# Work done by a body

This is a fundamental doubt Can we use the formula W is equal to F*D only when the body is accelerated? For example if the acceleration does not remain constant can we use it?

• The formula doesn't contain a velocity or acceleration term. A better way to write it would be differentially $dW = Fds$. This tells you that you have to integrate the term $Fds$ to get the total work. If $F$ changes along the way $s$, then this may be a non-trivial integral. – CuriousOne Dec 23 '15 at 3:34
• But the formula is deived from w is equal to f*v^2/(2*a)? – N.S.JOHN Dec 23 '15 at 3:37
• The formula is the definition of work and completely independent of motion. It can be applied to moving bodies, which then gives us the usual expression for kinetic energy by integrating over the work necessary to accelerate a body from standstill to a given velocity $v$. – CuriousOne Dec 23 '15 at 3:39

## 1 Answer

Work done on a body by a variable force $F(x)$ in displacing an object from a point a to b is given by
$$W=\int_a^b{F(x).ds}$$ This is the general formula for finding work done, provided $F(x)$ must be resultant force on the body that causes acceleration of body. The above integral is a path integral. If you are dealing with a constant force, then acceleration is constant, and you can see that the above integral reduce to $W=F.S$ , where $S=b-a$ is the displacement. If you are dealing with bodies have variable acceleration, then the force is a variable force (such as $F(x)=-Kx$ ) you have to take the above path integral into consideration.

• Just to clarify, W is the work made by force F, the body might or might not be accelerated (it might not be accelerating if there are other forces acting upon it). – user83548 Dec 23 '15 at 4:18