Pulling the Heavier Body

solved NewtonsSecondLaw

Q. Two bodies A and B of masses 10 kg $m_{A} = 10$ kg and 20 kg $m_{B} = 20$ kg respectively kept on a smooth, horizontal surface are tied to the ends $m_{AB} = m_{A} + m_{B}$ of a light string $a_{A} (t) = a_{AB} (p)$ . A horizontal force F = 600 N $| F_{ABp} (p) | = 600$ N is applied to B along the direction of the string $F_{ABp} (p) = | F_{ABp} (p) |$ . What is the tension in the string, in newtons?

Inputs

mass a

m_{A}

10 kg

mass b

m_{B}

20 kg

force

| F_{ABp} |

600 N

Outputs

tension

| F_{As} |

How it's solved

Solved with NewtonsSecondLaw

Given

$m_{A} = 10$ kg
$m_{B} = 20$ kg
$| F_{ABp} (p) | = 600$ N
$m_{AB} = m_{A} + m_{B}$ from “tied to the ends”
$a_{A} (t) = a_{AB} (p)$ from “a light string”
$F_{ABp} (p) = | F_{ABp} (p) |$ from “along the direction of the string”

Find $| F_{As} (t) |$ .

Working

along the direction of the string $F_{ABp} (p) = | F_{ABp} (p) |$
$F_{ABp} (p) = 600$
$F_{ABp} (p) = 600$ N
net force is the resultant(pair, pulled, x) $F_{net,AB} (p) = F_{ABp} (p) + 0$
$F_{net,AB} (p) = 600 + 0$
$F_{net,AB} (p) = 600$ N
tied to the ends $m_{AB} = m_{A} + m_{B}$
$m_{AB} = 10 + 20$
$m_{AB} = 30$ kg
second law(pulled, x) $F_{net,AB} (p) = m_{AB} \cdot a_{AB} (p)$
$600 = 30 \cdot a_{AB} (p)$
$a_{AB} (p) = 20$ m/s²
a light string $a_{A} (t) = a_{AB} (p)$
$a_{A} (t) = 20$
$a_{A} (t) = 20$ m/s²
second law(towed, x) $F_{net,A} (t) = m_{A} \cdot a_{A} (t)$
$F_{net,A} (t) = 10 \cdot 20$
$F_{net,A} (t) = 200$ N
net force is the resultant(body a, towed, x) $F_{net,A} (t) = F_{As} (t) + 0$
$200 = F_{As} (t) + 0$
$F_{As} (t) = 200$ N
magnitude from components(force on body a by string during towed) $| F_{As} (t) | = \sqrt{{F_{As} (t)}^{2}}$
$| F_{As} (t) | = \sqrt{200^{2}}$
$| F_{As} (t) | = 200$ N

Conclusion

Now the string has only the lighter body to drag along, so it needs to pull less hard to keep it moving with the other.

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 :pulling_the_heavier_body do4  kind :numerical5  describes "Two bodies A and B of masses %{mass_a} and %{mass_b} respectively kept on a smooth, horizontal " \6            "surface are tied to the ends of a light string. A horizontal force F = %{force} is applied to B " \7            "along the direction of the string. %{tension}, in newtons?"8  given mass_a: "10 kg"9  given mass_b: "20 kg"10  given force: "600 N"11  asks tension: "What is the tension in the string"12end

solution.rb

1require_relative "../questions/pulling_the_heavier_body"2require_relative "../science/laws_of_motion/newtons_second_law"34solve :pulling_the_heavier_body, using: NewtonsSecondLaw do5  objects :string, :puller, body_a: :A, body_b: :B, pair: :AB6  axes :x7  pair.goes(:pulled)8  body_a.goes(:towed)910  take mass_a: body_a.mass11  take mass_b: body_b.mass12  take force: pair.force(by: puller).magnitude1314  deduce_from "tied to the ends", pair.mass => body_a.mass + body_b.mass15  deduce_from "a light string", body_a.acceleration => pair.acceleration16  deduce_from "along the direction of the string", pair.force(by: puller).direction => x1718  tension = body_a.force(by: string).magnitude19  answer tension:, in: "N"2021  conclusion "Now the string has only the lighter body to drag along, so it needs to pull less hard to keep " \22             "it moving with the other."23end

