$$ $$ $$ $$ $$ $$ $$ $$ $$ $\text{Extra close brace or missing open brace}$ $\text{Extra close brace or missing open brace}$ $$

Frequency and Period Formulas

If you have a block with mass $m$ on a spring with spring constant $k$, the period and frequency of its oscillation are $$T=2\pi \sqrt{\frac{m}{k}}f=\frac{1}{2\pi }\sqrt{\frac{k}{m}}$$

If you are given $k$ and $m$, it's easy to calculate these. If you are given the period or frequency, however, you can solve for $k$ or $m$ by first squaring both sides, getting $$T^2=4{\pi }^2\frac{m}{k}{f}^2=\frac{1}{4{\pi }^2}\frac{k}{m}$$ and then solving for the desired variable.

For a pendulum, the formulae are very similar, although we use the constant $g=9.8\mathrm{m}/{\mathrm{s}}^2$. $$T=2\pi \sqrt{\frac{L}{g}}f=\frac{1}{2\pi }\sqrt{\frac{g}{L}}$$ $$T^2=4{\pi }^2\frac{L}{g}{f}^2=\frac{1}{4{\pi }^2}\frac{g}{L}$$