A Spinning Solid Cylinder

Q. A solid cylinder of mass 20 kg rotates about its axis with angular speed 100 rad/s. The radius of the cylinder is 0.25 m. What is the kinetic energy associated with the rotation of the cylinder, in joules? expects 3125 J

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Q. A solid cylinder of mass 20 kg $M = 20$ kg rotates about its axis with angular speed 100 rad/s $ω_{0} = 100$ rad/s. The radius of the cylinder is 0.25 m $R = 0.25$ m. What is the kinetic energy associated with the rotation of the cylinder, in joules?

The playground

Inputs

mass

M

20 kg

speed

ω_{0}

100 rad/s

radius

R

0.25 m

Outputs

energy

K_{rot,0}

The solution

Solved with RotationalDynamics

Given

$M = 20$ kg
$ω_{0} = 100$ rad/s
$R = 0.25$ m

Find $K_{rot,0}$ .

Working

a disc s moment of inertia(spinning) $I = \frac{M \cdot R^{2}}{2}$
$I = \frac{20 \cdot {0.25}^{2}}{2}$
$I = 0.625$ kg m²
kinetic energy of turning(cylinder, spinning, 1) $K_{rot,0} = \frac{I \cdot {ω_{0}}^{2}}{2}$
$K_{rot,0} = \frac{0.625 \cdot 100^{2}}{2}$
$K_{rot,0} = 3125$ J

Conclusion

A solid cylinder resists turning with half its mass times its radius squared, and its energy of turning is half that times the square of its angular speed.

The Ruby

1require_relative "../physics"23question :a_spinning_solid_cylinder do4  kind :numerical5  describes "A solid cylinder of mass %{mass} rotates about its axis with angular speed %{speed}. The radius of the " \6            "cylinder is %{radius}. %{energy}, in joules?"7  given mass: "20 kg"8  given speed: "100 rad/s"9  given radius: "0.25 m"10  asks energy: "What is the kinetic energy associated with the rotation of the cylinder"11end

1require_relative "../questions/a_spinning_solid_cylinder"2require_relative "../science/systems_of_particles_and_rotational_motion/rotational_dynamics"34solve :a_spinning_solid_cylinder, using: RotationalDynamics do5  objects :cylinder6  cylinder.goes(:spinning)78  take mass: cylinder.mass9  take speed: cylinder.initial_angular_velocity10  take radius: cylinder.radius1112  energy = cylinder.rotational_kinetic_energy(at: spinning.start)13  answer energy:, in: "J"1415  conclusion "A solid cylinder resists turning with half its mass times its radius squared, and its energy of " \16             "turning is half that times the square of its angular speed."17end

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

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(spinning)

v_{av} = \frac{Δ x}{Δ t}

displacement a time(spinning)

c_{av} = \frac{d}{Δ t}

path a time(spinning)

ω_{1} = ω_{0} + α_{r} \cdot Δ t

angular velocity after a time(spinning)

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

angle after a time(spinning)

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

angular velocity after an angle(spinning)

n_{rev} = \frac{θ}{2 \cdot π}

turns in an angle(spinning)

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

a disc s moment of inertia(spinning)

τ = I \cdot α_{r}

second law of rotation(spinning)

l = R \cdot θ

cord unwound by the turn(spinning)

τ = F_{g} \cdot R

torque of a pull at the rim(cylinder, ground, spinning)

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

kinetic energy of turning(cylinder, spinning, 1)

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

kinetic energy of turning(cylinder, spinning, 2)

L_{0} = I \cdot ω_{0}

angular momentum of turning(cylinder, spinning, 1)

L_{1} = I \cdot ω_{1}

angular momentum of turning(cylinder, spinning, 2)