Question
The friction force of air on an object moving in air is directly proportional to the velocity of the object, what will be the dimension of constant of proportionality (b)?
✓
Answer
If the frictional force of the air is F and velocity of the object is $v$, then according to question,
$\begin{array}{l}
F \propto v
\\F=b v\end{array}$
Here, b proportionality constant
$\begin{aligned} b & =\frac{F}{v}
\\ \text { Dimension of force } & =\left[ M ^1 L^1 T^{-2}\right]
\\ \text { Dimension of velocity } & =\left[ LT ^{-1}\right]
\\ \text { Dimension of } b & =\frac{\left[ M ^1 L^{-1} T^{-2}\right]}{\left[ LT ^{-1}\right]}=\left[ MT ^{-1}\right]\end{aligned}$
$\begin{array}{l}
F \propto v
\\F=b v\end{array}$
Here, b proportionality constant
$\begin{aligned} b & =\frac{F}{v}
\\ \text { Dimension of force } & =\left[ M ^1 L^1 T^{-2}\right]
\\ \text { Dimension of velocity } & =\left[ LT ^{-1}\right]
\\ \text { Dimension of } b & =\frac{\left[ M ^1 L^{-1} T^{-2}\right]}{\left[ LT ^{-1}\right]}=\left[ MT ^{-1}\right]\end{aligned}$
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