playground.rb

1require_relative "../solutions/pulling_the_heavier_body"23play_with :pulling_the_heavier_body do4  title "Pulling the Heavier Body"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 pair during pulled along x	$Δ x_{AB} (p)$	m	—
position of pair at end of pulled along x	$x_{AB} (p₁)$	m	—
position of pair at start of pulled along x	$x_{AB} (p₀)$	m	—
displacement of body a during towed along x	$Δ x_{A} (t)$	m	—
position of body a at end of towed along x	$x_{A} (t₁)$	m	—
position of body a at start of towed along x	$x_{A} (t₀)$	m	—
average velocity of pair during pulled along x	$v_{av,AB} (p)$	m/s	—
time taken of pulled	$Δ t_{p}$	s	0 … 1e+05
average velocity of body a during towed along x	$v_{av,A} (t)$	m/s	—
time taken of towed	$Δ t_{t}$	s	0 … 1e+05
average speed of pair during pulled	$c_{av,AB} (p)$	m/s	—
path length of pair during pulled	$d_{AB} (p)$	m	—
average speed of body a during towed	$c_{av,A} (t)$	m/s	—
path length of body a during towed	$d_{A} (t)$	m	—
change of velocity of pair during pulled along x	$Δ v_{AB} (p)$	m/s	—
velocity of pair at end of pulled from ground along x	$v_{AB} (p₁)$	m/s	—
velocity of pair at start of pulled from ground along x	$v_{AB} (p₀)$	m/s	—
change of velocity of body a during towed along x	$Δ v_{A} (t)$	m/s	—
velocity of body a at end of towed from ground along x	$v_{A} (t₁)$	m/s	—
velocity of body a at start of towed from ground along x	$v_{A} (t₀)$	m/s	—
average acceleration of pair during pulled along x	$a_{av,AB} (p)$	m/s²	—
average acceleration of body a during towed along x	$a_{av,A} (t)$	m/s²	—
acceleration of pair during pulled along x	$a_{AB} (p)$	m/s²	—
acceleration of body a during towed along x	$a_{A} (t)$	m/s²	—
net force on pair during pulled along x	$F_{net,AB} (p)$	N	—
mass of pair	$m_{AB}$	kg	0 … ∞
net force on body a during towed along x	$F_{net,A} (t)$	N	—
mass of body a	$m_{A}$	kg	0 … ∞
mass of string	$m_{s}$	kg	0 … ∞
mass of puller	$m_{p}$	kg	0 … ∞
mass of body b	$m_{B}$	kg	0 … ∞
weight on ground	$\| W_{g} \|$	N	—
weight on string	$\| W_{s} \|$	N	—
weight on puller	$\| W_{p} \|$	N	—
weight on body a	$\| W_{A} \|$	N	—
weight on body b	$\| W_{B} \|$	N	—
weight on pair	$\| W_{AB} \|$	N	—
force on body a by string during towed	$\| F_{As} (t) \|$	N	—
force on body a by string during towed along x	$F_{As} (t)$	N	—
force on pair by puller during pulled	$\| F_{ABp} (p) \|$	N	—
force on pair by puller during pulled along x	$F_{ABp} (p)$	N	—
displacement of pair during pulled	$\| Δ x_{AB} (p) \|$	m	—
position of pair at end of pulled	$\| x_{AB} (p₁) \|$	m	—
position of pair at start of pulled	$\| x_{AB} (p₀) \|$	m	—
displacement of body a during towed	$\| Δ x_{A} (t) \|$	m	—
position of body a at end of towed	$\| x_{A} (t₁) \|$	m	—
position of body a at start of towed	$\| x_{A} (t₀) \|$	m	—
average velocity of pair during pulled	$\| v_{av,AB} (p) \|$	m/s	—
average velocity of body a during towed	$\| v_{av,A} (t) \|$	m/s	—
change of velocity of pair during pulled	$\| Δ v_{AB} (p) \|$	m/s	—
velocity of pair at end of pulled from ground	$\| v_{AB} (p₁) \|$	m/s	—
velocity of pair at start of pulled from ground	$\| v_{AB} (p₀) \|$	m/s	—
change of velocity of body a during towed	$\| Δ v_{A} (t) \|$	m/s	—
velocity of body a at end of towed from ground	$\| v_{A} (t₁) \|$	m/s	—
velocity of body a at start of towed from ground	$\| v_{A} (t₀) \|$	m/s	—
average acceleration of pair during pulled	$\| a_{av,AB} (p) \|$	m/s²	—
average acceleration of body a during towed	$\| a_{av,A} (t) \|$	m/s²	—
acceleration of pair during pulled	$\| a_{AB} (p) \|$	m/s²	—
acceleration of body a during towed	$\| a_{A} (t) \|$	m/s²	—
net force on pair during pulled	$\| F_{net,AB} (p) \|$	N	—
net force on body a during towed	$\| F_{net,A} (t) \|$	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_{AB} (p) = x_{AB} (p₁) - x_{AB} (p₀)

