A small mass 'm' is suspended at the end of a wire having (negligible mass) length 'L' and cross-sectional…

A small mass 'm' is suspended at the end of a wire having (negligible mass) length 'L' and cross-sectional area 'A'. The frequency of oscillation for the S.H.M. along the vertical line is $(\mathrm{Y}=$ Young's modulus of the wire)
  1. $\frac{1}{2 \pi}\left(\frac{\mathrm{YA}}{\mathrm{mL}}\right)^{\frac{1}{2}}$
  2. $\frac{2 \pi \mathrm{YA}}{\mathrm{mL}}$
  3. $\frac{\mathrm{YA}}{2 \pi \mathrm{mI}}$
  4. $2 \pi\left(\frac{\mathrm{YA}}{\mathrm{mL}}\right)^{\frac{1}{2}}$

Solution

If $x$ is the extension of the wire then $P=\left(\frac{Y A}{L}\right) x \quad F=k x \quad \text { where } k=\frac{Y A}{L}$ frequency of oscillation is given by $f=\frac{1}{2 \pi} \sqrt{\frac{k}{m}}=\frac{1}{2 \pi} \sqrt{\frac{Y A}{m L}}$

Asked in: MHT CET 2020 (14 Oct Shift 1)

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