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,
$\left[M^1{L}^0{T}^0\right]={\left[L{T}^{-1}\right]}^x{\left[L{T}^{-2}\right]}^y{\left[M{L}^{-3}\right]}^z$
$\left[M^1{L}^0{T}^0\right]=\left[M^Z{L}^{x+y-3z}{T}^{-x-2y}\right]$
Compairing the dimensions of $M,L$ and $T$ on both sides, we get
$z=1\dots (i)$
$x+y-3z=0\dots (ii)$
$-x-2y=0\dots (iii)$
$x+y-3\times 1=0$ [From Eq. (i)and (ii)]
$x+y=3\dots (iv)$
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 $(6,-3,1)$