change of position(pulled, x)

Δ x_{A} (t) = x_{A} (t₁) - x_{A} (t₀)

change of position(towed, x)

v_{av,AB} (p) = \frac{Δ x_{AB} (p)}{Δ t_{p}}

displacement a time(pulled, x)

v_{av,A} (t) = \frac{Δ x_{A} (t)}{Δ t_{t}}

displacement a time(towed, x)

c_{av,AB} (p) = \frac{d_{AB} (p)}{Δ t_{p}}

path a time(pulled, x)

c_{av,A} (t) = \frac{d_{A} (t)}{Δ t_{t}}

path a time(towed, x)

Δ v_{AB} (p) = v_{AB} (p₁) - v_{AB} (p₀)

velocity gained(pulled, x)

Δ v_{A} (t) = v_{A} (t₁) - v_{A} (t₀)

velocity gained(towed, x)

a_{av,AB} (p) = \frac{Δ v_{AB} (p)}{Δ t_{p}}

velocity gained a time(pulled, x)

a_{av,A} (t) = \frac{Δ v_{A} (t)}{Δ t_{t}}

velocity gained a time(towed, x)

v_{AB} (p₁) = v_{AB} (p₀) + a_{AB} (p) \cdot Δ t_{p}

velocity after a time(pulled, x)

v_{A} (t₁) = v_{A} (t₀) + a_{A} (t) \cdot Δ t_{t}

velocity after a time(towed, x)

x_{AB} (p₁) = x_{AB} (p₀) + v_{AB} (p₀) \cdot Δ t_{p} + \frac{a_{AB} (p) \cdot {Δ t_{p}}^{2}}{2}

position after a time(pulled, x)

x_{A} (t₁) = x_{A} (t₀) + v_{A} (t₀) \cdot Δ t_{t} + \frac{a_{A} (t) \cdot {Δ t_{t}}^{2}}{2}

position after a time(towed, x)

{v_{AB} (p₁)}^{2} = {v_{AB} (p₀)}^{2} + 2 \cdot a_{AB} (p) \cdot (x_{AB} (p₁) - x_{AB} (p₀))

velocity after a distance(pulled, x)

{v_{A} (t₁)}^{2} = {v_{A} (t₀)}^{2} + 2 \cdot a_{A} (t) \cdot (x_{A} (t₁) - x_{A} (t₀))

velocity after a distance(towed, x)

x_{AB} (p₁) - x_{AB} (p₀) = \frac{(v_{AB} (p₁) + v_{AB} (p₀)) \cdot Δ t_{p}}{2}

mean of the velocities(pulled, x)

x_{A} (t₁) - x_{A} (t₀) = \frac{(v_{A} (t₁) + v_{A} (t₀)) \cdot Δ t_{t}}{2}

mean of the velocities(towed, x)

F_{net,AB} (p) = m_{AB} \cdot a_{AB} (p)

second law(pulled, x)

