The Potential Energy of a Spring: Conservation of Mechanical Energy

Potential energy is the stored energy in a system defined by the relative locations of various system components. When a spring is compressed or stretched, its potential energy increases. When we raise a steel ball above the ground, it has more potential energy than when it was at the bottom. 

The law of conservation of energy states that the overall mechanical energy of a body is conserved when the conservative force is acting on the body.

The Potential Energy of a Spring 

Spring is a standard tool, and due to its small mass, we often overlook its inertia. We expect a spring to deform when it undergoes strain due to compression and then reaches the point of equilibrium. Therefore, a spring exerts an equal and opposite force when compressing or extending a body.

A compressible or stretchable item, such as a spring, rubber band, or molecule, stores energy. It is also known as elastic potential energy. It is the product of the force and the movement’s distance.

Work Done by a Spring

The spring has no potential energy when it is in its mean position; there is no stress on it. However, if we shift the spring from its usual place, it will retain energy because of its new location.  

The work done by the spring or any elastic object is the result of the deformation due to stretching. The spring constant k and the extended distance determine it.

The Formula for Potential Energy of a Spring

The potential energy of a string equals force × displacement.

In addition, the force is proportional to the displacement of the spring constant.

A spring’s potential energy is:

P.E = ½ ×kx²

Where,

P.E. is the spring’s potential energy 

k is the spring constant

X is the displacement in the spring

We must apply Hooke’s law to calculate the spring potential energy. Potential energy is equivalent to the work done by a spring, and work is the product of force and displacement. The term “displacement” refers to the change in the spring’s position.

Hooke’s Law

When we state Hooke’s law, it is essential to mention British physicist Robert Hooke who formulated it. He lived in the 17th century. 

Hooke’s equation holds in different situations when an elastic body is deformed, such as wind blowing on a tall building or a musician plucking a guitar string (to some extent). The body which satisfies this equation is called the linear-elastic or Hookean body.

In its most general form, Hooke’s law allows you to determine the relationship between strain and stress for complicated objects based on the intrinsic qualities of their constituent materials. 

When we stretch a string,

Fspring = -kx

Where, 

Fspring is the spring force

k is spring constant

x is spring stretch or compression

Conservation of Mechanical Energy

If we state the law of conservation of energy, the overall mechanical energy of a system is conserved if a conservative force is applied to the system. Due to the inability to create and destroy energy simultaneously, energy can only be changed internally by conservative forces acting on the system. 

If a part of an isolated system like the universe loses energy, then another part of the same system must gain it. Though we cannot prove the conservation of energy, there has never been a demonstrated violation of the principle. 

In any system, we can calculate the amount of energy by using the equation below:

Ut= Ui+W+Q

Where,

Ut=total energy of the system

Ui= A system’s initial energy

W= Work done by or on the system 

Q= Addition or removal of heat

An overall system with only conservative forces is characterised by each force’s associated potential energy form. 

When an object moves in the opposite direction of a conservative net force, its potential energy increases; however, its kinetic energy also changes i.e. the object’s speed changes. 

The conversion of mechanical energy between forms involves a variety of devices. Electrical energy is converted to mechanical energy by electric motors, for instance.

Conclusion

Law of conservation of energy says energy cannot be created nor be destroyed. For mechanical energy we can also see that the overall mechanical energy of a system is conserved if a conservative force is applied to the system. Mechanical energy is the sum of the total kinetic energy and the total potential energy. A system’s mechanical energy is its macroscopic property.