Question
The velocity of sound $v$ in the air depends on the air pressure, P and density of air $d$. Establish the possible formula by dimensional method :
$
v= K \sqrt{\frac{ P }{d}}
$
$
v= K \sqrt{\frac{ P }{d}}
$
✓
Answer
We know that velocity of sound $v$ depends on the density $d$ and air pressure P . There we can write :
$v \propto P ^x d^y$
Where $x$ and $y$ are the dimensions of pressure and density respectively. If K is constant then :
$v = K P ^x d^y$
Dimension of left side $(v)=\left[ M ^0 L^1 T^{-1}\right]$
Dimension of Pressure $=\left[ M ^1 L^{-1} T^{-2}\right]$
K is dimensionless.
$\therefore$ Dimension of right side
$
\begin{array}{l}
=\left[M^1 L^{-1} T^{-2}\right]^x \times\left[M^1 L^{-3} T^0\right]^y \\
=\left[M^{x+y} L^{-x-3 y} T^{-2 x}\right]
\end{array}
$
On comparing the powers of $M , L$ and T on both sides,
$
\begin{aligned}
x+y & =0 \\
-x-3 y & =1 \\
-2 x & =-1 \\
x & =\frac{1}{2}
\end{aligned}
$
Put value of $x$ in equation (2)
$
y=-\frac{1}{2}
$
By putting values of $x$ and $y$ in equation (1)
$
\begin{array}{l}
v=K P^{\frac{1}{2}} d^{-\frac{1}{2}} \\
v=K \sqrt{\frac{P}{d}}
\end{array}
$
By dimensional method, value of K can't be determined.
$v \propto P ^x d^y$
Where $x$ and $y$ are the dimensions of pressure and density respectively. If K is constant then :
$v = K P ^x d^y$
Dimension of left side $(v)=\left[ M ^0 L^1 T^{-1}\right]$
Dimension of Pressure $=\left[ M ^1 L^{-1} T^{-2}\right]$
K is dimensionless.
$\therefore$ Dimension of right side
$
\begin{array}{l}
=\left[M^1 L^{-1} T^{-2}\right]^x \times\left[M^1 L^{-3} T^0\right]^y \\
=\left[M^{x+y} L^{-x-3 y} T^{-2 x}\right]
\end{array}
$
On comparing the powers of $M , L$ and T on both sides,
$
\begin{aligned}
x+y & =0 \\
-x-3 y & =1 \\
-2 x & =-1 \\
x & =\frac{1}{2}
\end{aligned}
$
Put value of $x$ in equation (2)
$
y=-\frac{1}{2}
$
By putting values of $x$ and $y$ in equation (1)
$
\begin{array}{l}
v=K P^{\frac{1}{2}} d^{-\frac{1}{2}} \\
v=K \sqrt{\frac{P}{d}}
\end{array}
$
By dimensional method, value of K can't be determined.
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