Work Energy Theorem : variable force

Q State and prove work- energy theorem, when force is variable.

 Work - Energy Theorem  

Work energy theorem states that total work done by body is equal to the change produced in its kinetic energy,

i.e, ΔK=W 

Let us derive this, 

F be the variable force 

$\therefore$ Work done by this variable force ,

 $W={\int }_{x_i}^{x_f}F\cdot dx$

where $x_i$ is the initial position 

and $x_f$ is final position.

For dx displacement particle or object receives constant force.

Also Kinetic energy of an object ,

     $K=\frac{1}{2}m{v}^2$ 

If we differentiate this with respect to time we get,

dk/dt = d/dt ( 1/2 mv²) 

Multiplying and dividing by dv

dk/dt = 1/2 m dv d v²/dt dv   

= 1/2 m dv/dt ( d/dv v²) 

= 1/2 m dv/dt x 2 v 

$⟹\frac{dK}{dt}=mv\frac{dv}{dt}$

$⟹\frac{dK}{dt}=ma\frac{dx}{dt}$

$⟹\frac{dK}{dt}F\frac{dx}{dt}$

$⟹dK=Fdx$

$⟹{\int }_{K_i}^{K_f}dK={\int }_{x_i}^{x_f}F\cdot dx$

$⟹\Delta K=W$

i.e Net work done by object is equal to change in its kinetic energy.

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