A Flywheel's Energy

Q. A cord of negligible mass is wound round the rim of a flywheel of mass 20 kg and radius 20 cm, at rest on frictionless bearings. A steady pull of 25 N is applied on the cord until 2 m of cord is unwound. What is the kinetic energy of the wheel at this time, in joules? expects 50 J

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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, at rest $ω_{0} = 0$ rad/s on frictionless bearings. A steady pull of 25 N $F_{c} = 25$ N is applied on the cord until 2 m $l = 2$ m of cord is unwound. What is the kinetic energy of the wheel at this time, in joules?

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Inputs

mass

M

20 kg

radius

R

20 cm

pull

F_{c}

25 N

unwound

l

2.00 m

Outputs

energy

K_{rot,1}

The solution

Solved with RotationalDynamics

Given

$M = 20$ kg
$R = 20$ cm
$F_{c} = 25$ N
$l = 2$ m
$ω_{0} = 0$ rad/s from “at rest”

Find $K_{rot,1}$ .

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
cord unwound by the turn(turning) $l = R \cdot θ$
$2 = 0.2 \cdot θ$
$θ = 10$
angular velocity after an angle(turning) ${ω_{1}}^{2} = {ω_{0}}^{2} + 2 \cdot α_{r} \cdot θ$
${ω_{1}}^{2} = 0^{2} + 2 \cdot 12.5 \cdot 10$
$ω_{1} = 15.8113883$ rad/s
kinetic energy of turning(flywheel, turning, 2) $K_{rot,1} = \frac{I \cdot {ω_{1}}^{2}}{2}$
$K_{rot,1} = \frac{0.4 \cdot {15.8113883}^{2}}{2}$
$K_{rot,1} = 50$ J

Conclusion

The pull does work equal to the force times the cord it unwinds, and with no friction all of it becomes the wheel's energy of turning.

The Ruby

1require_relative "../physics"23question :a_flywheel_s_energy do4  kind :numerical5  describes "A cord of negligible mass is wound round the rim of a flywheel of mass %{mass} and radius %{radius}, " \6            "at rest on frictionless bearings. A steady pull of %{pull} is applied on the cord until %{unwound} of " \7            "cord is unwound. %{energy}, in joules?"8  given mass: "20 kg"9  given radius: "20 cm"10  given pull: "25 N"11  given unwound: "2 m"12  asks energy: "What is the kinetic energy of the wheel at this time"13end

1require_relative "../questions/a_flywheel_s_energy"2require_relative "../science/systems_of_particles_and_rotational_motion/rotational_dynamics"34solve :a_flywheel_s_energy, using: RotationalDynamics do5  objects :flywheel, :cord6  flywheel.goes(:turning)78  take mass: flywheel.mass9  take radius: flywheel.radius10  take pull: flywheel.pull(by: cord)11  take unwound: flywheel.cord_unwound1213  deduce_from "at rest", flywheel.initial_angular_velocity => "0 rad/s"1415  energy = flywheel.rotational_kinetic_energy(at: turning.end)16  answer energy:, in: "J"1718  conclusion "The pull does work equal to the force times the cord it unwinds, and with no friction all " \19             "of it becomes the wheel's energy of turning."20end

1require_relative "../solutions/a_flywheel_s_energy"23play_with :a_flywheel_s_energy do4  title "A Flywheel's Energy"5end

The laws

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)