Question

Find an expression for the work done by the spring force and hence show that the

spring force is a conservative force. Also deduce an expression for the potential energy

of an stretched spring.​

Answer 1

We know that spring’s force is directly proportional to the amount of elongation or compression and is in an opposing direction to that of compression or elongation.

Spring force is a variable force.

Mathematically:

F ∝ -x

F = -kx

Since force is in a direction opposite to the x vector.

The work done by a variable force is:

W = ∫ F.ds =  ∫F. -kxdx

Limits of displacement “x” in  integration are from 0 to x.

W = -1/2 kx²

The above expression gives the amount of work done by the ideal spring, whether compressed or elongated by a distance “x”.

A conservative force is a force, the work is done by which is independent of path. And thus, the work done on a round trip by the said force is zero.

To prove that the elastic force is a conservative force, let's consider a round trip by a spring where it moves a particle to a distance dl and back.

W total=WA→B+WB→A

W =∫dW = ∫F.dx

Now,

W total = ∫F.dl (A to B) + ∫F.dl (B to A)

(Where F and dl is a vector quantity)

W total = ∫-kxi .dxi + ∫-kxi. (-dxi)          (i is position vector)

∴ W total =0

As work is done on the round is zero. Therefore, it is proved that elastic force is a conservative force.

For the potential energy of an elastic spring. Let us consider a spring with spring constant k.

Let it be stretched by a very small distance dx. The work done by the spring in this process is dW, and that work is:

dW=Fdx

W = ∫dW =∫F.dx =∫-kxdx (limit zero to x)

Integrating:

W = -k∫xdx = [-kx²/2]ˣ₀

W = -1/2 kx²

And since we are moving the spring slowly. There is no kinetic energy. So the entire work done is stored as spring potential energy.

∴The expression for spring potential energy is:

U=−1/2kx²

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Answer 2

The expression for spring potential energy is U=−1/2kx².

The energy that has been converted from the work done on a body is stored. Mechanical energy primarily comes in two forms: kinetic energy and potential energy. When a body is not moving, all of the labour done on it is transformed into potential energy. In addition, a conservative force does nothing throughout a round-trip.

Formula Used:

W=∫ dW

W =  $\int\limits^x_0 {F} \, dx$

SpringForce = kx

A conservative force works regardless of the direction. Therefore, the specified force's effort for a circular journey is zero.

Let's look at a spring moving a particle round trip to a distance dl and return to demonstrate the conservatism of the elastic force.

Wtotal = WA → B + WB → A

We know that work is given as

W=∫ dW

W =  $\int\limits^x_0 {F} \, dx$

Now,

Wtotal = WA → B + WB → A

⇒Wtotal  =  $\int\limits^B_A {F} \, dl$  + $\int\limits^A_B {F} \, dl$

⇒Wtotal = ∫( −kxiˆ )⋅( dxiˆ )+∫( −kxiˆ )⋅( −dxiˆ )

⇒Wtotal=0

The round's work is zero as of this moment. It has been established as a result that elastic force is a conservative force.

In relation to an elastic spring's potential energy. Consider a spring that has a spring constant of k.

Permit a very slight dx distance stretch to be applied to it. This procedure requires the spring to perform dW of work, where

dW=Fdx

Total work done in stretching the spring from x=0 to x=x will be

W=∫ dW

W =  $\int\limits^x_0 {F} \, dx$  ------------------(1)

to stretch, we are applying force against it. Therefore, to move the force slowly.

F = −kx

substituting values in equation 1,

⇒W =$\int\limits^x_0 {(-kx)} \, dx$

⇒W=−k$\int\limits^x_0 {(-x)} \, dx$

⇒W=−k[x²/2]^x 0

⇒W=−1/2kx²

Work Done in moving the spring is W=−1/2kx².

Additionally, the spring is being moved gradually. No kinetic energy exists. The total amount of work is, therefore, conserved as spring potential energy. Consequently, spring potential energy is expressed as

U = −1/2kx².

 

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