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1.Units, Dimensions and Measurement
medium
In equation $y=x^2 \cos ^2 2 \pi \frac{\beta \gamma}{\alpha}$, the units of $x, \alpha, \beta$ are $m , s ^{-1}$ and $\left( ms ^{-1}\right)^{-1}$ respectively. The units of $y$ and $\gamma$ are
A$m ^2, ms ^{-2}$
B$m , ms ^{-1}$
C$m^2, m$
D$m , ms ^{-2}$
Solution
(a)
$y=x^2 \cos ^2 2 \pi\left(\frac{\beta \gamma}{\alpha}\right)$
The argument of a trigonometric ratio is always dimensionless.
$\frac{\beta \gamma}{\alpha}=\left[ M ^0 L ^0 T ^0\right] \text { or } \beta \gamma=\alpha \Rightarrow \gamma=\frac{ T }{ L ^2}$
$\text { and } y=x^2 \Rightarrow\left[ L ^2\right]$
$\alpha= s ^{-1} \Rightarrow\left[ T ^{-1}\right], \beta=\left[ LT ^{-1}\right]^{-1} \Rightarrow\left[ L ^{-1} T \right]$
$y=m^2$
$\gamma= ms ^{-2}$
$y=x^2 \cos ^2 2 \pi\left(\frac{\beta \gamma}{\alpha}\right)$
The argument of a trigonometric ratio is always dimensionless.
$\frac{\beta \gamma}{\alpha}=\left[ M ^0 L ^0 T ^0\right] \text { or } \beta \gamma=\alpha \Rightarrow \gamma=\frac{ T }{ L ^2}$
$\text { and } y=x^2 \Rightarrow\left[ L ^2\right]$
$\alpha= s ^{-1} \Rightarrow\left[ T ^{-1}\right], \beta=\left[ LT ^{-1}\right]^{-1} \Rightarrow\left[ L ^{-1} T \right]$
$y=m^2$
$\gamma= ms ^{-2}$
Std 11
Physics
