A piano wire 50 cm long has a total mass of 10 ... - JAMB Physics 2023 Question

Given:
- Length of the wire (\(L\)) = 50 cm = 0.5 m
- Total mass of the wire (\(m\)) = 10 g = 0.01 kg
- Tension in the wire (\(T\)) = 800 N

First, calculate the linear mass density (\(\mu\)):

\[ \mu = \frac{m}{L} = \frac{0.01 \, \text{kg}}{0.5 \, \text{m}} = 0.02 \, \text{kg/m} \]

Now, calculate the fundamental frequency (\(f_0\)) using the formula:

\[ f_0 = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]

\[ f_0 = \frac{1}{2 \times 0.5} \sqrt{\frac{800}{0.02}} \]

\[ f_0 = \frac{1}{1} \sqrt{40000} \]

\[ f_0 = 200 \, \text{Hz} \]

Now, for the third overtone:
\[ f_3 = 4 \times f_0 \]

\[ f_3 = 4 \times 200 = 800 \, \text{Hz} \]

Therefore, the correct option is 800 Hz