The length of string of a musical instrument is 90 cm and has fundamental frequency of 120 Hz where should it be pressed to produce fundamental frequency of 180 Hz

Options

A. 75 cm
B. 60 cm
C. 45 cm
D. 80 cm

Answer

To solve the problem of where to press the string of a musical instrument to produce a fundamental frequency of 180 Hz, we can follow these steps:

### Step 1: Understand the relationship between frequency and length
The fundamental frequency (f) of a vibrating string is inversely proportional to its length (L). This relationship can be expressed as:
\[ f \propto \frac{1}{L} \]
This means that if the frequency increases, the length of the vibrating portion of the string must decrease.

### Step 2: Set up the equation for the frequencies
We know the initial fundamental frequency (f1) and length (L1):
- \( f_1 = 120 \, \text{Hz} \)
- \( L_1 = 90 \, \text{cm} \)

We want to find the new length (L2) when the frequency is increased to:
- \( f_2 = 180 \, \text{Hz} \)

Using the relationship between frequency and length, we have:
\[ \frac{f_1}{f_2} = \frac{L_2}{L_1} \]

### Step 3: Substitute the known values
Substituting the known values into the equation:
\[ \frac{120}{180} = \frac{L_2}{90} \]

### Step 4: Simplify the equation
Simplifying the left side:
\[ \frac{2}{3} = \frac{L_2}{90} \]

### Step 5: Solve for L2
Now, we can solve for \( L_2 \):
\[ L_2 = 90 \times \frac{2}{3} \]
\[ L_2 = 60 \, \text{cm} \]

### Step 6: Determine where to press the string
Since the total length of the string is 90 cm, to produce a fundamental frequency of 180 Hz, the string must be effectively shortened to 60 cm. Therefore, the distance from the fixed end to where the string should be pressed is:
\[ \text{Distance from fixed end} = 90 \, \text{cm} - 60 \, \text{cm} = 30 \, \text{cm} \]

### Final Answer
To produce a fundamental frequency of 180 Hz, the string should be pressed 30 cm from the fixed end.

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