Question

A piano wire weighing 6⋅00 g and having a length of 90⋅0 cm emits a fundamental frequency corresponding to the "Middle C" \[\left( \nu = 261 \cdot 63  Hz \right)\]. Find the tension in the wire.

Answer 1

Given: Length of the piano wire (L) = 90.0 cm = 0.90 m
Mass of the wire = 6.00 g = 0.006 kg
Fundamental frequency (f_o) = 261.63 Hz

Linear  mass  density,   m = `6/90` gm/cm

\[= \left\{ \frac{6 \times {10}^{- 3}}{90 \times {10}^{- 2}} \right\}  kg/m\]

\[ = \left( \frac{6}{900} \right)  kg/m\]

= 6.67 × 10⁻³

\[\text{ Fundamental  frequency, }   f_o  = \frac{1}{2L}\sqrt{\left( \frac{T}{m} \right)}\]

\[\Rightarrow   261 . 63 = \frac{1}{\left( 2 \times 0 . 09 \right)}  \sqrt{\left\{ \frac{T \times 900}{6} \right\}}\]

\[\Rightarrow 0.18 \times 261.63 = \sqrt{150  T}\]

\[\Rightarrow 150  T =  \left( 261.63 \times 0.18 \right)^2 \]

\[\Rightarrow T = \frac{\left( 261.63 \times 0.18 \right)^2}{150}\]

\[ = 1478.52  N \approx 1480  N\]

Hence, the tension in the piano wire is 1480 N.