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Consider a spongy block of mass $m$ floating on a flowing river. The maximum mass of the block is related to the speed of the river flow $v$, acceleration due to gravity $g$ and the density of the block $\rho$ such that $m_{\max }=k v^x g^y \rho^z(k$ is constant $)$. The values of $x, y$ and $z$ should then respectively be
(Mass of the spongy block is assumed to vary due to absorption of water by it)
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(Mass of the spongy block is assumed to vary due to absorption of water by it)
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Verified Answer
The correct answer is:
$6,-3,1$
Since, the maximum mass of the block
floating on river depends, speed of flow of the river $=v$, acceleration due to gravity $=g$ and density of the block $=\rho$,
$m_{\max }=k v^x g^y \rho^z$
Compairing the dimensions of $\mathrm{M}, \mathrm{L}$ and $\mathrm{T}$ on both
sides, we get
$z=1$
$\begin{aligned} x+y-3 z & =0 \\ -x-2 y & =0 \\ x+y-3 \times 1 & =0[\text { [From Eq. (i) and (ii)] } \\ x+y & =3\end{aligned}$
From Eqs. (iii) and (iv), we get
$-y=3 \Rightarrow y=-3$
From Eq. (iv), we have
$\begin{aligned} & & x-3 & =3 \\ \Rightarrow & & x & =6\end{aligned}$
Hence, the value of $x, y$ and $z$ will be $(6,-3,1)$.
floating on river depends, speed of flow of the river $=v$, acceleration due to gravity $=g$ and density of the block $=\rho$,
$m_{\max }=k v^x g^y \rho^z$
Compairing the dimensions of $\mathrm{M}, \mathrm{L}$ and $\mathrm{T}$ on both
sides, we get
$z=1$
$\begin{aligned} x+y-3 z & =0 \\ -x-2 y & =0 \\ x+y-3 \times 1 & =0[\text { [From Eq. (i) and (ii)] } \\ x+y & =3\end{aligned}$
From Eqs. (iii) and (iv), we get
$-y=3 \Rightarrow y=-3$
From Eq. (iv), we have
$\begin{aligned} & & x-3 & =3 \\ \Rightarrow & & x & =6\end{aligned}$
Hence, the value of $x, y$ and $z$ will be $(6,-3,1)$.
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