Question

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

Answer 1

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

1. We now take the block from its un-stretched position to a point P by stretching the spring
2. At this point P restoring force is exerted by the spring on the block trying it bring it back to the equilibrium position.
3. Similar restoring force developed in the spring when we try to compress it
4. For an ideal spring ,this restoring force F is proportional to displacement x and direction of restoring force is opposite to that displacement
5. Thus force and displacement are related as
6. F α x
7. or F= - kx           (16)
8. 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
9. 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
10. Fext=Kx Similarly for compressing the spring
11. - Fext= - Kx
12. or Fext=Kx (both F and x are being negative)
13. Work done in both elongation and compression of spring is stored in the spring as its PE which can be easily calculated
14. If the spring is stretched through a distance x from its equilibrium position x=0 then
15. W=∫Fextdx
16. Since both Fext and dx have same direction Now
17. W=∫Kxdx
18. On integrating with in the limits x=0 to x=x
19. We have
20. W=Kx2/2           (17)
21. This work done is positive as force is towards the right and spring also moves towards the right
22. Same amount of external is done on the spring when it is compressed through a distance x
23. Work done as calculated in equation (17) is stored as Potential Energy of the spring.Therefore
24. U=Kx2/2

Answer 2

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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

Consider the figure given below

• 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

• 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)