Force (F) and density (d) are related as F = | vedclass

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(A) According to the principle of homogeneity of dimensions,quantities added together must have the same dimensions.Since $\beta$ is added to $\sqrt{d

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$[M^{3/2} L^{-1/2} T^{-2}]$

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(A) According to the principle of homogeneity of dimensions,quantities added together must have the same dimensions.Since $\beta$ is added to $\sqrt{d}$,the dimensions of $\beta$ must be equal to the dimensions of $\sqrt{d}$.Density $d = [M L^{-3}]$,so $\sqrt{d} = [M^{1/2} L^{-3/2}]$.Thus,$[\beta] = [M^{1/2} L^{-3/2}]$.The denominator $(\beta + \sqrt{d})$ will also have the same dimensions: $[\beta + \sqrt{d}] = [M^{1/2} L^{-3/2}]$.The given equation is $F = \frac{\alpha}{\beta + \sqrt{d}}$.Rearranging for $\alpha$,we get $\alpha = F 	imes (\beta + \sqrt{d})$.Force $F = [M L T^{-2}]$.Substituting the dimensions: $[\alpha] = [M L T^{-2}] 	imes [M^{1/2} L^{-3/2}]$.$[\alpha] = [M^{1 + 1/2} L^{1 - 3/2} T^{-2}] = [M^{3/2} L^{-1/2} T^{-2}]$.

Force $(F)$ and density $(d)$ are related as $F = \frac{\alpha}{\beta + \sqrt{d}}$. The dimensions of $\alpha$ are:

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