Question

A thin rectangular magnet suspended freely has a period of oscillation equal to T. Now it is broken into two equal halves (each having half of the original length) and one-piece is made to oscillate freely in the same field. if its period of oscillation is T', the ratio $\frac{\mathrm{T}\text{'}}{\mathrm{T}}$is_______

Answer 1

Step 1: Given data

Old time period$=\mathrm{T}$

New time period$={\mathrm{T}}^{\text{'}}$

Old length$=\mathrm{l}$

New length$=\frac{\mathrm{l}}{2}$

Step 2: To Find

Ratio $\frac{{\mathrm{T}}^{\text{'}}}{\mathrm{T}}$.

Step 3: Calculate the ratio

$$\begin{gathered} \frac{{\mathrm{T}}^{\text{'}}}{\mathrm{T}}=\sqrt{\frac{{\mathrm{I}}_{\mathrm{new}}}{{\mathrm{M}}_{\mathrm{new}}}\times \frac{\mathrm{M}}{\mathrm{I}}} \\ =\sqrt{\frac{{\displaystyle \frac{{\displaystyle \frac{{\mathrm{M}}^{\text{'}}}{2}\times {\left(\frac{\mathrm{L}}{2}\right)}^2}}{12}\times \mathrm{M}}}{{\displaystyle \frac{{\mathrm{M}}^{\text{'}}}{2}\times \frac{{\mathrm{ML}}^2}{12}}}} \\ =\sqrt{\frac{2}{8}} \\ =\frac{1}{2} \end{gathered}$$