Question

A block of known mass is suspended from a fixed support through a light spring. Can you find the time period of vertical oscillation only by measuring the extension of the spring when the block is in equilibrium?

Answer 1

Yes.

Time period of a spring mass system is given by,
$T=2\pi \sqrt{\frac{m}{k}}$ ...(1) .
where m is mass of the block, and
k is the spring constant

Time period is also given by the relation,
$T=2\pi \sqrt{\frac{x_0}{g}}$ ...(2)
where, x₀ is extension of the spring, and
g is acceleration due to gravity

From the equations (1) and (2), we have:
$mg=k{x}_0$

$\Rightarrow k=\frac{mg}{x_0}$

Substituting the value of k in the above equation, we get:
$T=2\pi \sqrt{\frac{m}{{\displaystyle \frac{mg}{x_0}}}}=2\mathrm{\pi }\sqrt{\frac{{\mathrm{x}}_0}{\mathrm{g}}}$
Thus, we can find the time period if the value of extension x₀ is known.