A Steel Wire Under Tension

solved WavesOnAString

Q. A steel wire 0.72 m $L = 0.72$ m long has a mass of 5.0 × 10⁻³ kg $m = 5.0 \times 10^{−3}$ kg. If the wire is under a tension of 60 N $T = 60$ N, what is the speed of transverse waves on the wire, in m/s?

Inputs

length

L

0.72 m

mass

m

0.005 kg

tension

T

60 N

Outputs

speed

v

How it's solved

Solved with WavesOnAString

Given

$L = 0.72$ m
$m = 5.0 \times 10^{−3}$ kg
$T = 60$ N

Find $v$ .

Working

mass per unit length(wire) $μ = \frac{m}{L}$
$μ = \frac{0.005}{0.72}$
$μ = 0.00694444444$ kg/m
speed of a transverse wave(wire) $v = \sqrt{\frac{T}{μ}}$
$v = \sqrt{\frac{60}{0.00694444444}}$
$v = 93.0 m/s$
$v \approx 93$ m/sto 2 significant figures

Conclusion

Tension pulls a disturbance along and the wire's mass per metre holds it back; the speed is the square root of their ratio.

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_steel_wire_under_tension do4  kind :numerical5  describes "A steel wire %{length} long has a mass of %{mass}. If the wire is under a tension of %{tension}, " \6            "%{speed}, in m/s?"7  given length: "0.72 m"8  given mass: "5.0e-3 kg"9  given tension: "60 N"10  asks speed: "what is the speed of transverse waves on the wire"11end

solution.rb

1require_relative "../questions/a_steel_wire_under_tension"2require_relative "../science/waves/waves_on_a_string"34solve :a_steel_wire_under_tension, using: WavesOnAString do5  objects :wire67  take length: wire.length_of_the_string8  take mass: wire.mass9  take tension: wire.tension1011  speed = wire.wave_speed12  answer speed:, in: "m/s", figures: 21314  conclusion "Tension pulls a disturbance along and the wire's mass per metre holds it back; the speed is " \15             "the square root of their ratio."16end

playground.rb

1require_relative "../solutions/a_steel_wire_under_tension"23play_with :a_steel_wire_under_tension do4  title "A Steel Wire Under Tension"5end

The laws it drew on

quantity	written	units	within	from
mass	$m$	kg	0 … ∞	WavesOnAString
length of the string	$L$	m	—	WavesOnAString
linear mass density	$μ$	kg/m	—	WavesOnAString
tension	$T$	N	—	WavesOnAString
wave speed	$v$	m/s	—	WavesOnAString
travel time	$t$	s	—	WavesOnAString
harmonic	$n$	—	—	WavesOnAString
frequency	$ν$	Hz	0 … ∞	WavesOnAString
wave number	$k$	m⁻¹	—	WavesOnAString
angular frequency	$ω$	rad/s	—	WavesOnAString
linear mass density of ground	$μ_{g}$	kg/m	—
mass of ground	$m_{g}$	kg	0 … ∞
length of the string of ground	$L_{g}$	m	—
linear mass density of wire	$μ$	kg/m	—
mass of wire	$m$	kg	0 … ∞
length of the string of wire	$L$	m	—
wave speed of ground	$v_{g}$	m/s	—
tension of ground	$T_{g}$	N	—
wave speed of wire	$v$	m/s	—
tension of wire	$T$	N	—
angular frequency of ground	$ω_{g}$	rad/s	—
wave number of ground	$k_{g}$	m⁻¹	—
angular frequency of wire	$ω$	rad/s	—
wave number of wire	$k$	m⁻¹	—
travel time of ground	$t_{g}$	s	—
travel time of wire	$t$	s	—
frequency of ground	$ν_{g}$	Hz	0 … ∞
harmonic of ground	$n_{g}$	—	—
frequency of wire	$ν$	Hz	0 … ∞
harmonic of wire	$n$	—	—

μ_{s} = \frac{m_{s}}{L_{s}}

mass per unit length

v_{s} = \sqrt{\frac{T_{s}}{μ_{s}}}

speed of a transverse wave

v_{s} = \frac{ω_{s}}{k_{s}}

speed from the wave s numbers

t_{s} = \frac{L_{s}}{v_{s}}

time to run the length

ν_{s} = \frac{n_{s} \cdot v_{s}}{2 \cdot L_{s}}

harmonics of a string fixed at both ends

μ_{g} = \frac{m_{g}}{L_{g}}

mass per unit length(ground)

μ = \frac{m}{L}

mass per unit length(wire)

v_{g} = \sqrt{\frac{T_{g}}{μ_{g}}}

speed of a transverse wave(ground)

v = \sqrt{\frac{T}{μ}}

speed of a transverse wave(wire)

v_{g} = \frac{ω_{g}}{k_{g}}

speed from the wave s numbers(ground)

v = \frac{ω}{k}

speed from the wave s numbers(wire)

t_{g} = \frac{L_{g}}{v_{g}}

time to run the length(ground)

t = \frac{L}{v}

time to run the length(wire)

ν_{g} = \frac{n_{g} \cdot v_{g}}{2 \cdot L_{g}}

harmonics of a string fixed at both ends(ground)

ν = \frac{n \cdot v}{2 \cdot L}

harmonics of a string fixed at both ends(wire)