A wire of length 2.00 m is stretched to a tension of 160 N. If the fundamental frequency of vibration is 100 Hz, find its linear mass density.

Explanation

To solve the problem of finding the linear mass density of a wire given its length, tension, and fundamental frequency, we can follow these steps:

### Step 1: Identify the given values
- Length of the wire (L) = 2.00 m
- Tension in the wire (T) = 160 N
- Fundamental frequency (f) = 100 Hz

### Step 2: Use the formula for the fundamental frequency of a vibrating string
The fundamental frequency (f) of a string fixed at both ends is given by the formula:
\[ f = \frac{V}{2L} \]
where \( V \) is the wave velocity on the string.

### Step 3: Rearrange the formula to find the wave velocity
From the formula, we can express the wave velocity \( V \) as:
\[ V = 2Lf \]

### Step 4: Substitute the known values to calculate the wave velocity
Substituting the values of \( L \) and \( f \):
\[ V = 2 \times 2.00 \, \text{m} \times 100 \, \text{Hz} = 400 \, \text{m/s} \]

### Step 5: Relate the wave velocity to tension and linear mass density
The wave velocity \( V \) on a string is also related to the tension \( T \) and the linear mass density \( \mu \) by the formula:
\[ V = \sqrt{\frac{T}{\mu}} \]

### Step 6: Rearrange the formula to solve for linear mass density \( \mu \)
Squaring both sides gives:
\[ V^2 = \frac{T}{\mu} \]
Rearranging for \( \mu \):
\[ \mu = \frac{T}{V^2} \]

### Step 7: Substitute the known values to find \( \mu \)
Substituting \( T = 160 \, \text{N} \) and \( V = 400 \, \text{m/s} \):
\[ \mu = \frac{160 \, \text{N}}{(400 \, \text{m/s})^2} = \frac{160}{160000} = 0.001 \, \text{kg/m} \]

### Step 8: Convert the linear mass density to grams per meter
Since \( 1 \, \text{kg} = 1000 \, \text{g} \):
\[ \mu = 0.001 \, \text{kg/m} = 1 \, \text{g/m} \]

### Final Answer:
The linear mass density \( \mu \) of the wire is \( 1 \, \text{g/m} \).

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