Two identical strings of length ' $\ell$ ' and '2 $\ell$ ' vibrate with fundamental frequencies 'N' hertz…

Two identical strings of length ' $\ell$ ' and '2 $\ell$ ' vibrate with fundamental frequencies 'N' hertz and '1.5 N' hertz, respectively. The ratio of tensions for smaller length to larger length is
  1. 9:1
  2. 3:1
  3. 1:9
  4. 1:3

Solution

$\mathrm{N}=\frac{1}{2 \ell} \sqrt{\frac{\mathrm{T}_{1}}{\mathrm{~m}}}$ $1.5 \mathrm{~N}=\frac{1}{4 \ell} \sqrt{\frac{\mathrm{T}_{2}}{\mathrm{~m}}}$ $\therefore \sqrt{\frac{\mathrm{T}_{1}}{\mathrm{~T}_{2}}}=\frac{\mathrm{N} \times 2 \ell}{1.5 \mathrm{~N} \times 4 \ell}=\frac{1}{3}$ $\frac{\mathrm{T}_{1}}{\mathrm{~T}_{2}}=\frac{1}{9}$

Asked in: MHT CET 2020 (16 Oct Shift 1)

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