Find dimensional formula of $\alpha$

Question

The power delivered by a force is given by the relation $$P=\frac{\alpha}{\beta}e^{-\beta t},$$ where $t$ is time. Find the dimensional formula for $\alpha$.

So, $-\beta t$ doesn't make sense for the dimensional formula of $P$. But then we have two unknowns and one equation since we do not know if $\beta$'s dimensions. But then if I put $\beta t$ as dimensionless, it fits just right with the answer $([ML^2 T^{-4}])$ . But does this happen exactly. Why is $\beta t$ dimensionless, because we are just using it's numerical value. Please help. I don't understand it.

Possible duplicate of dimensional analysis - how to find the dimensions of a variable inside an angle — sammy gerbil, Jan 11 '18 at 16:23

score 1 · Answer 1 · edited Jan 11 '18 at 19:33

1

$\beta t$ must be dimensionless because it is the argument of a function (read this thread). Therefore

$$ [\beta] = t^{-1} $$

edited Jan 11 '18 at 19:33

user121330

2,037

answered Jan 11 '18 at 15:48

caverac

6,744

1

So is it like exponential functions have to be dimensionless?? Compulsory?? – Sri Jan 11 '18 at 15:52
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@Sri The argument must be dimensionless – caverac Jan 11 '18 at 15:54

Find dimensional formula of $\alpha$

1 Answers1