## What is elastic force?

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

The elastic force is also known as a restoring force because it opposes deformation to return objects to their equilibrium position. The transfer of 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 spring particles, decreasing the separation between them, at the same time, the particles resist being pushed by exerting a force contrary to the compression.

If instead of compressing the spring it is pulled, stretching, the particles that make it up separate more. Also, the particles resist separating by exerting a force contrary to stretching.

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

**What is the elastic force?**

The elastic force (*F**what*) is the force that an object exerts to recover its state of natural equilibrium when affected by an external force.

To analyze the elastic force, the ideal mass-spring system will be taken into account, which consists of a spring placed horizontally attached to the wall at one end and at the other end to a block of negligible mass. The other forces acting on the system, such as the force of friction 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 to 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 been 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 equilibrium state.

In the same way it 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.

**Elastic force 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**yes **[1]

*F**what** = *elastic force

*what *= Constant of proportionality

*Δ yes* = offset

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 **[2]

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 proportionality constant k is a constant that depends on the type of material the spring is made of. The unit of the constant k is N/m.

Elastic objects have a yield strength that depends on the strain constant. If it is stretched beyond the elastic limit, it will deform permanently.

The equations [1] and [2] they apply to small displacements of the spring. When the displacements are greater, terms with greater power of Δ are added.**x**.

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

The elastic force does work on the spring, displacing 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 the equation [3].

U=½k. Δx2 [3]

Potential energy is expressed in joules (J).

When the deformation force is no longer applied, the spring accelerates toward 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 the following equation [4].

Ek= ½ m.v2 [4]

*m* = mass

*v* = spring rate

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

**Solved exercises of elastic force**

**Obtaining the deformation force**

How much force is necessary to apply to a spring to make it stretch 5 cm if the spring constant is 35 N/m?

Since the applied force is opposite to the elastic force, we can determine *F**what* assuming the spring is stretched horizontally. The result does not require a negative sign, since only the application force is needed.

Hooke’s law

Fk = -k.Δx

The spring constant k is 35 N/m.

Δx = 5cm = 0.05m

Fk = -35 N/m . 0.05m

Fk = – 1.75 N = – F

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

* ***Obtaining the strain constant**

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

Δx** =**20cm=0.2m

F = 60N

Fk = -60N = –F

k = –Fk / Δx

= -(-60N)/0.2m

k = 300 N/m

The spring constant is 300 N/m.

**Obtaining potential energy**

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

Δx** =**10cm=0.1m

k =20 N/m

Fk = -20 N/m . 0.1m

Fk = -200 N

The elastic force of the spring is -200 N.

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 [3]

U=½k. Δx2

U= ½(20 N/m) . (0.1m)2

U = 0.1 joules

**References**

Kittel, C., Knight, W.D., and Ruderman, M.A. Mechanics. Mc Graw Hill.

Rama Reddy, K., Badami, SB, and Balasubramanian, V. Oscillations and Waves. India. Universities Press.

Murphy, J. Physics: understanding the properties of matter and energy. New York: Britannica Educational Publishing.

Giordano, NJ College Physics: Reasoning and Relationships. Canada: Brooks/Cole.

Walker, J., Halliday, D., and Resnick, R. Fundamentals of Physics. wiley.