A Wire's Tension in Its Fundamental Mode

Q. A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of 45 Hz. The mass of the wire is 3.5 × 10⁻² kg and its linear mass density is 4.0 × 10⁻² kg/m. What is the tension in the string, in newtons? expects 248 N

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Q. A wire stretched between two rigid supports vibrates in its fundamental mode $n = 1$ with a frequency of 45 Hz $ν = 45$ Hz. The mass of the wire is 3.5 × 10⁻² kg $m = 3.5 \times 10^{−2}$ kg and its linear mass density is 4.0 × 10⁻² kg/m $μ = 4.0 \times 10^{−2}$ kg/m. What is the tension in the string, in newtons?

The playground

Inputs

frequency

ν

45 Hz

mass

m

0.035 kg

density

μ

0.04 kg/m

Outputs

tension

T

The solution

Solved with WavesOnAString

Given

$ν = 45$ Hz
$m = 3.5 \times 10^{−2}$ kg
$μ = 4.0 \times 10^{−2}$ kg/m
$n = 1$ from “fundamental mode”

Find $T$ .

Working

mass per unit length(wire) $μ = \frac{m}{L}$
$0.04 = \frac{0.035}{L}$
$L = 0.875$ m
harmonics of a string fixed at both ends(wire) $ν = \frac{n \cdot v}{2 \cdot L}$
$45 = \frac{1 \cdot v}{2 \cdot 0.875}$
$v = 78.75$ m/s
speed of a transverse wave(wire) $v = \sqrt{\frac{T}{μ}}$
$78.75 = \sqrt{\frac{T}{0.04}}$
$T = 248.1$ N
$T \approx 248$ Nto 3 significant figures

Conclusion

In the fundamental mode the wire's length is half a wavelength, which fixes how fast the wave runs; the tension is the mass per metre times that speed squared.

The Ruby

1require_relative "../physics"23question :a_wire_s_tension_in_its_fundamental_mode do4  kind :numerical5  describes "A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of " \6            "%{frequency}. The mass of the wire is %{mass} and its linear mass density is %{density}. %{tension}, " \7            "in newtons?"8  given frequency: "45 Hz"9  given mass: "3.5e-2 kg"10  given density: "4.0e-2 kg/m"11  asks tension: "What is the tension in the string"12end

1require_relative "../questions/a_wire_s_tension_in_its_fundamental_mode"2require_relative "../science/waves/waves_on_a_string"34solve :a_wire_s_tension_in_its_fundamental_mode, using: WavesOnAString do5  objects :wire67  take frequency: wire.frequency8  take mass: wire.mass9  take density: wire.linear_mass_density1011  deduce_from "fundamental mode", wire.harmonic => 11213  tension = wire.tension14  answer tension:, in: "N", figures: 31516  conclusion "In the fundamental mode the wire's length is half a wavelength, which fixes how fast the wave " \17             "runs; the tension is the mass per metre times that speed squared."18end

1require_relative "../solutions/a_wire_s_tension_in_its_fundamental_mode"23play_with :a_wire_s_tension_in_its_fundamental_mode do4  title "A Wire's Tension in Its Fundamental Mode"5end

The laws

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)