related to pressure under the book. Pressure is defined as:

**Pressure = Force / Area**

The normal force is distributed over the area of contact between the book and table. Let’s call this contact area A.

Then the pressure P is:

**P = FN / A**

A larger normal force or smaller contact area leads to higher pressure.

For a flexible object like a pillow, increasing the normal force causes the contact area to increase as well. So the pressure rises less quickly.

This is why a bed of nails trick works – the weight of the person is distributed over many small contact points, keeping the pressure at each point low.

## Using a Scale to Measure Normal Force

A bathroom scale is a good example of measuring the normal force when you stand on it. The scale contains a spring or strain gauge that compresses under your weight.

When you stand on a scale, your mass exerts a downward gravitational force equal to mg. The scale responds by pushing up with an equal normal force FN. This force causes the spring or gauge to compress in proportion to your weight.

The scale is calibrated to convert this compression to a reading of your mass in kg, lb, etc.

So even though the scale reads your mass, what it is actually measuring is the normal force between you and the scale. This indirectly determines your mass since FN = mg.

## Examples of Normal Forces

Here are some additional examples of normal forces:

- Feet pressing against the floor
- Tires pressing on the road
- Weight of a bridge causing compression in the piers supporting it
- Force on the bottom of a dam due to water pressure behind it
- Upwards buoyant force on objects submerged in a fluid
- Force on a rocket nozzle due to high pressure exhaust gases

In all cases, the normal force arises perpendicular to a surface, in response to loads or pressures inducing compression or deformation. Static structures like bridges and dams must be designed so the normal force on all supporting elements does not exceed the failure limits.

## Using Free Body Diagrams

Drawing a free body diagram is helpful for understanding normal forces and analyzing the forces acting on an object.

To draw a free body diagram:

- Isolate the object you wish to analyze.
- Draw the object and label it.
- Draw vectors representing all forces acting on the object. Label the vectors.
- Make the length of each vector proportional to the force’s magnitude.

Here is an example free body diagram for the book on an incline:

From this diagram, you can apply Newton’s laws of motion to solve for the unknown forces, accelerations, etc.

## Conclusion

To summarize, a normal force arises between two solid surfaces in contact. For a book resting on a table, this normal force counteracts the downward gravitational pull, supporting the book at rest or equilibrium. Using free body diagrams and Newton’s laws facilitates analyzing the normal force and other physics in a system.