A Body in SHM — phryzby

solved SimpleHarmonicMotion

Q. A body oscillates with SHM according to the equation x = (5 m) cos $A = 5$ m[(2π rad/s) t $ω = 1$ rev/s + π/4 $φ = 45 °$ ]. At t = 1.5 s $Δ t = 1.5$ s, what is the speed of the body, in m/s?

Inputs

time

Δ t

1.50 s

Outputs

speed

| v_{1} |

How it's solved

Solved with SimpleHarmonicMotion

Given

$Δ t = 1.5$ s
$A = 5$ m from “(5 m) cos”
$ω = 1$ rev/s from “(2π rad/s) t”
$φ = 45 °$ from “+ π/4”

Find $| v_{1} |$ .

Working

$ω = 1$ rev/s $= 6.28318531$ rad/s
velocity in shm(body, oscillating) $| v_{1} | = - A \cdot ω \cdot \sin (ω \cdot Δ t + φ)$
$| v_{1} | = - 5 \cdot 6.28318531 \cdot \sin (6.28318531 \cdot 1.5 + 0.785398163)$
$| v_{1} | = 22.2 m/s$
$| v_{1} | \approx 22.2$ m/sto 3 significant figures

Conclusion

The speed is the amplitude times the angular frequency times the sine of the phase, largest as the body passes the middle.

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_body_in_shm do4  kind :numerical5  describes "A body oscillates with SHM according to the equation x = (5 m) cos[(2π rad/s) t + π/4]. At t = %{time}, " \6            "%{speed}, in m/s?"7  given time: "1.5 s"8  asks speed: "what is the speed of the body"9end

solution.rb

1require_relative "../questions/a_body_in_shm"2require_relative "../science/oscillations/simple_harmonic_motion"34solve :a_body_in_shm, using: SimpleHarmonicMotion do5  objects :body6  axes :x7  body.goes(:oscillating)89  take time: oscillating.time_taken1011  deduce_from "(5 m) cos", body.amplitude => "5 m"12  deduce_from "(2π rad/s) t", body.angular_frequency => "1 rev/s"13  deduce_from "+ π/4", body.phase_constant => "45 deg"1415  speed = body.final_velocity.magnitude16  answer speed:, in: "m/s", figures: 31718  conclusion "The speed is the amplitude times the angular frequency times the sine of the phase, " \19             "largest as the body passes the middle."20end

playground.rb

1require_relative "../solutions/a_body_in_shm"23play_with :a_body_in_shm do4  title "A Body in SHM"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
displacement of body during oscillating along x	$Δ x$	m	—
position of body at end of oscillating along x	$x_{1}$	m	—
position of body at start of oscillating along x	$x_{0}$	m	—
average velocity of body during oscillating along x	$v_{av}$	m/s	—
time taken of oscillating	$Δ t$	s	0 … 1e+05
average speed of body during oscillating	$c_{av}$	m/s	—
path length of body during oscillating	$d$	m	—
change of velocity of body during oscillating along x	$Δ v$	m/s	—
velocity of body at end of oscillating from ground along x	$v_{1}$	m/s	—
velocity of body at start of oscillating from ground along x	$v_{0}$	m/s	—
average acceleration of body during oscillating along x	$a_{av}$	m/s²	—
stroke of ground	$s_{g}$	m	—
amplitude of ground	$A_{g}$	m	0 … ∞
stroke of body	$s$	m	—
amplitude of body	$A$	m	0 … ∞
angular frequency of ground	$ω_{g}$	rad/s	0 … ∞
frequency of ground	$ν_{g}$	Hz	0 … ∞
angular frequency of body	$ω$	rad/s	0 … ∞
frequency of body	$ν$	Hz	0 … ∞
maximum speed of ground	$v_{max,g}$	m/s	—
maximum speed of body	$v_{\max}$	m/s	—
maximum acceleration of ground	$a_{max,g}$	m/s²	—
maximum acceleration of body	$a_{\max}$	m/s²	—
position of body at end of oscillating	$\| x_{1} \|$	m	—
phase constant of body	$φ$	—	—
velocity of body at end of oscillating from ground	$\| v_{1} \|$	m/s	—
acceleration of body at end of oscillating	$\| a_{1} \|$	m/s²	—

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_{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(body)

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(body)

v_{max,g} = ω_{g} \cdot A_{g}

fastest through the middle(ground)

v_{\max} = ω \cdot A

fastest through the middle(body)

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(body)

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

displacement in shm(body, oscillating)

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

velocity in shm(body, oscillating)

| a_{1} | = - ω^{2} \cdot | x_{1} |

acceleration in shm(body, oscillating)

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

amplitude from where and how fast(body, oscillating)