Potential Energy of Spring: Laws, Formula and Uses

Potential Energy of a spring can be determined when a spring is being compressed or extended.

• Potential energy is the energy stored in an object due to its position with respect to some zero position.
• When we compress or extend a stretched spring, we experience the same force that we applied in the opposite direction.
• When a spring deviates from its mean position, it tries to regain equilibrium by exerting a force equal to and opposite to the external force.

The formula for the potential energy of a spring is given by

\(F=\frac{1}{2}kx^2\)

Where

• k is known as the spring constant
• x is the displacement from the mean position

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What is Potential Energy?

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Elastic potential energy is energy stored as a result of applying a force to deform an elastic object.

• The energy is stored until the force is removed and the object springs back to its original shape, doing work in the process.
• The deformation could involve compressing, stretching or twisting the object.
• We have seen the use of the spring force in bike carriers, where the energy generated by disrupting the spring equilibrium is used as its potential energy and transformed into various forms.

This form of energy is caused by a force acting on an object temporarily which results in the object changing its shape. A good example of such objects is: when a squishy ball is squeezed with the hand or when an elastic band is stretched and released.

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Laws Relating to Potential Energy of Spring

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When a stretched spring is compressed or extended, we feel a power in the opposite direction. The explanation behind this is that when a spring deviates from its mean position, it uses an equal and opposite force to restore its balance. 

Hooke’s Law of Spring Constant

Hooke’s law states that the force that requires stretching an elastic object like a metal spring is always directly proportional to the spring extension for small-scale distances. The force applied back by the spring is known as Hooke's Law. It is expressed as:

F_s = -kx

Where,

• F_s → Force of the spring
• x → Unstretched length of the spring
• k → Spring constant.

The spring force may be dubbed as the restorative force since the spring force is always opposed to the displacement, this is why the Hooke law equation has a negative sign. When the spring is stretched downwards, an upward force is generated in the spring.

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Formulas for Potential Energy of a Spring 

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When we pull the spring to a displacement, the work done by the spring is:

The work done by pulling force is:

When the displacement is less than 0, the displacement done is:

W_s = –\(\frac{k(xc^2)}{2}\)

The external strength work W_s = –\(\frac{k(xc)^2}{2}\) is F. In the transition from the initial transition x_i to the final transformation, x_f, the work is done.

It is evident from the equation that the work of spring forces depends on displacement endpoints solely.

• We can also observe that the work performed by the spring force is zero in a cyclic process.
• Therefore, the spring force is said to be a conservative force, because it only depends on beginning and ending locations.
• This job is therefore carried out in the form of the potential energy for the Spring.

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Uses of Potential Energy of Spring

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There are multiple uses of the Potential Energy of Spring. Below listed are some of the uses:

• In many mechanical systems such as the shock absorbers used in vehicles, and spring mattresses, elastic potential energy is stored by a spring. 

• Elastic energy may be employed in numerous ways, as the spring may stay compressed or expanded without wasting energy for lengthy durations. 

 

• The elastic stretch energy is used by balloons, rubber belts, bungees and trampolines. 
• In squishy balls, a bow and arrow and spinning springs, we can find elastic energy applications.
• Also, elastic energy is used by catapults and slingshots.

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Things to Remember

• Elastic potential energy is the energy stored as a result of applying a force to deform an elastic object.
• This form of energy is caused by a force acting on an object temporarily which results in the object changing its shape.
• Hooke’s law expresses potential energy for a spring as F_s = -kx. 
• This type of spring potential energy is used in many applications such as vehicle suspension systems, spring mattresses, shock absorbers, etc.
• Spring force is said to be a conservative force because it only depends on beginning and ending locations.