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

Question

Aicharya · Accepted Answer

$f=C m^{x} k^{3}$. Writing dimensions on both sides: $\begin{aligned} [M^{0} L^{0} T^{-1}]&=M^{x}[M L^{0} T^{-2}]^{3} \\ [M^{0} L^{0} T^{-1}]&=[M^{x+3} T^{-6}] \end{aligned}$ Comparing dimensions on both sides, we have
$0=x+y \text { and }-1=-2 y \Rightarrow y=\frac{1}{2}, x=-\frac{1}{2}$
Aliter. Remembering that frequency of oscillation of loaded spring is
$f=\frac{1}{2 \pi} \sqrt{\frac{k}{m}}=\frac{1}{2 \pi}(k)^{1 / 2} m^{-1 / 2}$
which gives $x=-\frac{1}{2}$ and $y=\frac{1}{2}$

The frequency \(f\) of vibrations of a mass \(m\) suspended from a spring of spring consant \(k\) is given…

Solution