Match each physical modification of a vibrating string or air pipe system on the left with… — JAMB UTME

Match each physical modification of a vibrating string or air pipe system on the left with its corresponding effect on the system's frequency on the right.

Quadrupling the tension ( $T$ ) of a stretched string while keeping its length and linear mass density constantFrequency increases by a factor of $2$
Quadrupling the linear mass density ( $\mu$ ) of a stretched string while keeping its length and tension constantFrequency is reduced to half its original value (factor of $0.5$ )
Doubling the tension ( $T$ ) of a stretched string while keeping its length and linear mass density constantFrequency increases by a factor of $\sqrt{2}$
Transitioning a pipe closed at one end from its fundamental resonant mode to its first overtoneFrequency increases by a factor of $3$

Answer

Quadrupling tension corresponds to increasing frequency by a factor of 2; quadrupling linear mass density corresponds to reducing frequency to half; doubling tension corresponds to increasing frequency by a factor of

\sqrt{2}

; transitioning a closed pipe from fundamental mode to first overtone corresponds to increasing frequency by a factor of 3.

Each physical modification correctly maps to its quantitative outcome based on wave mechanics: string frequency scales with

\sqrt{T}

and

1/\sqrt{\mu}

, while closed pipe overtones follow odd harmonic multipliers (

1, 3, 5, \dots

Step-by-Step Solution

Examine the fundamental frequency formula for a stretched string under tension:

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

Frequency is directly proportional to

\sqrt{T}

and inversely proportional to

\sqrt{\mu}

This establishes how changes in tension and mass per unit length scale the fundamental frequency.

Calculate scaling factors for the string modifications.

Quadrupling

T

multiplies frequency by

\sqrt{4} = 2

. Quadrupling

\mu

multiplies frequency by

1/\sqrt{4} = 0.5

. Doubling

T

multiplies frequency by

\sqrt{2}

Applying square roots to the parameter change factors gives the resultant frequency change.

Analyze harmonic ratios for air columns in pipes closed at one end.

The fundamental mode frequency is

f_1 = \frac{v}{4L}

. The first overtone is the third harmonic (

f_3 = \frac{3v}{4L} = 3f_1

Closed air columns produce only odd harmonics (

n = 1, 3, 5, \dots

Key Concept

Parameter scaling of transverse waves on stretched strings and harmonic modes in closed air columns

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