A Lift Falling Freely

Q. A man of mass 70 kg stands on a weighing scale in a lift whose mechanism has failed, so that it hurtles down freely under gravity. Taking g as 10 m/s², what would be the reading on the scale, in newtons? expects 0 N

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The web of concepts

given read from the words equation asked

The equations it used

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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?

The playground

Inputs

mass

m

70 kg

gravity

g

10 m/s²

Outputs

reading

| F_{s} |

The solution

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 Ruby

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

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

1require_relative "../solutions/a_lift_falling_freely"23play_with :a_lift_falling_freely do4  title "A Lift Falling Freely"5end

The laws

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)