A linear spring of force constant $k$ is used in a physics lab experiment. A block of mass m is attached to the spring and the resulting frequency, $f$, of the simple harmonic oscillations is measured. Blocks of various masses are used in different trials, and in each case, the corresponding frequency is measured and recorded. If $f^2$ is plotted versus $\displaystyle \frac{1}{m}$, the graph will be a straight line with slope
A) $\displaystyle \frac{4\pi^2}{k^2}$
B) $\displaystyle \frac{4\pi^2}{k}$
C) $4\pi^2k$
D) $\displaystyle \frac{k}{4\pi^2}$
A) $\displaystyle \frac{4\pi^2}{k^2}$
B) $\displaystyle \frac{4\pi^2}{k}$
C) $4\pi^2k$
D) $\displaystyle \frac{k}{4\pi^2}$
The frequency of a spring‑block simple harmonic oscillator is given by the equation
$$f=\left(\displaystyle \frac{1}{2\pi }\right)\sqrt{\displaystyle \frac{k}{m}}$$
Squaring both sides of this equation, you get
$$f^2=\left(\displaystyle \frac{k}{\ 4\pi^2}\right)\left(\displaystyle \frac{1}{m}\right)$$
Therefore, if $f^2$ is plotted versus $\left(\displaystyle \frac{1}{m}\right)$, then the graph will be a straight line with slope $\displaystyle \frac{k}{\ 4\pi^2}$.
(Note: The slope of the line whose equation is $y=ax$ is $a$.)