Why is small work done always taken as dW=Fdx and not dW=xdF?

Question

Why is small work done always taken as

d W = F \cdot d x

and not

d W = x \cdot d F

?

Answer 1

You can see this from the second Newton's law:

m \ddot{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

\dot{x}

:

m \ddot{x} \cdot \dot{x} = F (x) \cdot \dot{x}

and finally integrate:

m \int d t \ddot{x} \cdot \dot{x} = \int d t F (x) \cdot \dot{x}

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

\frac{1}{2} m {\dot{x}}^{2} = \int d x \cdot 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).

Answers (1)