The potential energy of a particle as function of its position ‘ $x$ ’ is given by the expression $U (x) = - α x + β \sin \frac{x}{γ} .$ Among the following a dimensionless combination of the constants $α, β$ and $γ$ is:

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$\frac{α^{2}}{β α}$

$\frac{α γ}{β}$

$\frac{γ}{α β}$

$\frac{α}{β γ}$

answer is D.

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Detailed Solution

$\Rightarrow [γ] = [x] = [L] \Rightarrow [β] = [U] = [M^{1} L^{2} T^{- 2}] \Rightarrow [α x] = [U] = [M^{1} L^{2} T^{- 2}] [α] = [M^{1} L^{1} T^{- 2}] [\frac{α γ}{β}] = [M^{\circ} L^{\circ} T^{\circ}]$

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