# Hooke’s law

1. Hooke’s law for springs

If the spring is pulled to the right, the spring will stretch and increase in length (figure 1). If the pull force is not very large, it is found that the increase in spring length (Δx) is proportional to the magnitude of the pull force (F). In other words, the greater the pull force, the greater the length of the spring. Comparison of the magnitude of the pull force (F) and the increase in the spring length (Δx) is constant.

The spring is pulled to the right so that it has an increase in the length of Δx. The increase in spring length is proportional to the pull force.

Graph the relationship between Force (F) and increase in spring length (Δx), where F is proportional to Δx. The comparison of F with Δx is constant.

The ratio of force (F) to the increase in spring length (Δx) is indicated by the same graph slope (figure 2).

F / Δx = k

F = k Δx

k is a spring constant or the coefficient of spring elasticity. This relationship was first observed by Robert Hooke (1635 – 1703) in 1678, hence it was known as Hooke’s law.

If the force applied to the spring exceeds the spring elasticity limit, after the force is removed the spring length does not return to its original length. Hooke’s law only applies to the elasticity limit. The spring elasticity limit is the maximum force that can be given to the spring before the spring changes form permanently and the spring length cannot return to its original length. If the force continues to increase, the spring is damaged.

2. Hooke’s law for non-spring

Hooke’s law also applies to all solid objects. If on a solid object the external force is given then the object undergoes a change in shape. We discuss changes in the shape of these solid objects using the concept of stress and strain. Stress states the power of forces that causes the shape of the object to change. Strain states changes in the shape of objects due to stress. It was found that for very small stress and strain, the stress is proportional to the strain. The ratio of stress and strain is constant, where this comparative constant is called the modulus of elasticity.

Stress / Strain = elastic modulus

This relationship is also called Hooke’s law with conditions stress proportional to strain and the ratio between stress and strain is constant. If the force acting on an object exceeds the elasticity of objects, Hooke’s law does not apply. Hooke’s law applies only to the limits of object elasticity.

There are three types of elastic modulus, Young modulus, shear modulus, and bulk modulus.

2.1 Young’s Modulus

When both ends of the cable, wire, or rope are pulled by force with the same magnitude and opposite direction, the cable, wire, or rope is in a tension state. Examples in everyday life related to this include rope that holds rock climbers, the wire that holds elevators, wires or luggage lines or containers on loading and unloading systems on ships, etc. Tension cables, wires, or ropes experience tensile strain due to tensile pressure. We define tensile pressure as a ratio of the tensile strength (F) to the cross-sectional area of an object (A). While the tensile strain is defined as the ratio of length increase (Δl) to the initial length of the object (lo). Young’s Modulus is the ratio of tensile pressure to tensile strain.

Tensile pressure / tensile strain = Young’s modulus

F/A : Δl/lo = Y

If the tensile pressure is small, the length of the object will return to normal after the force is removed. If the tensile pressure exceeds the elasticity of the object, the length of the object does not return to normal after the force is removed. If the tensile pressure continues to increase, the object will break.

2.2 Shear modulus

Place a thick book on the table surface. Place your hand on the surface of the book and push the surface of the book forward. Your push works on the top surface of the book and its direction forward, parallel to the surface of the book. The bottom surface of the book is kept rest by static friction in the opposite direction to the push. At first, the book shape was square or rectangular, after being pushed, the shape of the book turned into a parallelogram. Changes in the shape of the book are one example of the occurrence of shear strain due to the presence of shear pressure.

Shear pressure is defined as the ratio of force (F) to the surface area (A) that is shifted. Shear strain is defined as the ratio of Δx to the height of the object (h). The ratio of shear pressure to shear strain is called shear modulus.

Shear pressure/shear strain = Shear modulus

F/A : Δx/h = Shear modulus

2.3 Bulk Modulus

In a documentary film about World War II, there are war scenes in the sea using submarines. Because they want to hide from enemy submarines, a nation’s naval submarine dives deeper and reaches almost the seafloor. Surprisingly, the wall of the submarine was cracked so that seawater entered the inside of the submarine. The wall of the submarine was cracked due to the pressure that seawater worked on the surface of the submarine. Seawater pressure is proportional to the depth of seawater. The deeper the dive, the greater the seawater pressure it experiences. If the submarine wall construction is not strong, the submarine wall will crack.

The story of a submarine that is cracked due to the pressure of seawater on its entire surface is an example of an object experiencing a volume strain due to the volume pressure. Volume-pressure is defined as the ratio of the total force (F) acting on the entire surface of the object to the surface area (A) of the object. Volume strain is defined as a ratio of volume reduction (-ΔV) to the initial volume (Vo) of an object. The pressure ratio of the volume to the volume strain is called bulk modulus.

Volume-pressure / volume strain = bulk modulus

– ΔF/A : ΔV/V_{o} = bulk modulus

Solid and liquid objects have the bulk modulus, but only solid objects have Young’s modulus and shear modulus.