Why is small work done always taken as dW=F*dx and not dW=x*dF?

Mariah Sparks

Mariah Sparks

Answered question


Why is small work done always taken as d W = F d x and not d W = x d F ?

Answer & Explanation

Steven Bates

Steven Bates

Beginner2022-07-24Added 15 answers

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 d t x ¨ x ˙ = d t F ( 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 = d x 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).

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