Hooke’s Law

When we apply force to a material, we know that it either expands or compresses in response. The force applied per unit area is known as stress in mechanics and is symbolized by the symbol. Strain is the amount to which a substance compresses or stretches. Different materials react differently to stress. Engineers will need this knowledge while deciding on materials for their constructions. 

While studying springs and elasticity in the nineteenth century, English scientist Robert Hooke found that numerous materials displayed a similar characteristic when the stress-strain connection was studied. There was a linear area where the force required to stretch the material was proportionate to its extent. Hooke’s Law is the name for this. Let us go through Hooke’s law in depth in this post.

What is Stress?

We know, when a deforming force is applied to a body then the restoring forces are developed inside the body. Therefore, the restoring force per unit area of a body is called stress. The restoring force is equal and the opposite to the deforming force applied to the body. 

Thus, stress can be defined as,

Stress refers to the body’s resistance to deformation. It may be represented mathematically as the restoring force per unit area.

i.e.

Stress = Restoring Force /Area

          = F / A

Units of Stress:

In SI, the unit of stress is N/m² or Nm^-2.

Another unit of stress is Pascal (Pa) where 1 Pa = Nm^-2.

The dimensional formula of stress is [ML^-1T^-2].

The dimensional formula of stress and pressure are the same.

Stress is a scalar quantity.

What is Strain?

The ratio of the change in the configuration (i.e. shape, length, or volume) to the original configuration of the body is called strain.

The strain definition defines it as the amount of deformation experienced by the body in the direction of force applied, divided by the original dimensions of the body. The relationship for deformation in terms of a solid’s length is provided below.

Strain = Change in the configuration / Original configuration

 = dX / X

The strain has no unit.

It has no dimensions, it is just a number.

Hooke’s Law

According to Hooke’s law, the force required to extend or compress a spring by a certain distance is directly proportional to that distance. The spring stiffness is a constant factor feature. The property of elasticity indicates that stretching a spring twice as long requires twice as much power. Hooke’s law is the linear relationship of displacement on stretching.

A linearly elastic material is one that acts elastically and has a linear connection between stress and strain. Stress is precisely proportional to strain in this situation.

The strain will remain in the body for as long as the stress is present, and when the tension is eliminated, the body will return to its original shape. Elasticity is the name given to this characteristic of materials. Hooke’s law, in essence, provides the foundation for elasticity, and hence it is known as the elasticity principle or law of elasticity.

Mathematically,

Stress ∝ Strain

or

Stress = Constant × Strain

Constant = Stress / Strain

or

Modulus of elasticity = Stress / Strain

This constant is known as the modulus of elasticity. Thus, the modulus of elasticity is defined as the ratio of stress and strain.

In the SI unit, the unit of modulus of elasticity is Nm^-2. The dimensional formula of modulus of elasticity is [ML^-1T^-2]

Modulus of elasticity depends upon the nature of the material of the body. The modulus of elasticity of a body is independent of its dimensions.

The dimensional formula of the Modulus of elasticity is the same as that of stress or pressure.

Hooke’s Law Equation

According to Hooke’s Law in an elastic body, extension and tension are proportional to each another. This relationship was discovered by Robert Hooke. Hooke’s law experiment is a good way to understand the behavior of materials when the degree of deformation is very small. This law is frequently demonstrated using a coil spring with weights suspended from it. The change in the length of the spring is proportional to the force of gravity F on the suspended weight. 

Hooke’s Law Equation is given as,

F = -K x

where F is the amount of force applied in N, x is the displacement in the spring in m and k is the spring constant or force constant.

