Question

Obtain expression for potential energy of a spring attached to mass m moving on a frictionless horizontal surface by graphical method.​

Answer 1

Answer:

Hope this attachment helps you.

Explanation:

This method is called graphical method.

Answer 2

To study this ,consider an electric spring of negligibly small mass .One end of the spring is attached to the rigid wall and another end of spring is attached to a block of mass m which can move on smooth frictionless horizontal surface

Here un-stretched or un-compressed position of the spring is taken at x=0

We now take the block from its un-stretched position to a point P by stretching the spring

At this point P restoring force is exerted by the spring on the block trying it bring it back to the equilibrium position.

Similar restoring force developed in the spring when we try to compress it

For an ideal spring ,this restoring force F is proportional to displacement x and direction of restoring force is opposite to that displacement

Thus force and displacement are related as

F α x

or F= - kx           (16)

where K is called the spring constant and this equation (16) is known as Hook's law.negative sign indicates that force oppose the motion of the block along x

To stretch a spring we need to apply the external force which should be equal in magnitude and opposite to the direction of the restoring force mentioned above i.e for stretching the spring

Fext=Kx Similarly for compressing the spring

- Fext= - Kx

or Fext=Kx (both F and x are being negative)

Work done in both elongation and compression of spring is stored in the spring as its PE which can be easily calculated

If the spring is stretched through a distance x from its equilibrium position x=0 then

W=∫Fextdx

Since both Fext and dx have same direction Now

W=∫Kxdx

On integrating with in the limits x=0 to x=x

We have

W=Kx2/2           (17)

This work done is positive as force is towards the right and spring also moves towards the right

Same amount of external is done on the spring when it is compressed through a distance x

Work done as calculated in equation (17) is stored as Potential Energy of the spring.Therefore

U=Kx2/2