# Elastic Force: What It Consists Of, Formulas And Exercises

The elastic force  is the force that an object exerts to resist a change in its shape. It manifests itself in an object that tends to regain its shape when it is under the action of a deformation force.

The elastic force is also called the restoring force because it opposes deformation to return objects to their equilibrium position. The transfer of the elastic force is through the particles that make up the objects. For example, when a metal spring is compressed, a force is exerted that pushes the particles of the spring, decreasing the separation between them, at the same time, the particles resist being pushed by exerting a force contrary to compression.

If instead of compressing the spring it is pulled, stretching, the particles that make it up are further separated. Likewise, the particles resist being separated by exerting a force contrary to stretching.

Objects that have the property of recovering their original shape when opposing the deformation force are called elastic objects. Springs, rubber bands, and bungee cords are examples of elastic objects.

Article index

• one

What is the elastic force?

• two

Formulas

• 2.1

Kinetic energy and potential energy referred to an elastic force

• 3

Practical exercises of examples

• 3.1

Obtaining the deformation force

• 3.2

Obtaining the strain constant

• 3.3

Obtaining potential energy

• 4

References

## What is the elastic force?

The elastic force ( F k ) is the force that an object exerts to regain its state of natural balance after being affected by an external force.

To analyze the elastic force, the ideal spring mass system will be taken into account, which consists of a horizontally placed spring attached at one end to the wall and at the other end to a block of negligible mass. The other forces acting on the system, such as the friction force or the force of gravity, will not be taken into account.

If a horizontal force is exerted on the mass, directed towards the wall, it is transferred towards the spring, compressing it. The spring moves from its equilibrium position to a new position. As the object tends to remain in equilibrium, the elastic force in the spring that opposes the applied force is manifested.

The displacement indicates how much the spring has deformed and the elastic force is proportional to that displacement. As the spring is compressed, the variation in position increases and consequently the elastic force increases.

The more the spring is compressed, the more opposing force it exerts until it reaches a point at which the applied force and the elastic force balance, consequently the spring-mass system stops moving. When you stop applying force, the only force that acts is the elastic force. This force accelerates the spring in the opposite direction to the deformation until it recovers the state of equilibrium.

The same happens when stretching the spring pulling the mass horizontally. The spring is stretched and immediately exerts a force proportional to the displacement opposing the stretch.

## Formulas

The elastic force formula is expressed by Hooke’s Law. This Law states that the linear elastic force exerted by an object is proportional to the displacement.

F k = -k.Δ s         

F k = Elastic force

k = constant of proportionality

Δ s = Displacement

When the object is displaced horizontally, as in the case of the spring attached to the wall, the displacement is Δ x , and the expression for Hooke’s Law is written:

F k = -k.Δ x The negative sign in the equation indicates that the elastic force of the spring is in the opposite direction to the force that caused the displacement. The constant of proportionality k is a constant that depends on the type of material from which the spring is made. The unit of the constant k is N / m .

Elastic objects have an elasticity limit that will depend on the deformation constant. If it is stretched beyond the elastic limit, it will permanently deform.

Equations  and  apply to small displacements of the spring. When the displacements are greater, terms with greater power of Δ x are added .

### Kinetic energy and potential energy referred to an elastic force

The elastic force does work on the spring by moving it toward its equilibrium position. During this process the potential energy of the spring mass system increases. The potential energy due to the work done by the elastic force is expressed in equation .

U = ½ k.  Δx 2               

Potential energy is expressed in Joules (J).

When the deformation force is not applied, the spring accelerates towards the equilibrium position, decreasing the potential energy and increasing the kinetic energy.

The kinetic energy of the spring-mass system, when it reaches the equilibrium position, is determined by equation .

E k = ½ mv 2                  

m = mass

v = spring speed

To solve the spring-mass system, Newton’s second law is applied taking into account that the elastic force is a variable force.

## Practical exercises of examples

### Obtaining the deformation force

How much force is necessary to apply to a spring for it to stretch 5cm if the spring constant is 35N / m? Since the application force is opposite to the elastic force, F k is determined assuming that the spring is stretched horizontally. The result does not require the negative sign since only the application force is needed.

Hooke’s law

F k = -k.Δx

The spring constant k is 35N / m.

Δ x = 5cm = 0.05m

F k = -35N / m. 0.05m

F k = – 1.75N = – F

It takes 1.75 N of force to deform the spring by 5cm .

### Obtaining the strain constant

What is the strain constant of a spring that is stretched 20 cm by the action of a 60N force ?

Δx = 20cm = 0.2m

F = 60N

F k = -60N = – F

k = – F k / Δx

= – (- 60N) / 0.2m

k = 300 N / m

The spring constant is 300N / m

### Obtaining potential energy

What is the potential energy referred to the work done by the elastic force of a spring that is compressed 10cm and its strain constant is 20N / m?

Δ x = 10 cm = 0.1m

k = 20 N / m

F k = -20N / m. 0.1m

F k = -200N

The elastic force of the spring is -200N.

This force does work on the spring to move it toward its equilibrium position. Doing this work increases the potential energy of the system.

The potential energy is calculated with the equation 

U = ½ k.  Δx 2

U = ½ (20N / m). (0.1m) 2

U = 0.1Joules

## References

1. Kittel, C, Knight, WD and Ruderman, M A. Mechanics. US: Mc Graw Hill, 1973, Vol. I.
2. Rama Reddy, K, Badami, SB and Balasubramanian, V. Oscillations and Waves. India: Universities Press, 1994.
3. Murphy, J. Physics: understanging the properties of matter and energy. New York: Britannica Educational Publishing, 2015.
4. Giordano, N J. College Physics: Reasoning and Relationships. Canada: Brooks / Cole, 2009.
5. Walker, J, Halliday, D and Resnick, R. Fundamentals of Physics. US: Wiley, 2014.