Application of Hooke’s Law

• Retractable Pen: Click pens are another name for retractable pens. A click pen is typically made up of springs connected to the top and bottom of the ink cartridge. A plastic tube is present between this arrangement and is fastened in a certain location. The springs attached to the internal mechanical arrangement of a retractable pen’s plunger and cam body function on Hooke’s principle and are responsible for locking and releasing the ink cartridge as needed.
• Recoil of a Toy Gun: A spring is linked to the rear of the toy pistol. When you pull the trigger on a toy gun, it fires a plastic bullet and immediately recoils thanks to a spring attached to the base. Hooke’s law underpins this motion of recoil.
• Inflating a Balloon: The nature of a balloon is elastomeric. It swells when air molecules are blasted into it. Similarly, as it is emptied, its size decreases. The balloon’s expansion and compression are determined by the force with which the air is forced into it; hence, it operates based on Hooke’s law.
• Manometer: A manometer is a device that measures and displays liquid pressure. It features a ‘U’ shaped tube attached to a stand with a graded scale. The tube is just half-full of water. One end of the tube is open, while the other is connected to a funnel through a flexible tube. The funnel’s mouth is wrapped with an elastic sheet that can demonstrate Hooke’s law. When the funnel is immersed in a liquid-filled container, the fluid molecules exert pressure on the rubber film, displacing the water in the tube to one side. The pressure presented by the liquid is shown by the scale attached to the setup.
• Spring Scale: Spring scales are not usually used by vegetable and fruit merchants, but rather to weigh big items such as trucks, storage silos, and so on. It is made up of a hook that is suspended with the entity to be weighed. The hook is internally linked to two big springs that are bolted to the top of the device. A gear arrangement occurs between the two springs that are linked to the dial and pointer. The gear shifts according to the weight of the object hooked to the hook, and the pointer deflects correspondingly, allowing the graduated scale to point to the correct weight. Hooke’s law may be simply applied because the scale’s internal mechanical arrangement is made up of springs.
• Balance Wheel of Clock: With the aid of a spring, the balance wheel moves in a continuous motion. It allows the watch’s needle to move at a consistent rate on a regular basis. On one end, the spring is linked to the centre of the balancing wheel, while the other end is fixed. As a result, the balancing wheel of a clock is a notable implementation of Hooke’s law.

Disadvantages of Hooke’s Law

1. Hooke’s law can be applied past the elastic limit of a material.
2. Hooke’s law is only applicable to solid bodies only if the deformation force is very small.
3. Hooke’s law isn’t a universal law.
4. Hooke’s Law only applies to the materials as long as they aren’t stretched way past their capacity.

Sample Problems 

Problems 1: A body is under tensile stress, its original length was L m, after applying tensile stress its length becomes L/4 m. Calculate the tensile strain applied to the body.

Solution:

Given, 

The original length =L

Change in length = L-L/4=3L/4

Longitudinal strain=change in length/original length = △L/L

                                                                                   = (3L/4)/L

                                                                                  = 0.75

Problem 2: A copper wire of length 2.5 m has a percentage strain of 0.012 % under a tensile force. Calculate the extension in the wire.

Solution: 

Original length = 2.5m

Strain= △L/L = 0.012%

                     = 0.012/100

or 

△L = (0.012/100) x 2.5

     = 0.3 m

Problem 3: Given the deforming force as 150 N applied on a body of area of cross-section as 10 m². Calculate the stress in the body.

Solution: 

Stress = deforming force/Area of the body = F/A

                                                                  = 150/10

                                                                  = 15 N/m²

Problem 4: Why are the bridges declared unsafe after a long time of use?

Solution:

Due to the repeated stress and strain, the material used in bridges loses elastic strength and ultimately may be collapsed. That is why bridges are declared unsafe after long time of use.

Problem 5: A force of 500N causes an increase of 0.5% in the length of a wire. Find out the longitudinal strain.

Solution:

Since, Longitudinal strain = Change in length/original length

or 

△L/L = 0.5%

         = 5/1000 

         = 0.005

Problem 7: Write down the cause of restoring stress in a stretched wire and compressed wire.

Solution:

 The restoring stress is caused by the interatomic attraction in a stretched wire and by inter atomic repulsion in a compressed wire. 

Problem 8: What are the Limitations of Hooke’s Law?

Solution:

Although Hooke’s law is often utilised in engineering, it is not a universal principle. When a material’s elastic limit is surpassed, the legislation is no longer relevant. Hooke’s law usually gives correct findings for sold particles when the deformations are minimal. Many materials depart from Hooke’s rule long before they reach the elastic limit.