A uniform wire of length 0.60\text{ m} fixed at both ends vibrates in its fundamental mode… — JAMB UTME

A uniform wire of length $0.60\text{ m}$ fixed at both ends vibrates in its fundamental mode with a frequency of $150\text{ Hz}$ . If the length of the wire is reduced to $0.40\text{ m}$ while the tension in the wire is increased by a factor of 4, what is the new fundamental frequency of vibration?

A
$200\text{ Hz}$
B
$300\text{ Hz}$
$450\text{ Hz}$ Answer
D
$900\text{ Hz}$

Answer

The new fundamental frequency of the vibrating wire is

450\text{ Hz}

The fundamental frequency of a stretched string is inversely proportional to its length and directly proportional to the square root of its tension (

f \propto \frac{\sqrt{T}}{L}

). Reducing the length from

0.60\text{ m}

0.40\text{ m}

increases frequency by a factor of

\frac{0.60}{0.40} = 1.5

. Quadrupling the tension increases frequency by a factor of

\sqrt{4} = 2

. Combining both effects gives an overall frequency multiplier of

1.5 \times 2 = 3.0

, leading to

3.0 \times 150\text{ Hz} = 450\text{ Hz}

Step-by-Step Solution

Write the general formula for the fundamental frequency of a stretched string.

f = \frac{1}{2L} \sqrt{\frac{T}{\mu}}

, where

L

is length,

T

is tension, and

\mu

is linear density.

Establishes the functional dependence of frequency on length and tension.

Set up the ratio between the new frequency

f_2

and initial frequency

f_1

\frac{f_2}{f_1} = \frac{L_1}{L_2} \sqrt{\frac{T_2}{T_1}}

Since linear mass density

\mu

remains constant, comparing ratios isolates the changing variables.

Substitute the given parameters into the ratio equation.

\frac{f_2}{150} = \frac{0.60}{0.40} \times \sqrt{4} = 1.5 \times 2 = 3.0

Calculates the scaling factor for the new fundamental frequency.

Solve for the new fundamental frequency

f_2

f_2 = 3.0 \times 150 = 450\text{ Hz}

Yields the final numerical value of the modified fundamental frequency.

Key Concept

Fundamental frequency of vibrating stretched strings under varying length and tension

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