A Tube with a Movable Piston

solved ClosedPipe

Q. A metre-long tube open at one end, with a movable piston at the other end, shows resonance with a fixed frequency source (a tuning fork of frequency 340 Hz $ν = 340$ Hz) when the tube length is $n = 1$ 25.5 cm $L = 25.5$ cm or 79.3 cm. The edge effects may be neglected. Estimate the speed of sound in air at the temperature of the experiment, in m/s?

Inputs

frequency

ν

340 Hz

length

L

25.5 cm

Outputs

speed

v

How it's solved

Solved with ClosedPipe

Given

$ν = 340$ Hz
$L = 25.5$ cm
$n = 1$ from “when the tube length is”

Find $v$ .

Working

$L = 25.5$ cm $= 0.255$ m
harmonics of a pipe closed at one end(tube) $ν = \frac{n \cdot v}{4 \cdot L}$
$340 = \frac{1 \cdot v}{4 \cdot 0.255}$
$v = 346.8 m/s$
$v \approx 347$ m/sto 3 significant figures

Conclusion

Closed by the piston, the tube first resonates when it holds a quarter of a wavelength, a node at the piston and an antinode at the open end.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :a_tube_with_a_movable_piston do4  kind :numerical5  describes "A metre-long tube open at one end, with a movable piston at the other end, shows resonance with a fixed " \6            "frequency source (a tuning fork of frequency %{frequency}) when the tube length is %{length} or 79.3 cm. " \7            "The edge effects may be neglected. %{speed}, in m/s?"8  given frequency: "340 Hz"9  given length: "25.5 cm"10  asks speed: "Estimate the speed of sound in air at the temperature of the experiment"11end

solution.rb

1require_relative "../questions/a_tube_with_a_movable_piston"2require_relative "../science/waves/closed_pipe"34solve :a_tube_with_a_movable_piston, using: ClosedPipe do5  objects :tube67  take frequency: tube.frequency8  take length: tube.length_of_the_pipe910  deduce_from "when the tube length is", tube.harmonic => 11112  speed = v13  answer speed:, in: "m/s", figures: 31415  conclusion "Closed by the piston, the tube first resonates when it holds a quarter of a wavelength, a node at " \16             "the piston and an antinode at the open end."17end

playground.rb

1require_relative "../solutions/a_tube_with_a_movable_piston"23play_with :a_tube_with_a_movable_piston do4  title "A Tube with a Movable Piston"5end

The laws it drew on

quantity	written	units	within	from
speed of sound	$v$	m/s	—	ClosedPipe
length of the pipe	$L$	m	—	ClosedPipe
harmonic	$n$	—	—	ClosedPipe
frequency	$ν$	Hz	0 … ∞	ClosedPipe
frequency of ground	$ν_{g}$	Hz	0 … ∞
harmonic of ground	$n_{g}$	—	—
length of the pipe of ground	$L_{g}$	m	—
frequency of tube	$ν$	Hz	0 … ∞
harmonic of tube	$n$	—	—
length of the pipe of tube	$L$	m	—

ν_{p} = \frac{n_{p} \cdot v}{4 \cdot L_{p}}

harmonics of a pipe closed at one end

ν_{g} = \frac{n_{g} \cdot v}{4 \cdot L_{g}}

harmonics of a pipe closed at one end(ground)

ν = \frac{n \cdot v}{4 \cdot L}

harmonics of a pipe closed at one end(tube)