F_{net,A} (t) = m_{A} \cdot a_{A} (t)

second law(towed, x)

F_{net,A} (t) = F_{As} (t) + 0

net force is the resultant(body a, towed, x)

F_{net,AB} (p) = F_{ABp} (p) + 0

net force is the resultant(pair, pulled, x)

| Δ x_{AB} (p) | = \sqrt{{Δ x_{AB} (p)}^{2}}

magnitude from components(displacement of pair during pulled)

| x_{AB} (p₁) | = \sqrt{{x_{AB} (p₁)}^{2}}

magnitude from components(position of pair at end of pulled)

| x_{AB} (p₀) | = \sqrt{{x_{AB} (p₀)}^{2}}

magnitude from components(position of pair at start of pulled)

| Δ x_{A} (t) | = \sqrt{{Δ x_{A} (t)}^{2}}

magnitude from components(displacement of body a during towed)

| x_{A} (t₁) | = \sqrt{{x_{A} (t₁)}^{2}}

magnitude from components(position of body a at end of towed)

| x_{A} (t₀) | = \sqrt{{x_{A} (t₀)}^{2}}

magnitude from components(position of body a at start of towed)

| v_{av,AB} (p) | = \sqrt{{v_{av,AB} (p)}^{2}}

magnitude from components(average velocity of pair during pulled)

| v_{av,A} (t) | = \sqrt{{v_{av,A} (t)}^{2}}

magnitude from components(average velocity of body a during towed)

| Δ v_{AB} (p) | = \sqrt{{Δ v_{AB} (p)}^{2}}

magnitude from components(change of velocity of pair during pulled)

| v_{AB} (p₁) | = \sqrt{{v_{AB} (p₁)}^{2}}

magnitude from components(velocity of pair at end of pulled from ground)

| v_{AB} (p₀) | = \sqrt{{v_{AB} (p₀)}^{2}}

magnitude from components(velocity of pair at start of pulled from ground)

| Δ v_{A} (t) | = \sqrt{{Δ v_{A} (t)}^{2}}

magnitude from components(change of velocity of body a during towed)

| v_{A} (t₁) | = \sqrt{{v_{A} (t₁)}^{2}}

magnitude from components(velocity of body a at end of towed from ground)

| v_{A} (t₀) | = \sqrt{{v_{A} (t₀)}^{2}}

magnitude from components(velocity of body a at start of towed from ground)

| a_{av,AB} (p) | = \sqrt{{a_{av,AB} (p)}^{2}}

magnitude from components(average acceleration of pair during pulled)

| a_{av,A} (t) | = \sqrt{{a_{av,A} (t)}^{2}}

magnitude from components(average acceleration of body a during towed)

| a_{AB} (p) | = \sqrt{{a_{AB} (p)}^{2}}

magnitude from components(acceleration of pair during pulled)

| a_{A} (t) | = \sqrt{{a_{A} (t)}^{2}}

magnitude from components(acceleration of body a during towed)

| F_{net,AB} (p) | = \sqrt{{F_{net,AB} (p)}^{2}}

magnitude from components(net force on pair during pulled)

| F_{net,A} (t) | = \sqrt{{F_{net,A} (t)}^{2}}

magnitude from components(net force on body a during towed)

| F_{As} (t) | = \sqrt{{F_{As} (t)}^{2}}

magnitude from components(force on body a by string during towed)

| F_{ABp} (p) | = \sqrt{{F_{ABp} (p)}^{2}}

magnitude from components(force on pair by puller during pulled)

m_{AB} = m_{A} + m_{B}

tied to the ends

a_{A} (t) = a_{AB} (p)

a light string

F_{ABp} (p) = | F_{ABp} (p) |

along the direction of the string

m_{s} \neq 0

has mass?(string)

m_{p} \neq 0

has mass?(puller)

m_{A} \neq 0

has mass?(body a)

m_{B} \neq 0

has mass?(body b)

m_{AB} \neq 0

has mass?(pair)

F_{net,A} (t) = 0

free of forces?(body a, towed)

F_{net,AB} (p) = 0

free of forces?(pair, pulled)