Mass Kinetic Energy from Virtual Work
From Newton's second law, (introduced in Eq.(B.1)), we can use d'Alembert's idea of virtual work to derive the formula for the kinetic energy of a mass given its speed . Let denote a small (infinitesimal) displacement of the mass in the direction. Then we have, using the calculus of differentials,
Thus, by Newton's second law, a differential of work applied to a mass by force through distance boosts the kinetic energy of the mass by . The kinetic energy of a mass moving at speed is then given by the integral of all such differential boosts from 0 to :
The quantity is classically called the virtual work associated with force , and a virtual displacement .
Energy in the Mass-Spring Oscillator
Kinetic Energy of a Mass