Spring Force, Hooke’s Law, and Examples for JEE

Spring force is the elastic force that helps the spring regain its original shape after the removal of the force.  When an external force is applied to the spring, then the spring gets into the compressed state with the stored energy. When the external force is removed, the spring stretches and gets back to its original shape and size due to the elastic nature of the spring or due to the spring force. The elastic nature of the spring provides flexibility in the area of application of the spring. Due to its elastic nature, the spring is used in various mechanical equipment, like medical instruments, small ballpoint pens, suspensions used in automobiles, etc. 

As we are now aware, what is spring force? We will discuss the concept of spring force, spring equation, spring formula, work done by spring force, etc in this article. This article attempted to give an overview of the spring force and better clarity for the understanding of the concept.  

Hooke’s Law

According to Hooke’s law, when an external force is applied to the spring, then deformation produced in the spring whether it is extension or compression in spring length is directly proportional to the restoring force.  In simple terms, the restoring force is directly proportional to the displacement.

Mathematically, the spring force equation is written as,

$F\propto x$

$F=-kx$

Here, k is the constant of proportionality which is called the stiffness of the spring, F is the restoring force and x is the displacement of the spring due to the applied force. 

As the spring force is generally the type of force. Therefore, the unit of the spring force is the same as the force. Hence, the unit of the spring force is Newton (N). 

The dimensional formula of the spring force formula is also the same as force, i.e. (M1L1T-2). The obtained equation is also known as the spring equation.

Stiffness – A Spring Constant 

According to Hooke’s law, stiffness is the constant of proportionality which is formed from the direct relationship of the restoring force to the displacement made by the spring when an external force is applied.

Therefore, stiffness will be the ratio of restoring force to the displacement made by the spring. Mathematically, it is shown as,

$k=\dfrac{F}{x}$

According to the formula, spring constant (or stiffness) is defined as the force required to move the spring per unit displacement. The unit of spring constant is newton/meter $\left(\dfrac{N}{m}\right)$. 

Mathematical Analysis of Work Done by Spring Force

Consider an object attached to the spring resting on the horizontal surface. The original position of the block is taken as the mean position (x1). An external force F is applied to compress the spring x2.

Image: An assembly of spring and block system 

Description: This image describes the displacement of a block that is attached to a spring due to the application of external force F.

Then the amount of work done by the spring is $W$ and It is given as the product of force and displacement.

Work done = Force $\times$ displacement

Consider that when the force is applied on the block, it makes a displacement of dx. Therefore, the work done will be, 

dW=Fdx

The total amount of work done will be evaluated by the integration. The integration will be done from x1 to x2 with respect to x, 

$\begin{align} &d W=-k x d x \\ &W=\int d w=-k \int_{x_{1}}^{x_{2}} x d x \\ &W=-k\left(\dfrac{x^{2}}{2}\right)_{x_{1}}^{x_{2}} \\ &W=-\dfrac{1}{2} k\left(x_{2}^{2}-x_{1}^{2}\right) \\ &W=\dfrac{1}{2} k\left(x_{1}^{2}-x_{2}^{2}\right) \end{align}$ 

Hence, the work done by the spring force is $\dfrac{1}{2}k(x_1^2-x_2^2)$

Examples of Spring Force 

There are various examples of the spring force is given below:

• When the person jumps on a trampoline, then the body weight pushes the trampoline in a downward direction, which acts as an external force applied to the springs of the trampoline. The springs of the trampoline developed a counterforce that acts on the person and throws the person in an upward direction.

• The suspension system of the automobile is made by the spring so that any resistance or shock or jerk that comes in the path of the vehicle is minimized and the journey of the vehicle remains smooth.

• The door latch assembly used in the houses and house appliances is also equipped with a spring for their smooth working. 

• Most medical equipment uses the spring for providing a squeezing action of the element. Compression springs are used in surgical staplers.

• Buckling springs are used below the keys of the keyboard for the coming back action of keys after pressing them. 

Conclusion

This article concludes with brief interesting information about the spring force, spring force formula, Hooke’s law of the spring, spring force equation, and work done by spring. This article also discussed the concept of stiffness. It also discusses the examples of spring force used in everyday life for smoothing the working of the equipment. It also discusses the derivation of the work done by the spring force, the dimensional formula of stiffness, and the spring force. This article tried to give a better understanding of the concept of spring force and an example of spring with clarity in an elaborate manner.