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
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