A Body in SHM — phryzby

Q. A body oscillates with SHM according to the equation x = (5 m) cos[(2π rad/s) t + π/4]. At t = 1.5 s, what is the speed of the body, in m/s? expects 22.2 m/s

What the draft establishes

Fills in once a draft is checked.

The web of concepts

given read from the words equation asked

The equations it used

None yet.

Check an answer

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?

The playground

Inputs

time

Δ t

1.50 s

Outputs

speed

| v_{1} |

The solution

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 Ruby

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

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

1require_relative "../solutions/a_body_in_shm"23play_with :a_body_in_shm do4  title "A Body in SHM"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
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)