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 step-by-step, we will follow the principles of wave frequency and string length.

### Step 1: Understand the relationship between frequency and length
The fundamental frequency of a vibrating string is inversely proportional to its length. The formula for the fundamental frequency \( f \) of a string fixed at both ends is given by:

\[ f = \frac{v}{2L} \]

where \( v \) is the velocity of the wave on the string and \( L \) is the length of the string.

### Step 2: Calculate the velocity of the wave on the string
Given that the fundamental frequency \( f_1 = 120 \, \text{Hz} \) and the length of the string \( L = 90 \, \text{cm} = 0.9 \, \text{m} \):

Using the formula:

\[ v = 2L f_1 \]

Substituting the values:

\[ v = 2 \times 0.9 \, \text{m} \times 120 \, \text{Hz} = 216 \, \text{m/s} \]

### Step 3: Determine the new length for the new frequency
We want to find the new length \( L' \) that will produce a fundamental frequency \( f_2 = 180 \, \text{Hz} \).

Using the same formula:

\[ f_2 = \frac{v}{2L'} \]

Rearranging for \( L' \):

\[ L' = \frac{v}{2f_2} \]

Substituting the values:

\[ L' = \frac{216 \, \text{m/s}}{2 \times 180 \, \text{Hz}} = \frac{216}{360} = 0.6 \, \text{m} = 60 \, \text{cm} \]

### Step 4: Conclusion
To produce a fundamental frequency of 180 Hz, the string should be pressed at a length of 60 cm from one end.

### Final Answer
The string should be pressed at **60 cm** from one end to produce a fundamental frequency of **180 Hz**.

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