Obtain expression for potential energy of a spring attached to mass m moving on a frictionless horizontal surface by graphical method.
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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)
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The area under the force in the spring vs. displacement curve is the work done on the spring. a plot of force on the spring vs. displacement, where displacement is 000 when the spring is unstretched. The work done on a spring stores elastic potential energy U
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