Is Spring Force Conservative: 5 Important Examples


Do you know what force will known as Conservative force?

Firstly you must have to understand what is conservative force. When an object is moving from one point to another the net work by a force exerting on the object is depends only on its starting point and the ending point not on the path traced then the force is called as the Conservative Force.

In the above figure the object traces three different path, but the work done on the object is same at three path also, it is because the work done is independent of the path through which it is travelling.

Now we shall move to the concept of  spring force. When an elastic body is stretched or compressed by an object having certain mass the force exerted on the elastic body to displace some distance is called Spring force. Here in this article we are going to know is spring force is a conservative or not.  

Is Spring Force Conservative
Conservation of force

Is spring force conservative

Conservation of potential energy:

 A conservative force gives rise to the concept of potential energy (P.E) of the system. If the potential energy of any force is zero then it must be a non conservative. To know is spring force is conservative or not we must check whether the potential energy of the spring is zero or not.

Graphical representation of the spring constant varying with spring force and the displacement

Let us consider a spring which stretches or compresses or elongate at certain distance. A stretched object is considered as elastic medium which obeys Hooke’s Law. The force acting on the spring to stretch from its original position is given by

                          Fspring =  kx

Where x is the displacement when the spring is elongated or compressed, k is the spring constant.

Since the compressed spring stretches by applying some force in opposite direction then the force will be

                      F =  – kx

Here we consider the spring as massless object whose force will be same or constant at every point on the spring.

The total work done to stretch the spring from its initial position to its final position is given by

The net work done on the stretched or compressed spring is equal to the potential energy of the spring is it is given by\

(Here we neglect the negative sign because energy can not be negative)

The potential energy of the spring force is non zero quantity. It shows that potential energy of the spring force is conserved.

Conservation of kinetic energy:

The energy caused by the motion of the system is called the kinetic energy. From the theory of  work-energy the is equal to its change kinetic energy.

As we know the kinetic energy is due to the motion of the system given by

Where v is the speed at which the spring is displaced. And m is the mass of the spring.

The net work done on the spring system when is displaced from original position to final position is given by the change in kinetic energy  as

Where ∆KE is the change in the kinetic energy, v0 and the v is the speed of displacement of the spring from original point to final point.

If only conservative force is acts on the spring system then the work done will  be

Wnet = Wcon  ; where  Wcon  is the total work done by all the forces of the system.

i.e;  Wcon  = ∆KE

The kinetic energy of the spring system is non zero.

It shows that the kinetic energy of the spring is a conserved quantity.

When the spring begins to displace from original position it loses the potential energy. Then the network done will be  

Wcon = -∆PE

Or -∆PE = ∆KE

Or ∆PE+∆KE = 0

The above equation implies that the total energy of the system ( i.e; potential energy and the kinetic energy ) is constant for the spring force system. The total energy of the any system is conserved force.

Hence it shows that the spring force is also a conservative force.

Some solved examples:

A spring is stretched at a distance of 0.65m whose spring constant is 150Nm-1. calculate the potential energy of the spring system.

Solution:

           Given : Displacement of the spring = 0.65m

                         Spring constant k = 150Nm-1

The potential energy of the spring is given by

P.E = 31.687J

The spring constant of a stretched spring is 84Nm-1 and the potential energy is calculated as 53J. find the displacement of the spring.

Solution :

            Given :  The spring constant k = 84Nm-1

                           Potential energy (P.E) = 53J

The potential energy of spring is given by

To find the displacement we have to rearrange the above equation as

Substituting the values

Taking the square root

The displacement x = 1.12m

A spring is attached to a slab. It uses the energy of 33J to stretch 45cm. Calculate the spring constant using potential energy formula of the spring force .

Solution:

            Given : Potential energy of the spring = 33J

                          Displacement of the spring = 45m = 0.45m

To calculate the spring constant, the potential energy of the spring is



Rearranging the equation

k = 325.92Nm-1

A toy is pulled by a spring whose force constant is 134N/m . It is displaced a distance of 6cm. Calculate the kinetic energy and the speed of the toy that displaced through?

Solution:

           Given : Force constant k = 134N/m

                         Displacement of the toy = 6cm = 0.06m

 The total energy of the spring system is given by

 KE = – PE

The potential energy energy of the system is

i.e; KE= PE

KE = 0.2412

Here we neglect the negative sign because the kinetic energy can not be negative.

Kinetic energy is given byKinetic energy is given by

The speed or the velocity is given by


Taking the square root on both side

v = 0.6 m/s2

The toy moves with a speed of 0.6 m/s2 .

Two spring of spring constant k1 and k2 are attached to a rigid support vertically. It has the displacement of x1 and x2 respectively. What will be the net force acting on the springs? And what is the potential energy of the system?

Here we just have to resolve the spring force equation.

Let the two spring attached to a rigid support be s1 and s2.

The force acting on s1 will be

F1 =  k1x1     …..(1)

The force acting on  s2 will be

F2 = k2x2    ……(2)

From equation (1) and (2) the net force acting on the spring is given by

F = F1 + F2

F =  k1x1 + k2x2

The total potential energy of the system can be given as

By knowing values we can solve the potential energy.

Calculate the force required for a spring to stretch when the it is expanded at a distance of 26cm and having the spring constant 93N/m

Given : Displacement of the spring = 26cm = 0.26m

              Spring constant k = 93N/m

By the formula of spring force

F = kx

F = 93 × 0.26

F = 24.18 N

Frequently Asked Questions on spring Force:

what do you mean by Spring Constant?

The spring constant is a measurement of stretching ability of the spring.

It can also be defined as The force that required for a spring to compress or to elongate or to stretch by certain distance is meant as spring constant.

What is Hooke’s Law?

Hooke’s law is stated as the amount of force required to expand or compress  an elastic body is directly proportional to the distance at which the body is expanded or stretched.

Dose the gravity affect the spring constant?

Gravity does not affect the spring constant however  Gravity definitely affect the net force of spring as the gravity acts as a restoring force when the spring is suspended vertically from its equilibrium position.

What are the factors that influences the spring constant? 

Factors that influences the spring constants are

  • diameter of the each coil of the spring.
  • diameter of the suspension wire.
  • length of the spring when at rest.

Keerthi Murthi

I am Keerthi K Murthy, I have completed post graduation in Physics, with the specialization in the field of solid state physics. I have always consider physics as a fundamental subject which is connected to our daily life. Being a science student I enjoy exploring new things in physics. As a writer my goal is to reach the readers with the simplified manner through my articles. Reach me – keerthikmurthy24@gmail.com

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