Physics

If $$F$$ is given by $$F$$ = ($${\alpha / \beta}$$) ($${1 - e ^{{- \beta t^2} / m}}$$), where $$F$$ is force and $$m$$ & $$t$$ are mass and time, then dimension of $$\alpha$$ will be :


SOLUTION
Since, $$F=\cfrac { \alpha  }{ \beta  } \left( 1-{ e }^{ -\cfrac { \beta { t }^{ 2 } }{ m }  } \right) $$

Since exponential is an dimensionless quantity

Hence,
$$-\cfrac { \beta { t }^{ 2 } }{ m } $$ is a dimensionless quantity

$$\therefore$$  $$\left[ \cfrac { \beta { t }^{ 2 } }{ m }  \right] =\left[ { M }^{ 0 }{ L }^{ 0 }{ T }^{ 0 } \right] $$

$$\Rightarrow \beta \left[ \cfrac { { T }^{ 2 } }{ M }  \right] =\left[ { M }^{ 0 }{ L }^{ 0 }{ T }^{ 0 } \right] $$

Hence, $$\quad \beta =\cfrac { \left[ { M }^{ 0 }{ L }^{ 0 }{ T }^{ 0 } \right] \left[ M \right]  }{ \left[ { T }^{ 2 } \right]  } $$

and the two quantities cannot subtracted added until they don't have same dimension

$$\left[ F \right] =\cfrac { \alpha  }{ \beta  } $$

since, $$\left[ F \right] =\left[ { M }^{ 1 }{ L }^{ 1 }{ T }^{ -2 } \right] $$

$$\left[ { M }^{ 1 }{ L }^{ 1 }{ T }^{ -2 } \right] =\cfrac { \left[ \alpha  \right]  }{ \left[ { M }^{ 1 }{ L }^{ 0 }{ T }^{ -2 } \right]  } $$

$$\therefore \left[ \alpha  \right] =\left[ { M }^{ 1 }{ L }^{ 1 }{ T }^{ -2 } \right] \left[ { M }^{ 1 }{ L }^{ 0 }{ T }^{ -2 } \right] $$

$$\left[ \alpha  \right] =\left[ { M }^{ 2 }{ L }^{ 1 }{ T }^{ -4 } \right] $$
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