Power delivered by a force is given by relation P = (alpha/beta)e^-beta × t. t = time. Find dimensional formula for alpha and beta.

Dear Student,

Please find below the solution to the asked query:

Here, the given relation is,

$P=\left(\frac{\alpha }{\beta }\right) e^{-\beta t}$
Here, in exponent, the term should be unit less. Therefore,


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

And, the ratio of alpha and beta should have the dimensions of power. Therefore,

$$\begin{gathered} \left[\frac{\alpha }{\beta }\right]=\left[M{L}^2{T}^{-3}\right] \Rightarrow \left[\alpha \right]=\left[\beta \right]\left[M{L}^2{T}^{-3}\right] \Rightarrow \left[\alpha \right]=\left[M^0{L}^0{T}^{-1}\right]\left[M{L}^2{T}^{-3}\right] \\ \Rightarrow \left[\alpha \right]=\left[M{L}^2{T}^{-4}\right] \end{gathered}$$

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