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Work Energy Theorem

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According to this theorem, the work done by all the forces (conservative or non-conservative, external or internal) acting on a particle or an object is equal to the change in kinetic energy of it.

 \Rightarrow W_{net} = \Delta K.E. = K_f - K_i

Let  \overrightarrow{F_1},  \overrightarrow{F_2} ... be the individual forces acting on a particle. The resultant force is  \overrightarrow{F} = \overrightarrow{F_1} + \overrightarrow{F_2} + ... and the work done by the resultant force is:

 W = \int{ \overrightarrow{F} \cdot \overrightarrow{dr} } = \int {( \overrightarrow{F_1} + \overrightarrow{F_2} + ... ) \cdot \overrightarrow{dr} }
 = \int{ \overrightarrow{F_1} \cdot \overrightarrow{dr} } + \int{ \overrightarrow{F_2} \cdot \overrightarrow{dr} + ...}

where  \int{ \overrightarrow{F} \cdot \overrightarrow{dr} } is the work done on the particle by  \overrightarrow{F_1} and so on. Thus, work energy theorem can also be written as: Work done by the resultant force is equal to the sum of the work done by the individual forces. There are a few points worth noting:

  1. ) If Wnet is positive then Kf - Ki = +ve (positive), i.e, Kf > Ki or kinetic energy will increase and vice versa.
  2. ) This theorem can be applied to non-inertial frames also. In a non-inertial fram it can be written as: Work done by all the forces (including pseudo forces) = change in kinetic energy in non-inertial frame.


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