Question

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 $m_{\text{max}}=k v^{x}\, g^{y}\, \rho^{z}(k$ 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)

• A. 6,3,2
• B. 6,-3,1
• C. 3,6,1
• D. 6,1,3

Answer 1

Answer: (B)

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 $=\rho$,
$m_{\max }=k v^{x} g^{y} \rho^{z}$
Write the dimensional formula of the both side, we get,
$\left[M^{1} L^{0} T^{0}\right]=\left[L T^{-1}\right]^{x}\left[L T^{-2}\right]^{y}\left[M L^{-3}\right]^{z}$
$\left[M^{1} L^{0} T^{0}\right]=\left[M^{Z} L^{x+y-3 z} T^{-x-2 y}\right]$
Compairing the dimensions of $M, L$ and $T$ on both sides, we get
$z=1 \,\,\,\ldots(i)$
$x+y-3 z =0 \,\,\,\ldots (ii)$
$-x-2 y =0 \,\,\,\ldots(iii)$
$x+y-3 \times 1=0$ [From Eq. (i)and (ii)]
$x+y=3\,\,\,\ldots( iv )$
From Eqs. (iii) and (iv), we get
$-y=3 $
$\Rightarrow y=-3$
From Eq. (iv), we have
$= x-3=3 $
$x=6$
Hence, the value of $x, y$ and $z$ will be $(6,-3,1)$