Question

The frequency $f$ of vibration of 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 B

$-\frac{1}{2},\frac{1}{2}$

Step 1: Given

Frequency of object=$f$

Mass of object=$m$

Spring constant=$k$

Relation between frequency, mass, and spring constant, $f=c{m}^x{k}^y$

Step 2: Dimensions Used

Dimension of frequency$\left(f\right)$=$\left[T^{-1}\right]$

Dimension of mass$\left(m\right)=\left[M\right]$

Dimension of spring constant$\left(k\right)=\left[M{T}^{-2}\right]$

Step 3: Determine the value of $x$ and $y$

Substitute the known dimensions in the relation $f=c{m}^x{k}^y$,

Where $c$ is a dimensionless constant

$$\begin{gathered} \left[T^{-1}\right]={\left[M\right]}^x{\left[M{T}^{-2}\right]}^y \\ \left[T^{-1}\right]={\left[M\right]}^{x+y}{\left[T^{-2}\right]}^y \end{gathered}$$

Compare the dimension of both sides,

$$\begin{gathered} x+y=0\dots \dots \dots \dots \dots \left(1\right) \\ -2y=-1\dots \dots \dots \dots \dots \left(2\right) \end{gathered}$$

So, the value of $xandy$ will be $-\frac{1}{2},\frac{1}{2}$ respectively.

Hence, option B is the correct answer.