Question

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

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

Answer 1

The correct option is D $-\frac{1}{2},\frac{1}{2}$

We know $F=-KK\Rightarrow dim\left(K\right)=\left[{MLT}^{-2}\right]\left[L^{-1}\right]={ML}^0{T}^{-2}$

$dim\left(M\right)={ML}^0{T}^0$ $dim\left(f\right)=\left[M^0{L}^0{T}^{-1}\right]$

$f={Cm}^x{K}^y\Rightarrow M^0{L}^0{T}^{-1}={\left[{ML}^0{T}^0\right]}^k{\left[{ML}^0{T}^{-2}\right]}^y$

$\Rightarrow$ Comparing powers of $M,L$ and $T$ gives,

$x+y=0-2y=-1\Rightarrow y=\frac{1}{2}$

and $x=-1/2$