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Energy

 
 

Kinetic Energy

We define #~{kinetic energy} of a particle, mass ~m, velocity #~v:

~T _ #:= _ &hlf. ~m #~v &dot. #~v _ = _ &hlf. ~m #~v^2

Note that

deriv{~T} _ = _ &hlf. ~m fract{d ( #~v &dot. #~v ), d ~t} _ = _ &hlf. ~m ( deriv{#~v} &dot. #~v + #~v &dot. deriv{#~v} ) _ = _ ~m deriv{#~v} &dot. #~v _ = _ #~F &dot. #~v

Where #~F is the force acting on the particle. The quantity _ #~F &dot. #~v _ is called the #~{activity} of the force

Now by the fundamental theorem

int{#~F ( ~t ) &dot. #~v ( ~t ),~t_0,~t_1,d~t} _ = _ ~T ( ~t_1 ) - ~T ( ~t_0 )

The quantity _ ~W #:= ~{∫} #~F &dot. #~v d~t _ is called the ~#{work} done by the force in the given time interval.

int{#~F ( ~t ) &dot. #~v ( ~t ),~t_0,~t_1,d~t} _ = _ int{#~F ( ~t ) &dot. fract{d#~r, d~t} ,~t_0,~t_1,d~t} _ = _ int{#~F ( ~t ) &dot. ,~A,~B,d~#r}

Where _ ~A = ~#r ( ~t_0 ) _ and _ ~B = ~#r ( ~t_1 ) .

Conservative Fields

Note that work, as defined above, is a tangental line integral along the path taken by the particle, and, in general, is dependent on that path. If ~W is only dependent on the end-points, ~A and ~B, and not on the actual path between them, then the force field ~#F is said to be #~{conservative}.

For the rest of these notes we will assume that the force field is conservative.

Potential Energy

If #~F is a conservative field, then the work done when we move a particle from any point ~P say, to a fixed reference point, ~O, along ~{any} path is

~V ( ~P ) _ #:= _ int{#~F ( ~t ) &dot. ,~P,~O,d~#r}

This is called the #~{potential energy} of the particle at point ~P in the field #~F.

Conservation of Energy

A particle moves from ~A to ~B along a path &Gamma.. Then since the field is conservative

int{#~F ( ~t ) &dot. ,~A,~B,d~#r} _ _ = _ _ int{#~F ( ~t ) &dot. ,~A,~O,d~#r} _ + _ int{#~F ( ~t ) &dot. ,~O,~B,d~#r}

_

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = _ _ ~V ( ~A ) _ - _ ~V ( ~B )

But

int{#~F ( ~t ) &dot. ,~A,~B,d~#r} _ _ = _ _ ~T ( ~t_1 ) _ - _ ~T ( ~t_0 )

Now _ ~A = ~A ( ~t_0 ) , _ and _ ~B = ~B ( ~t_1 ), so we write

~V ( ~A ) _ = _ ~V ( ~t_0 ) , _ and _ ~V ( ~B ) _ = _ ~V ( ~t_1 )

Which means that

~V ( ~t_0 ) _ - _ ~V ( ~t_1 ) _ _ = _ _ ~T ( ~t_1 ) _ - _ ~T ( ~t_0 )

or

~V ( ~t_0 ) _ + _ ~T ( ~t_0 ) _ _ = _ _ ~V ( ~t_1 ) _ + _ ~T ( ~t_1 )

i.e. _ ~V ( ~t ) + ~T ( ~t ) _ is constant. _ This is the #~{principal of conservation of energy}.