An Elevator's Motor

solved Power

Q. An elevator can carry a maximum load of 1800 kg $m = 1800$ kg (elevator + passengers) and is moving up $F_{m} = | F_{m} |$ with a constant speed $a = 0$ m/s² of 2 m/s $v_{0} = 2$ m/s. The frictional force opposing the motion $F_{s} = - | F_{s} |$ is 4000 N $| F_{s} | = 4000$ N. Taking g as $W = - | W |$ 10 m/s² $g = 10$ m/s², what is the minimum power delivered by the motor to the elevator, in watts?

Inputs

mass

m

1800 kg

speed

v_{0}

2.0 m/s

friction

| F_{s} |

4000 N

gravity

g

10 m/s²

Outputs

power

P_{0,m}

How it's solved

Solved with Power

Given

$m = 1800$ kg
$v_{0} = 2$ m/s
$| F_{s} | = 4000$ N
$g = 10$ m/s²
$a = 0$ m/s² from “constant speed”
$W = - | W |$ from “Taking g as”
$F_{s} = - | F_{s} |$ from “opposing the motion”
$F_{m} = | F_{m} |$ from “moving up”

Find $P_{0,m}$ .

Working

second law(rising, y) $F_{net} = m \cdot a$
$F_{net} = 1800 \cdot 0$
$F_{net} = 0$ N
opposing the motion $F_{s} = - | F_{s} |$
$F_{s} = - 4000$
$F_{s} = −4000$ N
weight of a body(elevator) $| W | = m \cdot g$
$| W | = 1800 \cdot 10$
$| W | = 18000$ N
Taking g as $W = - | W |$
$W = - 18000$
$W = −18000$ N
net force is the resultant(elevator, rising, y) $F_{net} = F_{s} + F_{m} + W$
$0 = (−4000) + F_{m} + (−18000)$
$F_{m} = 22000$ N
power of a force(elevator, rising, 3) $P_{0,m} = F_{m} \cdot v_{0}$
$P_{0,m} = 22000 \cdot 2$
$P_{0,m} = 44000$ W

Conclusion

At a steady speed the motor only has to balance the weight and the friction; its power is that force times the speed it lifts at.

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 :an_elevator_motor do4  kind :numerical5  describes "An elevator can carry a maximum load of %{mass} (elevator + passengers) and is moving up with a " \6            "constant speed of %{speed}. The frictional force opposing the motion is %{friction}. Taking g as " \7            "%{gravity}, %{power}, in watts?"8  given mass: "1800 kg"9  given speed: "2 m/s"10  given friction: "4000 N"11  given gravity: "10 m/s2"12  asks power: "what is the minimum power delivered by the motor to the elevator"13end

solution.rb

1require_relative "../questions/an_elevator_motor"2require_relative "../science/work_energy_and_power/power"34solve :an_elevator_motor, using: Power do5  objects :elevator, :motor, :shaft6  axes :y7  elevator.goes(:rising)8  deduce_from "Taking g as", elevator.weight.direction => -y910  take mass: elevator.mass11  take speed: elevator.initial_velocity12  take friction: elevator.force(by: shaft).magnitude13  take gravity: g1415  deduce_from "constant speed", elevator.acceleration => "0 m/s2"16  deduce_from "opposing the motion", elevator.force(by: shaft).direction => -y17  deduce_from "moving up", elevator.force(by: motor).direction => y1819  power = elevator.power(by: motor, at: rising.start)20  answer power:, in: "W"2122  conclusion "At a steady speed the motor only has to balance the weight and the friction; its power " \23             "is that force times the speed it lifts at."24end

playground.rb

1require_relative "../solutions/an_elevator_motor"23play_with :an_elevator_motor do4  title "An Elevator's Motor"5end

