Two wires of same material are vibrating under the same tension. If the first overtone of first wire is…

Two wires of same material are vibrating under the same tension. If the first overtone of first wire is equal to the second overtone of second wire and radius of first wire is twice the radius of the second then the ratio of length of first wire to second wire is
  1. $1: 3$
  2. $1: 2$
  3. $2: 1$
  4. $3: 1$

Solution

$\mathrm{n}_{1}=2 \cdot \frac{1}{2 \ell_{1} \mathrm{r}_{1}} \sqrt{\frac{\mathrm{T}}{\pi \rho}} ; \mathrm{n}_{2}=3 \cdot \frac{1}{2 \ell_{2} \mathrm{r}_{2}} \sqrt{\frac{\mathrm{T}}{\pi \rho}}$ $\mathrm{n}_{1}=\mathrm{n}_{2} \quad \therefore \frac{2}{\ell_{1} \mathrm{r}_{1}}=\frac{3}{\ell_{2} \mathrm{r}_{2}} \quad \therefore \frac{\ell_{1}}{\ell_{2}} \quad=\frac{\mathrm{r}_{2}}{\mathrm{r}_{1}} \times \frac{2}{3}=\frac{1}{2} \times \frac{2}{3}=\frac{1}{3}$

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

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