Stationary Wave Practical

01Stationary Wave Practical

Experimental setup

This section covers the apparatus, variables and arrangement used to investigate stationary waves on a stretched string.

Aim and variables

The investigation measures how the frequency of the first harmonic changes when one of three properties of the stretched string is varied:

the vibrating length, $L$ ;
the tension, $T$ ;
the mass per unit length, $\mu$ .

The dependent variable is the frequency at which the first-harmonic stationary-wave pattern is observed.

Variable changed	Important control variables
Length	Keep the hanging mass and string unchanged, so $T$ and $\mu$ remain constant.
Tension	Keep the vibrating length and the string unchanged.
Mass per unit length	Keep the hanging mass and vibrating length unchanged.

Apparatus arrangement

Apparatus	Purpose
Signal generator	Controls the vibration generator and provides the frequency reading.
Vibration generator	Drives the string to produce the stationary wave.
Retort stand	Provides a stable support for the driving end of the apparatus.
G-clamp or heavy mass	Stabilises the retort stand.
String	Provides the medium in which the stationary wave is observed.
Wooden bridge	Sets the other end of the measured vibrating length.
Pulley	Allows hanging masses to provide tension in the string.
Mass hanger and masses	Allow the string tension to be varied.
Metre ruler	Measures the vibrating length.
Top-pan balance	Measures string mass when determining $\mu$ .

The string runs from the vibration generator, across the bridge and over the pulley to the mass hanger. The vibrating length $L$ is measured from the vibration generator to the bridge.

02Stationary Wave Practical

Investigating frequency

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Method

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Finding tension and mass per unit length

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$T=mg$

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$\mu=\frac{\text{mass of string}}{\text{length of string}}$

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Identifying the first harmonic

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03Stationary Wave Practical

Analysing results

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Linearising the length relationship

Vqy jiw mraom vseaiwxr:

$\lambda=2L$

Sbwiz mas onlx jutxmvju:

$v=f\lambda=2fL$

Cwjpzrwis:

$f=\frac{v}{2L}=\frac{v}{2}\left(\frac{1}{L}\right)$

Exge oxx eyk ecslhzfb-bzrd zrzp $y=mx$ :

Straight-line quantity	Experiment
$y$	$f$ hi Yn
$x$	$1/L$ ue p $^{-1}$
Ybbiaqgu	$v/2$

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$v=2\times\text{gradient}$

Checking the wave speed

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$v=\sqrt{\frac{T}{\mu}}$

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04Stationary Wave Practical

Uncertainty & evaluation

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Measurement uncertainty

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$\text{percentage uncertainty}= \frac{\text{absolute uncertainty}}{\text{measured value}}\times100\%$

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Uncertainty in calculated frequency

Qrz

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

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$2.5\%+\frac{2.5\%}{2}+\frac{3.0\%}{2}$

$=2.5\%+1.25\%+1.5\%=5.25\%$

Improving the experiment

Systematic effects

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Random variation

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Safety

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