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General Definitions of Impulse and Work

  1. Nov 19, 2011 #1
    In classical mechanics, if we consider the motion of a particle of mass [itex]m[/itex], then



    The mass [itex]m[/itex] is [itex]constant[/itex]


    The vector [itex]\vec{c}[/itex] can be: [itex]\ldots[/itex] or [itex]\vec{r}[/itex] or [itex]\vec{v}[/itex] or [itex]\vec{a}[/itex] or [itex]\vec{j}[/itex] or [itex]\ldots[/itex]

    [itex]\vec{c}_1 = d\vec{c} / {dt}[/itex]

    [itex]\vec{c}_2 = d^2 \, \vec{c} / {dt^2}[/itex]



    Definition of Impulse [itex]\vec{c}[/itex] [itex]\, ( \vec{I}_{\vec{c}} )[/itex]

    [tex]\vec{I}_{\vec{c}} \; = \int_a^b m \, \vec{c}_1 \, dt \; = \Delta \; m \, \vec{c}[/tex]
    [tex]\vec{I}_{\vec{c}} \; = \Delta \; \vec{P}_{\vec{c}}[/tex]
    where:

    [tex]\Delta \; \vec{P}_{\vec{c}} \; = \Delta \; m \, \vec{c}[/tex]

    If [itex]\; \vec{c}_1 = 0[/itex]

    [tex]\rightarrow \; \; \Delta \; \vec{P}_{\vec{c}} \; = 0[/tex]
    [tex]\rightarrow \; \; \vec{P}_{\vec{c}} \; = constant[/tex]


    Definition of Work [itex]\vec{c}[/itex] [itex]\, (W_{\vec{c}})[/itex]

    [tex]W_{\vec{c}} \; = \int_a^b m \; \vec{c}_2 \cdot d\vec{c} \; = \Delta \; {\textstyle \frac{1}{2}} \, m \; \vec{c}_1^2[/tex]
    [tex]W_{\vec{c}} \; = \Delta \; T_{\vec{c}_1} + \Delta \; V_{\vec{c}} \; = \int_a^b m \; \vec{c}_{2n\vec{c}} \cdot d\vec{c}[/tex]
    where:

    [tex]\Delta \; T_{\vec{c}_1} = \Delta \; {\textstyle \frac{1}{2}} \, m \; \vec{c}_1^2[/tex]
    [tex]\Delta \; V_{\vec{c}} = - \int_a^b m \; \vec{c}_{2\vec{c}} \cdot d\vec{c}[/tex]
    [tex]\vec{c}_2 = \vec{c}_{2\vec{c}} + \vec{c}_{2n\vec{c}}[/tex]
    [tex]\vec{c}_{2\vec{c}} \; \; \; is \; \; function \; \; of \; \; \; \vec{c}[/tex]
    [tex]\vec{c}_{2n\vec{c}} \; \; \; is \; \; not \; \; function \; \; of \; \; \; \vec{c}[/tex]

    If [itex]\; \vec{c}_{2n\vec{c}} = 0[/itex]

    [tex]\rightarrow \; \; \Delta \; T_{\vec{c}_1} + \Delta \; V_{\vec{c}} \; = 0[/tex]
    [tex]\rightarrow \; \; T_{\vec{c}_1} + V_{\vec{c}} \; = constant[/tex]

    If [itex]\; \vec{c}_2 = 0[/itex]

    [tex]\rightarrow \; \; \Delta \; T_{\vec{c}_1} \; = 0[/tex]
    [tex]\rightarrow \; \; T_{\vec{c}_1} \; = constant[/tex]
     
    Last edited: Nov 19, 2011
  2. jcsd
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