A Locomotive Piston

solved SimpleHarmonicMotion

Q. The piston in the cylinder head of a locomotive has a stroke (twice the amplitude) of 1.0 m $s = 1.0$ m. If the piston moves with simple harmonic motion with an angular frequency of 200 rad/min $ω = 200$ rad/min, what is its maximum speed, in m/s?

Inputs

stroke

s

1.00 m

angular frequency

ω

200 rad/min

Outputs

speed

v_{\max}

How it's solved

Solved with SimpleHarmonicMotion

Given

$s = 1.0$ m
$ω = 200$ rad/min

Find $v_{\max}$ .

Working

$ω = 200$ rad/min $= 3.33333333$ rad/s
stroke is twice the amplitude(piston) $s = 2 \cdot A$
$1 = 2 \cdot A$
$A = 0.5$ m
fastest through the middle(piston) $v_{\max} = ω \cdot A$
$v_{\max} = 3.33333333 \cdot 0.5$
$v_{\max} = 1.7 m/s$
$v_{\max} \approx 1.7$ m/sto 2 significant figures

Conclusion

The piston moves fastest through the middle of its stroke, at the angular frequency times half the stroke.

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_locomotive_piston do4  kind :numerical5  describes "The piston in the cylinder head of a locomotive has a stroke (twice the amplitude) of %{stroke}. If the " \6            "piston moves with simple harmonic motion with an angular frequency of %{angular_frequency}, %{speed}, " \7            "in m/s?"8  given stroke: "1.0 m"9  given angular_frequency: "200 rad/min"10  asks speed: "what is its maximum speed"11end

solution.rb

1require_relative "../questions/a_locomotive_piston"2require_relative "../science/oscillations/simple_harmonic_motion"34solve :a_locomotive_piston, using: SimpleHarmonicMotion do5  objects :piston67  take stroke: piston.stroke8  take angular_frequency: piston.angular_frequency910  speed = piston.maximum_speed11  answer speed:, in: "m/s", figures: 21213  conclusion "The piston moves fastest through the middle of its stroke, at the angular frequency times half " \14             "the stroke."15end

playground.rb

1require_relative "../solutions/a_locomotive_piston"23play_with :a_locomotive_piston do4  title "A Locomotive Piston"5end

The laws it drew on

quantity	written	units	within	from
pi	$π$	—	0 … ∞	SimpleHarmonicMotion
position	$x$	m	—	Velocity
displacement	$Δ x$	m	—	Velocity
path length	$d$	m	—	Velocity
time taken	$Δ t$	s	0 … 1e+05	Velocity
average velocity	$v_{av}$	m/s	—	Velocity
average speed	$c_{av}$	m/s	—	Velocity
starting position	$x_{0}$	m	—	Velocity
final position	$x$	m	—	Velocity
velocity	$v$	m/s	—	Acceleration
change of velocity	$Δ v$	m/s	—	Acceleration
average acceleration	$a_{av}$	m/s²	—	Acceleration
starting velocity	$v_{0}$	m/s	—	Acceleration
final velocity	$v$	m/s	—	Acceleration
amplitude	$A$	m	0 … ∞	SimpleHarmonicMotion
stroke	$s$	m	—	SimpleHarmonicMotion
angular frequency	$ω$	rad/s	0 … ∞	SimpleHarmonicMotion
frequency	$ν$	Hz	0 … ∞	SimpleHarmonicMotion
phase constant	$φ$	—	—	SimpleHarmonicMotion
maximum speed	$v_{\max}$	m/s	—	SimpleHarmonicMotion
maximum acceleration	$a_{\max}$	m/s²	—	SimpleHarmonicMotion
acceleration	$a$	m/s²	—	SimpleHarmonicMotion
stroke of ground	$s_{g}$	m	—
amplitude of ground	$A_{g}$	m	0 … ∞
stroke of piston	$s$	m	—
amplitude of piston	$A$	m	0 … ∞
angular frequency of ground	$ω_{g}$	rad/s	0 … ∞
frequency of ground	$ν_{g}$	Hz	0 … ∞
angular frequency of piston	$ω$	rad/s	0 … ∞
frequency of piston	$ν$	Hz	0 … ∞
maximum speed of ground	$v_{max,g}$	m/s	—
maximum speed of piston	$v_{\max}$	m/s	—
maximum acceleration of ground	$a_{max,g}$	m/s²	—
maximum acceleration of piston	$a_{\max}$	m/s²	—
phase constant of piston	$φ$	—	—

x_{0} \equiv x_{i} (start of motion)

starting position

x \equiv x_{i} (end of motion)

final position

v_{0} \equiv v_{i} (start of motion)

starting velocity

v \equiv v_{i} (end of motion)

final velocity

d_{i} (whole) = d_{i} (part 1) + d_{i} (part 2)

path in parts

Δ t_{w} = Δ t_{p} + Δ t_{p}

time in parts

s_{b} = 2 \cdot A_{b}

stroke is twice the amplitude

ω_{b} = 2 \cdot π \cdot ν_{b}

angular frequency is a turn each oscillation

v_{max,b} = ω_{b} \cdot A_{b}

fastest through the middle

a_{max,b} = {ω_{b}}^{2} \cdot A_{b}

hardest pulled at the ends

Δ x = x - x_{0}

change of position

v_{av} = \frac{Δ x}{Δ t}

displacement a time

c_{av} = \frac{d}{Δ t}

path a time

Δ v = v - v_{0}

velocity gained

a_{av} = \frac{Δ v}{Δ t}

velocity gained a time

s_{g} = 2 \cdot A_{g}

stroke is twice the amplitude(ground)

s = 2 \cdot A

stroke is twice the amplitude(piston)

ω_{g} = 2 \cdot π \cdot ν_{g}

angular frequency is a turn each oscillation(ground)

ω = 2 \cdot π \cdot ν

angular frequency is a turn each oscillation(piston)

v_{max,g} = ω_{g} \cdot A_{g}

fastest through the middle(ground)

v_{\max} = ω \cdot A

fastest through the middle(piston)

a_{max,g} = {ω_{g}}^{2} \cdot A_{g}

hardest pulled at the ends(ground)

a_{\max} = ω^{2} \cdot A

hardest pulled at the ends(piston)