A Spring's Mass at Its Fastest

Q. A spring having with a spring constant 1200 N/m is mounted on a horizontal table. A mass of 3 kg is attached to the free end of the spring. The mass is then pulled sideways to a distance of 2.0 cm and released. Determine the maximum speed of the mass, in m/s? expects 0.40 m/s

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Q. A spring having with a spring constant 1200 N/m $k = 1200$ N/m is mounted on a horizontal table. A mass of 3 kg $m = 3$ kg is attached to the free end of the spring. The mass is then pulled sideways to a distance of 2.0 cm $A = 2.0$ cm and released. Determine the maximum speed of the mass, in m/s?

The playground

Inputs

constant

k

1200 N/m

mass

m

3 kg

distance

A

2 cm

Outputs

speed

v_{\max}

The solution

Solved with SpringOscillator

Given

$k = 1200$ N/m
$m = 3$ kg
$A = 2.0$ cm

Find $v_{\max}$ .

Working

$A = 2.0$ cm $= 0.02$ m
angular frequency of a mass on a spring(block) $ω^{2} = \frac{k}{m}$
$ω^{2} = \frac{1200}{3}$
$ω = 20$ rad/s
fastest through the middle(block) $v_{\max} = ω \cdot A$
$v_{\max} = 20 \cdot 0.02$
$v_{\max} = 0.4 m/s$
$v_{\max} \approx 0.40$ m/sto 2 significant figures

Conclusion

All the spring's energy is the mass's motion as it passes the middle, where its speed is the angular frequency times the amplitude.

The Ruby

1require_relative "../physics"23question :a_spring_s_mass_at_its_fastest do4  kind :numerical5  describes "A spring having with a spring constant %{constant} is mounted on a horizontal table. A mass of %{mass} " \6            "is attached to the free end of the spring. The mass is then pulled sideways to a distance of " \7            "%{distance} and released. %{speed}, in m/s?"8  given constant: "1200 N/m"9  given mass: "3 kg"10  given distance: "2.0 cm"11  asks speed: "Determine the maximum speed of the mass"12end

1require_relative "../questions/a_spring_s_mass_at_its_fastest"2require_relative "../science/oscillations/spring_oscillator"34solve :a_spring_s_mass_at_its_fastest, using: SpringOscillator do5  objects :block67  take constant: block.spring_constant8  take mass: block.mass9  take distance: block.amplitude1011  speed = block.maximum_speed12  answer speed:, in: "m/s", figures: 21314  conclusion "All the spring's energy is the mass's motion as it passes the middle, " \15             "where its speed is the angular frequency times the amplitude."16end

1require_relative "../solutions/a_spring_s_mass_at_its_fastest"23play_with :a_spring_s_mass_at_its_fastest do4  title "A Spring's Mass at Its Fastest"5end

The laws

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
mass	$m$	kg	0 … ∞	SpringOscillator
spring constant	$k$	N/m	—	SpringOscillator
kinetic energy	$K$	J	—	SpringOscillator
potential energy	$U$	J	—	SpringOscillator
total energy	$E$	J	—	SpringOscillator
stroke of ground	$s_{g}$	m	—
amplitude of ground	$A_{g}$	m	0 … ∞
stroke of block	$s$	m	—
amplitude of block	$A$	m	0 … ∞
angular frequency of ground	$ω_{g}$	rad/s	0 … ∞
frequency of ground	$ν_{g}$	Hz	0 … ∞
angular frequency of block	$ω$	rad/s	0 … ∞
frequency of block	$ν$	Hz	0 … ∞
maximum speed of ground	$v_{max,g}$	m/s	—
maximum speed of block	$v_{\max}$	m/s	—
maximum acceleration of ground	$a_{max,g}$	m/s²	—
maximum acceleration of block	$a_{\max}$	m/s²	—
spring constant of ground	$k_{g}$	N/m	—
mass of ground	$m_{g}$	kg	0 … ∞
spring constant of block	$k$	N/m	—
mass of block	$m$	kg	0 … ∞
total energy of ground	$E_{g}$	J	—
total energy of block	$E$	J	—
phase constant of block	$φ$	—	—

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

{ω_{b}}^{2} = \frac{k_{b}}{m_{b}}

angular frequency of a mass on a spring

E_{b} = \frac{k_{b} \cdot {A_{b}}^{2}}{2}

energy of an oscillator

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(block)

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(block)

v_{max,g} = ω_{g} \cdot A_{g}

fastest through the middle(ground)

v_{\max} = ω \cdot A

fastest through the middle(block)

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(block)

{ω_{g}}^{2} = \frac{k_{g}}{m_{g}}

angular frequency of a mass on a spring(ground)

ω^{2} = \frac{k}{m}

angular frequency of a mass on a spring(block)

E_{g} = \frac{k_{g} \cdot {A_{g}}^{2}}{2}

energy of an oscillator(ground)

E = \frac{k \cdot A^{2}}{2}

energy of an oscillator(block)