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 find:

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

Solution:

First of all, let's draw the Force vs Displacement graph for spring attached to the mass.

$\therefore \: F \propto ( - x)∴F∝(−x)$

$\implies \: F = - kx \: \: \: . \: . \: . \: .(k \: is \: constant)$

So, F - x graph will be straight line passing through the origin with negative slope.

Now, as per ENERGY CONSERVATION PRINCIPLE, the potential energy stored in the spring will be equal to the total work done.

$\therefore \: PE = total \: work∴PE=totalwork$

$\implies \: PE = area \: under \: F - x \: graph$

$\implies \: PE = area \: of \: \Delta$

$\implies \: PE = \dfrac{1}{2} \times base \times height$

$\implies \: PE = \dfrac{1}{2} \times |F| \times x$

$\implies \: PE = \dfrac{1}{2} \times kx \times x$

$\implies \: PE = \dfrac{1}{2} k {x}^{2}$

So, potential energy in spring is ½kx². . .

Answer 2

Answer:

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)