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Notions of Work and Kinetic Energy: the Work-Energy Theorem

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  • Work-Energy Theorem

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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
`(dK)/(dt)=d/(dt)(1/2 mv^2)`
       `=m(dv)/(dt)v`
       `=Fv`           ...(from Newton's Second Law)
       `=F(dx)/(dt)`
Thus,
   `dK=F dx`
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
     `K_f-K_i =W`
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.

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