- Work-Energy Theorem
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.”
Kf – Ki = W
where Kf is the final Kinetic Energy and Ki is the initial Kinetic Energy
We know the equation in 3D: v2 – u2 = 2 a.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
½ mv2 – ½ mu2 = F.d (Since ma = F)
The Work-Energy Theorem for a Variable Force
The time rate for change of kinetic energy
`=Fv` ...(from Newton's Second Law)
Integrating from initial position (xi) to final position (xf), we have
`∫_(K_i)^(K_f) dK=∫_(x_i)^(x_f)F dx`
where, Kt and Kf are the initial and final kinetic energies corresponding to xi and xf,
or `K_f-K_i=∫_(x_i)^(x_f) F dx`
From earlier proved equation, it follows that
Thus, the W.E. theorem is proved for a variable force.
Regarding the work-energy theorem it is worth noting that
(i) If Wnet is positive, then Kf – Ki = positive, i.e., Kf > Ki or kinetic energy will increase and vice-versa.
(ii) This theorem can be applied to non-inertial frames also. In a non-inertial frame it can be written as:
Work done by all the forces (including the Pseudo force) = change in kinetic energy in non-inertial frame.