A Block on a Spring

solved SpringOscillator

Q. A block whose mass is 1 kg $m = 1$ kg is fastened to a spring. The spring has a spring constant of 50 N/m $k = 50$ N/m. The block is pulled to a distance x = 10 cm $A = 10$ cm from its equilibrium position at x = 0 on a frictionless surface from rest at t = 0. What is the kinetic energy of the block when it is 5 cm $x_{1} = 5$ cm away from the mean position, in joules?

Inputs

mass

m

1 kg

constant

k

50 N/m

amplitude

A

10 cm

place

x_{1}

5 cm

Outputs

energy

K_{1}

How it's solved

Solved with SpringOscillator

Given

$m = 1$ kg
$k = 50$ N/m
$A = 10$ cm
$x_{1} = 5$ cm

Find $K_{1}$ .

Working

$A = 10$ cm $= 0.1$ m
$x_{1} = 5$ cm $= 0.05$ m
energy of an oscillator(block) $E = \frac{k \cdot A^{2}}{2}$
$E = \frac{50 \cdot {0.1}^{2}}{2}$
$E = 0.25$ J
energies at an instant(block, oscillating, 2) $U_{1} = \frac{k \cdot {x_{1}}^{2}}{2}$
$U_{1} = \frac{50 \cdot {0.05}^{2}}{2}$
$U_{1} = 0.0625$ J
energies at an instant(block, oscillating, 3) $E = K_{1} + U_{1}$
$0.25 = K_{1} + 0.0625$
$K_{1} = 0.1875$ J
$K_{1} \approx 0.19$ Jto 2 significant figures

Conclusion

The oscillator's energy stays half the spring constant times the amplitude squared; halfway out, the spring holds a quarter of it and the rest is the block's motion.

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_block_on_a_spring do4  kind :numerical5  describes "A block whose mass is %{mass} is fastened to a spring. The spring has a spring constant of %{constant}. " \6            "The block is pulled to a distance x = %{amplitude} from its equilibrium position at x = 0 on a " \7            "frictionless surface from rest at t = 0. %{energy} when it is %{place} away from the mean position, in " \8            "joules?"9  given mass: "1 kg"10  given constant: "50 N/m"11  given amplitude: "10 cm"12  given place: "5 cm"13  asks energy: "What is the kinetic energy of the block"14end

solution.rb

1require_relative "../questions/a_block_on_a_spring"2require_relative "../science/oscillations/spring_oscillator"34solve :a_block_on_a_spring, using: SpringOscillator do5  objects :block6  axes :x7  block.goes(:oscillating)89  take mass: block.mass10  take constant: block.spring_constant11  take amplitude: block.amplitude12  take place: block.final_position1314  energy = block.kinetic_energy(at: oscillating.end)15  answer energy:, in: "J", figures: 21617  conclusion "The oscillator's energy stays half the spring constant times the amplitude squared; halfway " \18             "out, the spring holds a quarter of it and the rest is the block's motion."19end

playground.rb

1require_relative "../solutions/a_block_on_a_spring"23play_with :a_block_on_a_spring do4  title "A Block on a Spring"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
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
displacement of block during oscillating along x	$Δ x$	m	—
position of block at end of oscillating along x	$x_{1}$	m	—
position of block at start of oscillating along x	$x_{0}$	m	—
average velocity of block during oscillating along x	$v_{av}$	m/s	—
time taken of oscillating	$Δ t$	s	0 … 1e+05
average speed of block during oscillating	$c_{av}$	m/s	—
path length of block during oscillating	$d$	m	—
change of velocity of block during oscillating along x	$Δ v$	m/s	—
velocity of block at end of oscillating from ground along x	$v_{1}$	m/s	—
velocity of block at start of oscillating from ground along x	$v_{0}$	m/s	—
average acceleration of block during oscillating along x	$a_{av}$	m/s²	—
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	—
position of block at end of oscillating	$\| x_{1} \|$	m	—
phase constant of block	$φ$	—	—
velocity of block at end of oscillating from ground	$\| v_{1} \|$	m/s	—
acceleration of block at end of oscillating	$\| a_{1} \|$	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	—
kinetic energy of block at end of oscillating	$K_{1}$	J	—
potential energy of block at end of oscillating	$U_{1}$	J	—

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_{1} - x_{0}

change of position(oscillating, x)

v_{av} = \frac{Δ x}{Δ t}

displacement a time(oscillating, x)

c_{av} = \frac{d}{Δ t}

path a time(oscillating, x)

Δ v = v_{1} - v_{0}

velocity gained(oscillating, x)

a_{av} = \frac{Δ v}{Δ t}

velocity gained a time(oscillating, x)

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)

| x_{1} | = A \cdot \cos (ω \cdot Δ t + φ)

displacement in shm(block, oscillating)

| v_{1} | = - A \cdot ω \cdot \sin (ω \cdot Δ t + φ)

velocity in shm(block, oscillating)

| a_{1} | = - ω^{2} \cdot | x_{1} |

acceleration in shm(block, oscillating)

A^{2} = {x_{1}}^{2} + {(\frac{v_{1}}{ω})}^{2}

amplitude from where and how fast(block, oscillating)

{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)

K_{1} = \frac{m \cdot {v_{1}}^{2}}{2}

energies at an instant(block, oscillating, 1)

U_{1} = \frac{k \cdot {x_{1}}^{2}}{2}

energies at an instant(block, oscillating, 2)

E = K_{1} + U_{1}

energies at an instant(block, oscillating, 3)