We want to hang a thin hoop on a horizontal nail and have the hoop make one complete small-angle oscillation each \({\bf{2}}.{\bf{0}}\,{\bf{s}}\). What must the hoop’s radius be?

Time period of oscillation\(T = 2.0\,{\rm{s}}\)

The letter "\(T\)" stands for the period of time needed for the pendulum to complete one complete oscillation.

\(T = 2\pi \sqrt {\frac{I}{{mgR}}} \) …(i)

Where, \(m\) is mass, \(I\) is the moment of inertia, \(R\) is radius of the loop and \(g\) is the acceleration due to gravity

The moment of inertia about the center of mass is expressed as

\({I_{cm}} = m{R^2}\)

By using parallel axis theorem

\(\begin{aligned}{}I = {I_{cm}} + m{R^2}\\ = m{R^2} + m{R^2}\\ = 2m{R^2}\end{aligned}\)

Substitute \(I\) value in equation (i)

\(\begin{aligned}{}T = 2\pi \sqrt {\frac{{2m{R^2}}}{{mgR}}} \\ = 2\pi \sqrt {\frac{{2R}}{g}} \\R = \frac{{g{T^2}}}{{8{\pi ^2}}}\end{aligned}\)

Substitute all the values in above equation

\(\begin{aligned}{}R = \frac{{\left( {9.8\,{\rm{m/}}{{\rm{s}}^2}} \right) \times {{\left( {2.0\,{\rm{s}}} \right)}^2}}}{{8{\pi ^2}}}\\ = 0.496\,{\rm{m}}\end{aligned}\)

Hence, the hoop’s radius is \(0.496\,{\rm{m}}\)