# Mechanics: Chapter 10 – Work Done by Forces

There are some easy derivations for the work done by common forces. The work done by a constant gravitational force is pretty simple. The force exerted by a constant gravitational force is just $mg$ and the displacement in the direction of the force is just the change in height $\Delta h$, hence

• Work done by constant gravitational force: $W = mg \Delta h$

The force exerted by a spring is given by Hooke’s Law, $F = -kx$, now this force varies with extension $x$ (which is essentially the displacement). Hence, we can’t simply take the product, we have to take the integral of $F$ with respect to $dx$, i.e

• Work done by spring: $W = \int F \cdot dx = - \int kx \cdot dx = -\frac{1}{2}kx^2$

The work done to accelerate an object from rest to a velocity $v$ is called the Kinetic Energy of the object. Now the force needed to accelerate the object is, by Newton’s second law:

• $F = ma$
• $F = m \frac{dv}{dt}$

Since the work done by $F$ is

• $W = \int F \cdot ds$
• $\Rightarrow W = \int m \frac{dv}{dt} \cdot ds$
• $\Rightarrow W = \int m \frac{ds}{dt} \cdot dv$
• $\Rightarrow W = \int mv \cdot dv$
• $\Rightarrow W = \frac{1}{2}mv^2$

Hence,

• Kinetic Energy of an object: $E_{K} = \frac{1}{2}mv^2$