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:

\[T = 2\pi\sqrt{\frac{m}{\frac{mg}{x_0}}} = 2\pi\sqrt{\frac{x_0}{g}}\]

Thus, we can find the time period if the value of extension x₀ is known.