A Flywheel on a Cord

solved RotationalDynamics

Q. A cord of negligible mass is wound round the rim of a flywheel of mass 20 kg $M = 20$ kg and radius 20 cm $R = 20$ cm. A steady pull of 25 N $F_{c} = 25$ N is applied on the cord, and the flywheel turns on frictionless bearings. What is the angular acceleration of the wheel, in rad/s²?

Inputs

mass

M

20 kg

radius

R

20 cm

pull

F_{c}

25 N

Outputs

angular acceleration

α_{r}

How it's solved

Solved with RotationalDynamics

Given

$M = 20$ kg
$R = 20$ cm
$F_{c} = 25$ N

Find $α_{r}$ .

Working

$R = 20$ cm $= 0.2$ m
torque of a pull at the rim(flywheel, cord, turning) $τ = F_{c} \cdot R$
$τ = 25 \cdot 0.2$
$τ = 5$ J
a disc s moment of inertia(turning) $I = \frac{M \cdot R^{2}}{2}$
$I = \frac{20 \cdot {0.2}^{2}}{2}$
$I = 0.4$ kg m²
second law of rotation(turning) $τ = I \cdot α_{r}$
$5 = 0.4 \cdot α_{r}$
$α_{r} = 12.5$ rad/s²

Conclusion

The pull turns the wheel with the force times the radius, and a flywheel resists turning with half its mass times its radius squared; their ratio is the angular acceleration.

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_flywheel_on_a_cord do4  kind :numerical5  describes "A cord of negligible mass is wound round the rim of a flywheel of mass %{mass} and radius %{radius}. A " \6            "steady pull of %{pull} is applied on the cord, and the flywheel turns on frictionless bearings. " \7            "%{angular_acceleration}, in rad/s²?"8  given mass: "20 kg"9  given radius: "20 cm"10  given pull: "25 N"11  asks angular_acceleration: "What is the angular acceleration of the wheel"12end

solution.rb

1require_relative "../questions/a_flywheel_on_a_cord"2require_relative "../science/systems_of_particles_and_rotational_motion/rotational_dynamics"34solve :a_flywheel_on_a_cord, using: RotationalDynamics do5  objects :flywheel, :cord6  flywheel.goes(:turning)78  take mass: flywheel.mass9  take radius: flywheel.radius10  take pull: flywheel.pull(by: cord)1112  angular_acceleration = flywheel.angular_acceleration13  answer angular_acceleration:, in: "rad/s2"1415  conclusion "The pull turns the wheel with the force times the radius, and a flywheel resists turning " \16             "with half its mass times its radius squared; their ratio is the angular acceleration."17end

playground.rb

1require_relative "../solutions/a_flywheel_on_a_cord"23play_with :a_flywheel_on_a_cord do4  title "A Flywheel on a Cord"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
mass	$M$	kg	0 … ∞	RotationalDynamics
radius	$R$	m	0 … ∞	RotationalDynamics
moment of inertia	$I$	kg m²	—	RotationalDynamics
pull	$F$	N	—	RotationalDynamics
torque	$τ$	J	—	RotationalDynamics
cord unwound	$l$	m	—	RotationalDynamics
rotational kinetic energy	$K_{rot}$	J	—	RotationalDynamics
angular momentum	$L$	J s	—	RotationalDynamics
displacement of flywheel during turning	$Δ x$	m	—
position of flywheel at end of turning	$x_{1}$	m	—
position of flywheel at start of turning	$x_{0}$	m	—
average velocity of flywheel during turning	$v_{av}$	m/s	—
time taken of turning	$Δ t$	s	0 … 1e+05
average speed of flywheel during turning	$c_{av}$	m/s	—
path length of flywheel during turning	$d$	m	—
angular velocity of flywheel at end of turning	$ω_{1}$	rad/s	—
angular velocity of flywheel at start of turning	$ω_{0}$	rad/s	—
angular acceleration of flywheel during turning	$α_{r}$	rad/s²	—
angle turned of flywheel during turning	$θ$	—	—
turns of flywheel during turning	$n_{rev}$	—	—
moment of inertia of flywheel	$I$	kg m²	—
mass of flywheel	$M$	kg	0 … ∞
radius of flywheel	$R$	m	0 … ∞
torque on flywheel during turning	$τ$	J	—
cord unwound of flywheel during turning	$l$	m	—
pull on flywheel by ground during turning	$F_{g}$	N	—
pull on flywheel by cord during turning	$F_{c}$	N	—
rotational kinetic energy of flywheel at start of turning	$K_{rot,0}$	J	—
rotational kinetic energy of flywheel at end of turning	$K_{rot,1}$	J	—
angular momentum of flywheel at start of turning	$L_{0}$	J s	—
angular momentum of flywheel at end of turning	$L_{1}$	J s	—
mass of cord	$M_{c}$	kg	0 … ∞
radius of cord	$R_{c}$	m	0 … ∞
moment of inertia of cord	$I_{c}$	kg m²	—

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(turning)

v_{av} = \frac{Δ x}{Δ t}

displacement a time(turning)

c_{av} = \frac{d}{Δ t}

path a time(turning)

ω_{1} = ω_{0} + α_{r} \cdot Δ t

angular velocity after a time(turning)

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

angle after a time(turning)

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

angular velocity after an angle(turning)

n_{rev} = \frac{θ}{2 \cdot π}

turns in an angle(turning)

I = \frac{M \cdot R^{2}}{2}

a disc s moment of inertia(turning)

τ = I \cdot α_{r}

second law of rotation(turning)

l = R \cdot θ

cord unwound by the turn(turning)

τ = F_{g} \cdot R

torque of a pull at the rim(flywheel, ground, turning)

τ = F_{c} \cdot R

torque of a pull at the rim(flywheel, cord, turning)

K_{rot,0} = \frac{I \cdot {ω_{0}}^{2}}{2}

kinetic energy of turning(flywheel, turning, 1)

K_{rot,1} = \frac{I \cdot {ω_{1}}^{2}}{2}

kinetic energy of turning(flywheel, turning, 2)

L_{0} = I \cdot ω_{0}

angular momentum of turning(flywheel, turning, 1)

L_{1} = I \cdot ω_{1}

angular momentum of turning(flywheel, turning, 2)