A steel wire of mass 4.0 g and length 80 cm is fixed at the two ends. The tension in the wire is 50 N. Find the frequency and wavelength of the fourth harmonic of the fundamental.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Convert the mass and length into appropriate units
- Given mass of the wire = 4.0 g = 4.0 / 1000 kg = 0.004 kg
- Given length of the wire = 80 cm = 80 / 100 m = 0.8 m

### Step 2: Calculate the mass per unit length (μ)
- The mass per unit length (μ) is calculated using the formula:
\[ \mu = \frac{\text{mass}}{\text{length}} = \frac{0.004 \text{ kg}}{0.8 \text{ m}} = 0.005 \text{ kg/m} \]

### Step 3: Calculate the wave speed (v) in the wire
- The wave speed (v) in the wire can be calculated using the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
where T is the tension in the wire.
- Given tension (T) = 50 N, we can substitute the values:
\[ v = \sqrt{\frac{50 \text{ N}}{0.005 \text{ kg/m}}} = \sqrt{10000} = 100 \text{ m/s} \]

### Step 4: Determine the wavelength (λ) of the fourth harmonic
- For a wire fixed at both ends, the wavelength of the nth harmonic is given by:
\[ \lambda_n = \frac{2L}{n} \]
where L is the length of the wire and n is the harmonic number.
- For the fourth harmonic (n = 4):
\[ \lambda_4 = \frac{2 \times 0.8 \text{ m}}{4} = \frac{1.6 \text{ m}}{4} = 0.4 \text{ m} = 40 \text{ cm} \]

### Step 5: Calculate the frequency (f) of the fourth harmonic
- The frequency can be calculated using the formula:
\[ f = \frac{v}{\lambda} \]
- Substituting the values we found:
\[ f = \frac{100 \text{ m/s}}{0.4 \text{ m}} = 250 \text{ Hz} \]

### Final Answers:
- **Wavelength (λ)** = 40 cm
- **Frequency (f)** = 250 Hz

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