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1.Units, Dimensions and Measurement
hard
The potential energy of a point particle is given by the expression $V(x)=-\alpha x+\beta \sin (x / \gamma)$. A dimensionless combination of the constants $\alpha, \beta$ and $\gamma$ is
A
$\frac{\alpha}{\beta \gamma}$
B
$\frac{\alpha^2}{\beta \gamma}$
C
$\frac{\gamma}{\alpha \beta}$
D
$\frac{\alpha \gamma}{\beta}$
(KVPY-2012)
Solution
(d)
Potential energy of the particle is
$V(x)=-\alpha x+\beta \sin \left(\frac{x}{\gamma}\right)$
Clearly, dimensions of $\alpha, \beta$ and $\gamma$ are
$[\alpha]=\frac{[ V ]}{[ x ]}=\frac{\left[ ML ^2 T ^{-2}\right]}{[ L ]}=\left[ MLT ^{-2}\right]$
$\mid \beta]=[ V ]=\left[ ML ^2 T ^{-2}\right]$
and $[\gamma]=[ x ]=[ L ]$
$=\left[ M ^0 L ^0 T ^0\right]$
Standard 11
Physics
