Question

give the d.f of work done​

Answer 1

Very nice question! You can see this from the second Newton's law:

m

¨

x

=F(x)

Now I would like to integrate this equation of motion with respect to time, to arrive at the energy conservation. To do so I multiply both sides with

˙

x

:

m

¨

x

⋅

˙

x

=F(x)⋅

˙

x

and finally integrate:

m∫dt

¨

x

⋅

˙

x

=∫dtF(x)⋅

˙

x

The l.h.s. gives me the kinetic energy. The r.h.s. gives me exactly the integral in question:

1

2

m

˙

x

2=∫dx⋅F(x)

So the work done by the force is the kinetic energy of the particle (up to an integration constant representing its total energy).