Notions of Work and Kinetic Energy

• The work-energy theorem states that the change in kinetic energy of a particle is equal to the work done on it by the net force.
• K_f – K_i = W
• We know the equation in 3D : v² – u² = 2a.d (where u-initial velocity, v-final velocity, a-acceleration, d-displacement)
   now multiplying the equation by m/2 we have, ½ mv² – ½ mu² = ma.d = F.d   (Since ma = F)

Work

The work done by the force is defined to be the product of component of the force in the direction of the displacement and the magnitude of this displacement. W = F.d       .

 No work is done if:

• The displacement is zero

The force is zero.

Work done by a variable force

• The variable force is more commonly encountered than the constant force.
• If the displacement Dx is small, we can take the force F (x) as approximately constant and the work done is then
  DW =F (x) Dx

For total work, we add all work done along small displacements.

Example: A force F = 3x² start acting on a particle which is initially at rest. Find the work done by the force during the displacement of the particle from x =0m to x = 2m.

Kinetic energy

• The kinetic energy of an object is a measure of the work an object can do by the virtue of its motion.
• If an object of mass m has velocity v, its kinetic energy K is  
• Kinetic energy is a scalar quantity.