The laws it drew on

quantity	written	units	within	from
volume	$volume$	—	—	Vectors
mass	$m$	kg	0 … ∞	Vectors
speed	$speed$	—	—	Vectors
density	$density$	—	—	Vectors
number of moles	$n$	—	—	Vectors
angular frequency	${angular}_{frequency}$	—	—	Vectors
work	$W_{F}$	J	—	Vectors
current	$current$	—	—	Vectors
temperature	$temperature$	—	—	Vectors
pressure	$pressure$	—	—	Vectors
time	$t$	s	0 … 1e+05	Vectors
power	$P$	W	—	Vectors
total path length	$s$	—	—	Vectors
energy	$energy$	—	—	Vectors
gravitational potential	${gravitational}_{potential}$	—	—	Vectors
coefficient of friction	$μ$	—	—	Vectors
charge	$charge$	—	—	Vectors
acceleration	$a$	m/s²	—	Vectors
velocity	$v$	m/s	—	Vectors
displacement	$Δ x$	m	—	Vectors
angular velocity	${angular}_{velocity}$	—	—	Vectors
force	$F$	N	—	Vectors
angular momentum	${angular}_{momentum}$	—	—	Vectors
linear momentum	$p_{lin}$	—	—	Vectors
electric field	${electric}_{field}$	—	—	Vectors
average velocity	$v_{av}$	m/s	—	Vectors
magnetic moment	${magnetic}_{moment}$	—	—	Vectors
relative velocity	$v_{AB}$	—	—	Vectors
impulse	$impulse$	—	—	Vectors
first vector	$A$	—	0 … ∞	Vectors
second vector	$B$	—	0 … ∞	Vectors
angle between	$φ$	—	0 … 3.142	Vectors
resultant	$R$	—	0 … ∞	Vectors
resultant direction	$β$	—	0 … 3.142	Vectors
direction	$α$	—	-3.142 … 3.142	Vectors
x component	$A_{x}$	—	—	Vectors
y component	$A_{y}$	—	—	Vectors
position	$x$	m	—	Velocity
path length	$d$	m	—	Velocity
time taken	$Δ t$	s	0 … 1e+05	Velocity
average speed	$c_{av}$	m/s	—	Velocity
starting position	$x_{0}$	m	—	Velocity
final position	$x$	m	—	Velocity
change of velocity	$Δ v$	m/s	—	Acceleration
average acceleration	$a_{av}$	m/s²	—	Acceleration
starting velocity	$v_{0}$	m/s	—	Acceleration
final velocity	$v$	m/s	—	Acceleration
gravity	$g$	m/s²	—	UniformAcceleration
net force	$F_{net}$	N	—	NewtonsSecondLaw
weight	$W$	N	—	NewtonsSecondLaw
work by gravity	$W_{g}$	J	—	Work
net work	$W_{net}$	J	—	Work
displacement of elevator during rising along y	$Δ x$	m	—
position of elevator at end of rising along y	$x_{1}$	m	—
position of elevator at start of rising along y	$x_{0}$	m	—
average velocity of elevator during rising along y	$v_{av}$	m/s	—
time taken of rising	$Δ t$	s	0 … 1e+05
average speed of elevator during rising	$c_{av}$	m/s	—
path length of elevator during rising	$d$	m	—
change of velocity of elevator during rising along y	$Δ v$	m/s	—
velocity of elevator at end of rising from ground along y	$v_{1}$	m/s	—
velocity of elevator at start of rising from ground along y	$v_{0}$	m/s	—
average acceleration of elevator during rising along y	$a_{av}$	m/s²	—
acceleration of elevator during rising along y	$a$	m/s²	—
net force on elevator during rising along y	$F_{net}$	N	—
mass of elevator	$m$	kg	0 … ∞
mass of motor	$m_{m}$	kg	0 … ∞
mass of shaft	$m_{s}$	kg	0 … ∞
weight on ground	$\| W_{g} \|$	N	—
weight on elevator	$\| W \|$	N	—
weight on motor	$\| W_{m} \|$	N	—
weight on shaft	$\| W_{s} \|$	N	—
force on elevator by shaft during rising	$\| F_{s} \|$	N	—
force on elevator by motor during rising	$\| F_{m} \|$	N	—
weight on elevator along y	$W$	N	—
force on elevator by shaft during rising along y	$F_{s}$	N	—
force on elevator by motor during rising along y	$F_{m}$	N	—
work on elevator by shaft during rising	$W_{F,s}$	J	—
work on elevator by motor during rising	$W_{F,m}$	J	—
net work on elevator during rising	$W_{net}$	J	—
work by gravity on elevator during rising	$W_{g}$	J	—
power on elevator by shaft at start of rising	$P_{0,s}$	W	—
power on elevator by shaft at end of rising	$P_{1,s}$	W	—
power on elevator by motor at start of rising	$P_{0,m}$	W	—
power on elevator by motor at end of rising	$P_{1,m}$	W	—
displacement of elevator during rising	$\| Δ x \|$	m	—
position of elevator at end of rising	$\| x_{1} \|$	m	—
position of elevator at start of rising	$\| x_{0} \|$	m	—
average velocity of elevator during rising	$\| v_{av} \|$	m/s	—
change of velocity of elevator during rising	$\| Δ v \|$	m/s	—
velocity of elevator at end of rising from ground	$\| v_{1} \|$	m/s	—
velocity of elevator at start of rising from ground	$\| v_{0} \|$	m/s	—
average acceleration of elevator during rising	$\| a_{av} \|$	m/s²	—
acceleration of elevator during rising	$\| a \|$	m/s²	—
net force on elevator during rising	$\| F_{net} \|$	N	—

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

starting position

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

final position

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

starting velocity

v \equiv v_{i} (end of motion)

final velocity

t \equiv Δ t_{m}

time

A = \sqrt{{A_{x}}^{2} + {A_{y}}^{2}}

magnitude from components

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

a_{i} (part 1) = a_{i} (whole)

steady through its parts

m_{b} \neq 0

has mass?

