Question

The three physical quantities $x,y$ and $z$ have units $g{cm}^2{s}^{-5},g{s}^{-1}$ and $cm{s}^{-2}$ respectively.
The relation between $x,y$ and $z$ is

• A. $x=y{z}^2$
• B. $x=y^{2}x$
• C. $y^2=xz$
• D. $z=x^{2}y$

Answer 1

The correct option is A $x=y{z}^2$

$\text{Step 1 Determine the dimension for x, y and z}$

$x=g{cm}^2{s}^{-5}=\left[M{L}^2{T}^{-5}\right]$

$y=g{s}^{-1}=\left[M{L}^0{T}^{-1}\right]$

$z=cm{s}^{-2}=\left[M^{0}L{T}^{-2}\right]$

$\text{Step 2: Dimensional analysis}$

Let $x=y^a{z}^b$

$\Rightarrow \left[M{L}^2{T}^{-5}\right]=\left[M{L}^0{T}^{-1}{]}^a\right[M^{0}L{T}^{-2}{]}^b$

$\Rightarrow \left[M{L}^2{T}^{-5}\right]=\left[M^a{L}^b{T}^{-a-2b}\right]$

Comparing the above equation

$\Rightarrow a=1,b=2$

Hence, $x=y{z}^2$, Option (A) is correct.

$\text{Alternate solution:}$

We can directly check the answer from options.

Check option(A):

Dimensions of x, y and z are :

$x=g{cm}^2{s}^{-5}=\left[M{L}^2{T}^{-5}\right]$, $y=g{s}^{-1}=\left[M{L}^0{T}^{-1}\right]$, $z=cm{s}^{-2}=\left[M^{0}L{T}^{-2}\right]$

Now, $z^2=\left[M^0{L}^2{T}^{-4}\right]$

and $y{z}^2=\left[M{L}^0{T}^{-1}\right]\left[M^0{L}^2{T}^{-4}\right]=\left[M{L}^2{T}^{-5}\right]=x$

i.e., $x=y{z}^2$