MHT CET · Physics · Waves and Sound
A string is under tension of 180 N and mass per unit length \(2 \times 10^{-3} \mathrm{Kg} / \mathrm{m}\). It produces two consecutive resonant frequencies with a tuning fork, which are 375 Hz and 450 Hz . The mass of the string is
- A 1 gram
- B 2 gram
- C 3 gram
- D 4 gram
Answer & Solution
Correct Answer
(D) 4 gram
Step-by-step Solution
Detailed explanation
\(\begin{aligned} \text { Fundamental frequency } & =\mathrm{f}_2-\mathrm{f}_1 \\ & =450-375 \\ & =75 \mathrm{~Hz}\end{aligned}\)
\(\begin{aligned}
\mathrm{f} & =\frac{1}{2 \mathrm{~L}} \sqrt{\frac{\mathrm{~T}}{\mu}} \\
75 & =\frac{1}{2 \mathrm{~L}} \sqrt{\frac{180}{2 \times 10^{-3}}} \\
\mathrm{~L} & =2 \mathrm{~m} \\
\mu & =\frac{\mathrm{m}}{\mathrm{~L}} \\
\mathrm{~m} & =2 \times 10^{-3} \times 2 \\
& =4 \text { gram }
\end{aligned}\)
...where \(\mu\) is mass per unit length
\(\begin{aligned}
\mathrm{f} & =\frac{1}{2 \mathrm{~L}} \sqrt{\frac{\mathrm{~T}}{\mu}} \\
75 & =\frac{1}{2 \mathrm{~L}} \sqrt{\frac{180}{2 \times 10^{-3}}} \\
\mathrm{~L} & =2 \mathrm{~m} \\
\mu & =\frac{\mathrm{m}}{\mathrm{~L}} \\
\mathrm{~m} & =2 \times 10^{-3} \times 2 \\
& =4 \text { gram }
\end{aligned}\)
...where \(\mu\) is mass per unit length
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