Pushed Two Ways — phryzby

solved NewtonsSecondLaw

Q. A body of 5 kg $m = 5$ kg is pushed by two forces at right angles $θ_{f;s} = 90 °$ to each other, 8 N $F_{f} = 8$ N and 6 N $F_{s} = 6$ N. How strongly does it accelerate, in m/s²?

Inputs

mass

m

5 kg

first force

F_{f}

8 N

second force

F_{s}

6 N

Outputs

acceleration

a

How it's solved

Solved with NewtonsSecondLaw

Given

$m = 5$ kg
$F_{f} = 8$ N
$F_{s} = 6$ N
$θ_{f;s} = 90 °$ from “at right angles”
$F_{net}$ : the positive one of the values that fit from “How strongly”

Find $a$ .

Working

net force is the resultant(body, pushed) ${F_{net}}^{2} = {F_{f}}^{2} + {F_{s}}^{2} + 2 \cdot F_{f} \cdot F_{s} \cdot \cos θ_{f;s}$
${F_{net}}^{2} = 8^{2} + 6^{2} + 2 \cdot 8 \cdot 6 \cdot \cos 90 °$
$F_{net} = −10, 10$ Nthe positive one is taken
$F_{net} = 10$ N
second law(pushed) $F_{net} = m \cdot a$
$10 = 5 \cdot a$
$a = 2$ m/s²

Conclusion

At right angles the two forces add like the sides of a right triangle, and the body accelerates along the hypotenuse at that force over its mass.

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 :pushed_two_ways do4  kind :numerical5  describes "A body of %{mass} is pushed by two forces at right angles to each other, %{first_force} and " \6            "%{second_force}. %{acceleration}, in m/s²?"7  given mass: "5 kg"8  given first_force: "8 N"9  given second_force: "6 N"10  asks acceleration: "How strongly does it accelerate"11end

solution.rb

1require_relative "../questions/pushed_two_ways"2require_relative "../science/laws_of_motion/newtons_second_law"34solve :pushed_two_ways, using: NewtonsSecondLaw do5  objects :body, :first_pusher, :second_pusher6  body.goes(:pushed)78  take mass: body.mass9  take first_force: body.force(by: first_pusher).magnitude10  take second_force: body.force(by: second_pusher).magnitude1112  deduce_from "at right angles", first_force.perpendicular_to(second_force)13  deduce_from "How strongly", body.net_force => :positive1415  acceleration = body.acceleration.magnitude16  answer acceleration:, in: "m/s2"1718  conclusion "At right angles the two forces add like the sides of a right triangle, and the body " \19             "accelerates along the hypotenuse at that force over its mass."20end

playground.rb

1require_relative "../solutions/pushed_two_ways"23play_with :pushed_two_ways do4  title "Pushed Two Ways"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
angle between force on body by first pusher during pushed and force on body by second pusher during pushed	$θ_{f;s}$	—	0 … 3.142
displacement of body during pushed	$Δ x$	m	—
position of body at end of pushed	$x_{1}$	m	—
position of body at start of pushed	$x_{0}$	m	—
average velocity of body during pushed	$v_{av}$	m/s	—
time taken of pushed	$Δ t$	s	0 … 1e+05
average speed of body during pushed	$c_{av}$	m/s	—
path length of body during pushed	$d$	m	—
change of velocity of body during pushed	$Δ v$	m/s	—
velocity of body at end of pushed from ground	$v_{1}$	m/s	—
velocity of body at start of pushed from ground	$v_{0}$	m/s	—
average acceleration of body during pushed	$a_{av}$	m/s²	—
acceleration of body during pushed	$a$	m/s²	—
net force on body during pushed	$F_{net}$	N	—
mass of body	$m$	kg	0 … ∞
mass of first pusher	$m_{f}$	kg	0 … ∞
mass of second pusher	$m_{s}$	kg	0 … ∞
weight on ground	$W_{g}$	N	—
weight on body	$W$	N	—
weight on first pusher	$W_{f}$	N	—
weight on second pusher	$W_{s}$	N	—
force on body by first pusher during pushed	$F_{f}$	N	—
force on body by second pusher during pushed	$F_{s}$	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(pushed)

v_{av} = \frac{Δ x}{Δ t}

displacement a time(pushed)

c_{av} = \frac{d}{Δ t}

path a time(pushed)

Δ v = v_{1} - v_{0}

velocity gained(pushed)

a_{av} = \frac{Δ v}{Δ t}

velocity gained a time(pushed)

v_{1} = v_{0} + a \cdot Δ t

velocity after a time(pushed)

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

position after a time(pushed)

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

velocity after a distance(pushed)

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

mean of the velocities(pushed)

F_{net} = m \cdot a

second law(pushed)

{F_{net}}^{2} = {F_{f}}^{2} + {F_{s}}^{2} + 2 \cdot F_{f} \cdot F_{s} \cdot \cos θ_{f;s}

net force is the resultant(body, pushed)

m \neq 0

has mass?(body)

m_{f} \neq 0

has mass?(first pusher)

m_{s} \neq 0

has mass?(second pusher)

F_{net} = 0

free of forces?(body, pushed)