R^{2} = A^{2} + B^{2} + 2 \cdot A \cdot B \cdot \cos φ

law of cosines

R \cdot \sin β = B \cdot \sin φ

law of sines

R \cdot \cos β = A + B \cdot \cos φ

along the first

A_{x} = A \cdot \cos α

resolved along x

A_{y} = A \cdot \sin α

resolved along y

Δ x = x_{1} - x_{0}

change of position(rising, y)

v_{av} = \frac{Δ x}{Δ t}

displacement a time(rising, y)

c_{av} = \frac{d}{Δ t}

path a time(rising, y)

Δ v = v_{1} - v_{0}

velocity gained(rising, y)

a_{av} = \frac{Δ v}{Δ t}

velocity gained a time(rising, y)

v_{1} = v_{0} + a \cdot Δ t

velocity after a time(rising, y)

x_{1} = x_{0} + v_{0} \cdot Δ t + \frac{a \cdot {Δ t}^{2}}{2}

position after a time(rising, y)

{v_{1}}^{2} = {v_{0}}^{2} + 2 \cdot a \cdot (x_{1} - x_{0})

velocity after a distance(rising, y)

x_{1} - x_{0} = \frac{(v_{1} + v_{0}) \cdot Δ t}{2}

mean of the velocities(rising, y)

F_{net} = m \cdot a

second law(rising, y)

| W | = m \cdot g

weight of a body(elevator)

F_{net} = F_{s} + F_{m} + W

net force is the resultant(elevator, rising, y)

W_{F,s} = F_{s} \cdot Δ x

work of a steady force(elevator, rising, 1)

W_{F,m} = F_{m} \cdot Δ x

work of a steady force(elevator, rising, 2)

W_{net} = F_{net} \cdot Δ x

work of the net force(elevator, rising)

W_{g} = W \cdot Δ x

work of the weight(elevator, rising)

P_{0,s} = F_{s} \cdot v_{0}

power of a force(elevator, rising, 1)

P_{1,s} = F_{s} \cdot v_{1}

power of a force(elevator, rising, 2)

P_{0,m} = F_{m} \cdot v_{0}

power of a force(elevator, rising, 3)

P_{1,m} = F_{m} \cdot v_{1}

power of a force(elevator, rising, 4)

| Δ x | = \sqrt{{Δ x}^{2}}

magnitude from components(displacement of elevator during rising)

| x_{1} | = \sqrt{{x_{1}}^{2}}

magnitude from components(position of elevator at end of rising)

| x_{0} | = \sqrt{{x_{0}}^{2}}

magnitude from components(position of elevator at start of rising)

| v_{av} | = \sqrt{{v_{av}}^{2}}

magnitude from components(average velocity of elevator during rising)

| Δ v | = \sqrt{{Δ v}^{2}}

magnitude from components(change of velocity of elevator during rising)

| v_{1} | = \sqrt{{v_{1}}^{2}}

magnitude from components(velocity of elevator at end of rising from ground)

| v_{0} | = \sqrt{{v_{0}}^{2}}

magnitude from components(velocity of elevator at start of rising from ground)

| a_{av} | = \sqrt{{a_{av}}^{2}}

magnitude from components(average acceleration of elevator during rising)

| a | = \sqrt{a^{2}}

magnitude from components(acceleration of elevator during rising)

| F_{net} | = \sqrt{{F_{net}}^{2}}

magnitude from components(net force on elevator during rising)

| W | = \sqrt{W^{2}}

magnitude from components(weight on elevator)

| F_{s} | = \sqrt{{F_{s}}^{2}}

magnitude from components(force on elevator by shaft during rising)

| F_{m} | = \sqrt{{F_{m}}^{2}}

magnitude from components(force on elevator by motor during rising)

W = - | W |

Taking g as

F_{s} = - | F_{s} |

opposing the motion

F_{m} = | F_{m} |

moving up

m \neq 0

has mass?(elevator)

m_{m} \neq 0

has mass?(motor)

m_{s} \neq 0

has mass?(shaft)

F_{net} = 0

free of forces?(elevator, rising)