A Wheel's Spokes

solved MotionalEmf

Q. A wheel with 10 metallic spokes each 0.5 m $R = 0.5$ m long is rotated with a speed of 120 rpm $ω = 120$ rpm in a plane normal to the horizontal component of earth's magnetic field at a place, which is 0.4 G $B = 0.4$ G. What is the induced emf between the axle and the rim of the wheel, in volts?

Inputs

length

R

0.50 m

rate

ω

120 rpm

field

B

0.4 G

Outputs

emf

ε

How it's solved

Solved with MotionalEmf

Given

$R = 0.5$ m
$ω = 120$ rpm
$B = 0.4$ G

Find $ε$ .

Working

$ω = 120$ rpm $= 12.5663706$ rad/s
$B = 0.4$ G $= 4 \times 10^{−5}$ T
emf of a rod turning about one end(spoke) $ε = \frac{B \cdot ω \cdot R^{2}}{2}$
$ε = \frac{4 \times 10^{−5} \cdot 12.5663706 \cdot {0.5}^{2}}{2}$
$ε = 6.283 \times 10^{−5}$ V
$ε \approx 6.28 \times 10^{−5}$ Vto 3 significant figures

Conclusion

Every spoke turns through the field alike and they all join the axle to the rim side by side, so ten spokes give the same emf as one.

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_wheel_s_spokes do4  kind :numerical5  describes "A wheel with 10 metallic spokes each %{length} long is rotated with a speed of %{rate} in a plane normal " \6            "to the horizontal component of earth's magnetic field at a place, which is %{field}. %{emf}, in volts?"7  given length: "0.5 m"8  given rate: "120 rpm"9  given field: "0.4 G"10  asks emf: "What is the induced emf between the axle and the rim of the wheel"11end

solution.rb

1require_relative "../questions/a_wheel_s_spokes"2require_relative "../science/electromagnetic_induction/motional_emf"34solve :a_wheel_s_spokes, using: MotionalEmf do5  objects :spoke67  take length: spoke.length_of_the_rod8  take rate: spoke.angular_speed9  take field: B1011  emf = spoke.induced_emf12  answer emf:, in: "V", figures: 31314  conclusion "Every spoke turns through the field alike and they all join the axle to the rim side by side, " \15             "so ten spokes give the same emf as one."16end

playground.rb

1require_relative "../solutions/a_wheel_s_spokes"23play_with :a_wheel_s_spokes do4  title "A Wheel's Spokes"5end

The laws it drew on

quantity	written	units	within	from
magnetic field	$B$	T	—	MotionalEmf
length of the rod	$R$	m	0 … ∞	MotionalEmf
angular speed	$ω$	rad/s	—	MotionalEmf
induced emf	$ε$	V	—	MotionalEmf
induced emf of ground	$ε_{g}$	V	—
angular speed of ground	$ω_{g}$	rad/s	—
length of the rod of ground	$R_{g}$	m	0 … ∞
induced emf of spoke	$ε$	V	—
angular speed of spoke	$ω$	rad/s	—
length of the rod of spoke	$R$	m	0 … ∞

ε_{r} = \frac{B \cdot ω_{r} \cdot {R_{r}}^{2}}{2}

emf of a rod turning about one end

ε_{g} = \frac{B \cdot ω_{g} \cdot {R_{g}}^{2}}{2}

emf of a rod turning about one end(ground)

ε = \frac{B \cdot ω \cdot R^{2}}{2}

emf of a rod turning about one end(spoke)