Question

The frequency of vibration $f$ of a mass $m$ suspended from a spring of spring constant $k$ is given by a relation $f=a{m}^x{k}^y$, where $a$ is a dimensionless constant. The values of $x$ and $y$ are

• A. $x=\frac{1}{2},y=\frac{1}{2}$
• B. $x=-\frac{1}{2},y=-\frac{1}{2}$
• C. $x=\frac{1}{2},y=-\frac{1}{2}$
• D. $x=-\frac{1}{2},y=\frac{1}{2}$

Answer 1

Answer: (D)

$f=a{m}^x{k}^y....(i)$
Dimensions of frequency $f=[M^0{L}^0{T}^{-1}]$
Dimensions of constant $a=[M^0{L}^0{T}^0]$
Dimensions of mass $m=[M]$
Dimensions of spring constant $k=[M{T}^{-2}]$
Putting these value in equation $(i)$, we get
$[M^0{L}^0{T}^{-1}]=[M{]}^x[M{T}^{-2}{]}^y$
Applying principle of homogeneity of dimensions, we get
$x+y=\mathrm{0.....}(ii)$
$-2y=-\mathrm{1......}(iii)$
or $y=\frac{1}{2},x=-\frac{1}{2}$