A Block on a Spring

Q. A block whose mass is 1 kg is fastened to a spring. The spring has a spring constant of 50 N/m. The block is pulled to a distance x = 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 away from the mean position, in joules? expects 0.19 J

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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?

The playground

Inputs

mass

m

1 kg

constant

k

50 N/m

amplitude

A

10 cm

place

x_{1}

5 cm

Outputs

energy

K_{1}

The solution

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 Ruby

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

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

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

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)