A Motor Wheel Speeds Up

solved RotationalKinematics

Q. The angular speed of a motor wheel is increased from 1200 rpm $ω_{0} = 1200$ rpm to 3120 rpm $ω_{1} = 3120$ rpm in 16 s $Δ t = 16$ s. Assuming the angular acceleration is uniform, how many revolutions does the engine make during this time?

Inputs

start

ω_{0}

1200 rpm

finish

ω_{1}

3120 rpm

time

Δ t

16.00 s

Outputs

turns

n_{rev}

How it's solved

Solved with RotationalKinematics

Given

$ω_{0} = 1200$ rpm
$ω_{1} = 3120$ rpm
$Δ t = 16$ s

Find $n_{rev}$ .

Working

$ω_{0} = 1200$ rpm $= 125.663706$ rad/s
$ω_{1} = 3120$ rpm $= 326.725636$ rad/s
angular velocity after a time(speeding up) $ω_{1} = ω_{0} + α_{r} \cdot Δ t$
$326.725636 = 125.663706 + α_{r} \cdot 16$
$α_{r} = 12.5663706$ rad/s²
angle after a time(speeding up) $θ = ω_{0} \cdot Δ t + \frac{α_{r} \cdot {Δ t}^{2}}{2}$
$θ = 125.663706 \cdot 16 + \frac{12.5663706 \cdot 16^{2}}{2}$
$θ = 3619.11474$
turns in an angle(speeding up) $n_{rev} = \frac{θ}{2 \cdot π}$
$n_{rev} = \frac{3619.11474}{2 \cdot 3.14159265}$
$n_{rev} = 576$
$n_{rev} \approx 576$ to 3 significant figures

Conclusion

With a steady angular acceleration the wheel turns as if at the mean of its two speeds for the whole time.

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_motor_wheel_speeds_up do4  kind :numerical5  describes "The angular speed of a motor wheel is increased from %{start} to %{finish} in %{time}. Assuming the " \6            "angular acceleration is uniform, %{turns}?"7  given start: "1200 rpm"8  given finish: "3120 rpm"9  given time: "16 s"10  asks turns: "how many revolutions does the engine make during this time"11end

solution.rb

1require_relative "../questions/a_motor_wheel_speeds_up"2require_relative "../science/systems_of_particles_and_rotational_motion/rotational_kinematics"34solve :a_motor_wheel_speeds_up, using: RotationalKinematics do5  objects :wheel6  wheel.goes(:speeding_up)78  take start: wheel.initial_angular_velocity9  take finish: wheel.final_angular_velocity10  take time: speeding_up.time_taken1112  turns = wheel.turns13  answer turns:, figures: 31415  conclusion "With a steady angular acceleration the wheel turns as if at the mean of its two speeds " \16             "for the whole time."17end

playground.rb

1require_relative "../solutions/a_motor_wheel_speeds_up"23play_with :a_motor_wheel_speeds_up do4  title "A Motor Wheel Speeds Up"5end

The laws it drew on

quantity	written	units	within	from
pi	$π$	—	0 … ∞	RotationalKinematics
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
angular velocity	$ω$	rad/s	—	RotationalKinematics
angular acceleration	$α_{r}$	rad/s²	—	RotationalKinematics
angle turned	$θ$	—	—	RotationalKinematics
turns	$n_{rev}$	—	—	RotationalKinematics
starting angular velocity	$ω_{0}$	rad/s	—	RotationalKinematics
final angular velocity	$ω$	rad/s	—	RotationalKinematics
displacement of wheel during speeding up	$Δ x$	m	—
position of wheel at end of speeding up	$x_{1}$	m	—
position of wheel at start of speeding up	$x_{0}$	m	—
average velocity of wheel during speeding up	$v_{av}$	m/s	—
time taken of speeding up	$Δ t$	s	0 … 1e+05
average speed of wheel during speeding up	$c_{av}$	m/s	—
path length of wheel during speeding up	$d$	m	—
angular velocity of wheel at end of speeding up	$ω_{1}$	rad/s	—
angular velocity of wheel at start of speeding up	$ω_{0}$	rad/s	—
angular acceleration of wheel during speeding up	$α_{r}$	rad/s²	—
angle turned of wheel during speeding up	$θ$	—	—
turns of wheel during speeding up	$n_{rev}$	—	—

x_{0} \equiv x_{i} (start of motion)

starting position

x \equiv x_{i} (end of motion)

final position

ω_{0} \equiv ω_{i} (start of motion)

starting angular velocity

ω \equiv ω_{i} (end of motion)

final angular 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

Δ x = x_{1} - x_{0}

change of position(speeding up)

v_{av} = \frac{Δ x}{Δ t}

displacement a time(speeding up)

c_{av} = \frac{d}{Δ t}

path a time(speeding up)

ω_{1} = ω_{0} + α_{r} \cdot Δ t

angular velocity after a time(speeding up)

θ = ω_{0} \cdot Δ t + \frac{α_{r} \cdot {Δ t}^{2}}{2}

angle after a time(speeding up)

{ω_{1}}^{2} = {ω_{0}}^{2} + 2 \cdot α_{r} \cdot θ

angular velocity after an angle(speeding up)

n_{rev} = \frac{θ}{2 \cdot π}

turns in an angle(speeding up)