Question

what is the proof of this formula : w=1/2kx² ​

Answer 1

Explanation:

Experimentally, we find

This is known as Hooke's law

We might write this in equation form as F = k x. However, the force exerted by the spring is always in the opposite direction to the stretch (or compression) of the spring. Therefore, we write Hooke's law as

F = - k x

This is the force exerted by the spring. The external force we exert on the spring is Fext = + k x.

The proportionality constant k is known as the spring constant and describes how stiff the spring is.

Now we are again ready to ask

"How much work is done when we stretch a spring a distance x from its equilibrium position?"

We know how to handle a constant force. For a constant force, we know

W = F d

Our spring force varies, but we can think of it as being (nearly) constant as we move through a (very) small distance,

The work done by a variable force is the area under the "curve" on a Force - distance graph. For this Hooke's law force of Fext = k x, the work done to the spring by the external force Fext is

W = (1/2) k x2

This work done on the spring as it is stretched (or compressed) can be recovered. This is stored work that can be used to do work on something else by this spring. That means the stretched (or compressed) spring has energy -- potential energy. This is spring potential energy or elastic potential energy.

PEel = (1/2) k x2

or

Uel = (1/2) k x2