A Lift Falling Freely

solved NewtonsSecondLaw

Q. A man of mass 70 kg $m = 70$ kg stands on a weighing scale $F_{s} = | F_{s} |$ in a lift whose mechanism has failed, so that it hurtles down freely under gravity $W = - | W |$ . Taking g as 10 m/s² $g = 10$ m/s², what would be the reading on the scale, in newtons?

Inputs

mass

m

70 kg

gravity

g

10 m/s²

Outputs

reading

| F_{s} |

How it's solved

Solved with NewtonsSecondLaw

Given

$m = 70$ kg
$g = 10$ m/s²
$W = - | W |$ from “under gravity”
$a = - g$ from “freely under gravity”
$F_{s} = | F_{s} |$ from “stands on a weighing scale”

Find $| F_{s} |$ .

Working

freely under gravity $a = - g$
$a = - 10$
$a = −10$ m/s²
second law(riding, y) $F_{net} = m \cdot a$
$F_{net} = 70 \cdot (−10)$
$F_{net} = −700$ N
weight of a body(man) $| W | = m \cdot g$
$| W | = 70 \cdot 10$
$| W | = 700$ N
under gravity $W = - | W |$
$W = - 700$
$W = −700$ N
net force is the resultant(man, riding, y) $F_{net} = F_{s} + W$
$(−700) = F_{s} + (−700)$
$F_{s} = 0$ N
magnitude from components(force on man by scale during riding) $| F_{s} | = \sqrt{{F_{s}}^{2}}$
$| F_{s} | = \sqrt{0^{2}}$
$| F_{s} | = 0$ N

Conclusion

Falling freely, the man and the scale drop together with nothing to spare, so the scale need not push at all.

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_lift_falling_freely do4  kind :numerical5  describes "A man of mass %{mass} stands on a weighing scale in a lift whose mechanism has failed, so that it " \6            "hurtles down freely under gravity. Taking g as %{gravity}, %{reading}, in newtons?"7  given mass: "70 kg"8  given gravity: "10 m/s2"9  asks reading: "what would be the reading on the scale"10end

solution.rb

1require_relative "../questions/a_lift_falling_freely"2require_relative "../science/laws_of_motion/newtons_second_law"34solve :a_lift_falling_freely, using: NewtonsSecondLaw do5  objects :man, :scale6  axes :y7  man.goes(:riding)8  deduce_from "under gravity", man.weight.direction => -y910  take mass: man.mass11  take gravity: g1213  deduce_from "freely under gravity", man.acceleration => -g14  deduce_from "stands on a weighing scale", man.force(by: scale).direction => y1516  reading = man.force(by: scale).magnitude17  answer reading:, in: "N"1819  conclusion "Falling freely, the man and the scale drop together with nothing to spare, so the scale need " \20             "not push at all."21end

playground.rb

1require_relative "../solutions/a_lift_falling_freely"23play_with :a_lift_falling_freely do4  title "A Lift Falling Freely"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	$work$	—	—	Vectors
current	$current$	—	—	Vectors
temperature	$temperature$	—	—	Vectors
pressure	$pressure$	—	—	Vectors
time	$t$	s	0 … 1e+05	Vectors
power	$power$	—	—	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
displacement of man during riding along y	$Δ x$	m	—
position of man at end of riding along y	$x_{1}$	m	—
position of man at start of riding along y	$x_{0}$	m	—
average velocity of man during riding along y	$v_{av}$	m/s	—
time taken of riding	$Δ t$	s	0 … 1e+05
average speed of man during riding	$c_{av}$	m/s	—
path length of man during riding	$d$	m	—
change of velocity of man during riding along y	$Δ v$	m/s	—
velocity of man at end of riding from ground along y	$v_{1}$	m/s	—
velocity of man at start of riding from ground along y	$v_{0}$	m/s	—
average acceleration of man during riding along y	$a_{av}$	m/s²	—
acceleration of man during riding along y	$a$	m/s²	—
net force on man during riding along y	$F_{net}$	N	—
mass of man	$m$	kg	0 … ∞
mass of scale	$m_{s}$	kg	0 … ∞
weight on ground	$\| W_{g} \|$	N	—
weight on man	$\| W \|$	N	—
weight on scale	$\| W_{s} \|$	N	—
force on man by scale during riding	$\| F_{s} \|$	N	—
weight on man along y	$W$	N	—
force on man by scale during riding along y	$F_{s}$	N	—
displacement of man during riding	$\| Δ x \|$	m	—
position of man at end of riding	$\| x_{1} \|$	m	—
position of man at start of riding	$\| x_{0} \|$	m	—
average velocity of man during riding	$\| v_{av} \|$	m/s	—
change of velocity of man during riding	$\| Δ v \|$	m/s	—
velocity of man at end of riding from ground	$\| v_{1} \|$	m/s	—
velocity of man at start of riding from ground	$\| v_{0} \|$	m/s	—
average acceleration of man during riding	$\| a_{av} \|$	m/s²	—
acceleration of man during riding	$\| a \|$	m/s²	—
net force on man during riding	$\| F_{net} \|$	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(riding, y)

v_{av} = \frac{Δ x}{Δ t}

displacement a time(riding, y)

c_{av} = \frac{d}{Δ t}

path a time(riding, y)

Δ v = v_{1} - v_{0}

velocity gained(riding, y)

a_{av} = \frac{Δ v}{Δ t}

velocity gained a time(riding, y)

v_{1} = v_{0} + a \cdot Δ t

velocity after a time(riding, y)

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

position after a time(riding, y)

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

velocity after a distance(riding, y)

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

mean of the velocities(riding, y)

F_{net} = m \cdot a

second law(riding, y)

| W | = m \cdot g

weight of a body(man)

F_{net} = F_{s} + W

net force is the resultant(man, riding, y)

| Δ x | = \sqrt{{Δ x}^{2}}

magnitude from components(displacement of man during riding)

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

magnitude from components(position of man at end of riding)

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

magnitude from components(position of man at start of riding)

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

magnitude from components(average velocity of man during riding)

| Δ v | = \sqrt{{Δ v}^{2}}

magnitude from components(change of velocity of man during riding)

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

magnitude from components(velocity of man at end of riding from ground)

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

magnitude from components(velocity of man at start of riding from ground)

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

magnitude from components(average acceleration of man during riding)

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

magnitude from components(acceleration of man during riding)

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

magnitude from components(net force on man during riding)

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

magnitude from components(weight on man)

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

magnitude from components(force on man by scale during riding)

W = - | W |

under gravity

a = - g

freely under gravity

F_{s} = | F_{s} |

stands on a weighing scale

m \neq 0

has mass?(man)

m_{s} \neq 0

has mass?(scale)

F_{net} = 0

free of forces?(man, riding)