Two Forces At Right Angles

Part of eval multi-llm main, agent version f0f5267

solved NewtonsSecondLaw

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

Inputs

mass

m

5 kg

force a

F_{pa}

8 N

force b

F_{pb}

6 N

Outputs

acceleration

a

How it's solved

Solved with NewtonsSecondLaw

Given

$m = 5$ kg
$F_{pa} = 8$ N
$F_{pb} = 6$ N
$θ_{pa;pb} = 90 °$ from “at right angles to each other”

Find $a$ .

Working

net force is the resultant(body, accelerating, 1) ${F_{net}}^{2} = {F_{pa}}^{2} + {F_{pb}}^{2} + 2 \cdot F_{pa} \cdot F_{pb} \cdot \cos θ_{pa;pb}$
${F_{net}}^{2} = 8^{2} + 6^{2} + 2 \cdot 8 \cdot 6 \cdot \cos 90 °$
$F_{net} = 10$ N
second law in size(body, accelerating) $F_{net} = m \cdot a$
$10 = 5 \cdot a$
$a = 2$ m/s²

Conclusion

The two forces act at right angles, so their resultant is the hypotenuse of a right triangle; Newton's second law gives the acceleration.

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

1question :two_forces_at_right_angles do2  kind :numerical3  describes "A body of mass %{mass} is pushed by two forces at right angles to each other, %{force_a} and %{force_b}. %{acceleration}, in m/s²?"4  given mass: "5 kg"5  given force_a: "8 N"6  given force_b: "6 N"7  asks acceleration: "How strongly does it accelerate"8end

solution.rb

1solve :two_forces_at_right_angles, using: NewtonsSecondLaw do2  objects :body, :pusher_a, :pusher_b3  axes :x, :y4  body.goes(:accelerating)56  take mass: body.mass7  take force_a: body.force(by: pusher_a).magnitude8  take force_b: body.force(by: pusher_b).magnitude910  deduce_from "at right angles to each other", body.force(by: pusher_a).perpendicular_to(body.force(by: pusher_b))1112  net_force = body.net_force.magnitude13  acceleration = body.acceleration.magnitude1415  answer acceleration:, in: "m/s2"1617  conclusion "The two forces act at right angles, so their resultant is the hypotenuse of a right triangle; Newton's second law gives the acceleration."18end

playground.rb

1play_with :two_forces_at_right_angles do2  title "Two Forces At Right Angles"3end

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 pusher a during accelerating and force on body by pusher b during accelerating	$θ_{pa;pb}$	—	0 … 3.142
displacement of body during accelerating along x	$Δ x_{x}$	m	—
position of body at end of accelerating along x	$x_{1,x}$	m	—
position of body at start of accelerating along x	$x_{0,x}$	m	—
displacement of body during accelerating along y	$Δ x_{y}$	m	—
position of body at end of accelerating along y	$x_{1,y}$	m	—
position of body at start of accelerating along y	$x_{0,y}$	m	—
average velocity of body during accelerating along x	$v_{av,x}$	m/s	—
time taken of accelerating	$Δ t$	s	0 … 1e+05
average velocity of body during accelerating along y	$v_{av,y}$	m/s	—
average speed of body during accelerating	$c_{av}$	m/s	—
path length of body during accelerating	$d$	m	—
change of velocity of body during accelerating along x	$Δ v_{x}$	m/s	—
velocity of body at end of accelerating from ground along x	$v_{1,x}$	m/s	—
velocity of body at start of accelerating from ground along x	$v_{0,x}$	m/s	—
change of velocity of body during accelerating along y	$Δ v_{y}$	m/s	—
velocity of body at end of accelerating from ground along y	$v_{1,y}$	m/s	—
velocity of body at start of accelerating from ground along y	$v_{0,y}$	m/s	—
average acceleration of body during accelerating along x	$a_{av,x}$	m/s²	—
average acceleration of body during accelerating along y	$a_{av,y}$	m/s²	—
acceleration of body during accelerating along x	$a_{x}$	m/s²	—
acceleration of body during accelerating along y	$a_{y}$	m/s²	—
net force on body during accelerating along x	$F_{net,x}$	N	—
mass of body	$m$	kg	0 … ∞
net force on body during accelerating along y	$F_{net,y}$	N	—
mass of pusher a	$m_{pa}$	kg	0 … ∞
mass of pusher b	$m_{pb}$	kg	0 … ∞
weight on ground	$W_{g}$	N	—
weight on body	$W$	N	—
weight on pusher a	$W_{pa}$	N	—
weight on pusher b	$W_{pb}$	N	—
force on body by pusher a during accelerating	$F_{pa}$	N	—
force on body by pusher b during accelerating	$F_{pb}$	N	—
net force on body during accelerating	$F_{net}$	N	—
force on body by pusher a during accelerating along x	$F_{pa,x}$	N	—
force on body by pusher b during accelerating along x	$F_{pb,x}$	N	—
force on body by pusher a during accelerating along y	$F_{pa,y}$	N	—
force on body by pusher b during accelerating along y	$F_{pb,y}$	N	—
acceleration of body during accelerating	$a$	m/s²	—
displacement of body during accelerating	$Δ x$	m	—
position of body at end of accelerating	$x_{1}$	m	—
position of body at start of accelerating	$x_{0}$	m	—
average velocity of body during accelerating	$v_{av}$	m/s	—
change of velocity of body during accelerating	$Δ v$	m/s	—
velocity of body at end of accelerating from ground	$v_{1}$	m/s	—
velocity of body at start of accelerating from ground	$v_{0}$	m/s	—
average acceleration of body during accelerating	$a_{av}$	m/s²	—

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} = x_{1,x} - x_{0,x}

change of position(accelerating, x)

Δ x_{y} = x_{1,y} - x_{0,y}

change of position(accelerating, y)

v_{av,x} = \frac{Δ x_{x}}{Δ t}

displacement a time(accelerating, x)

v_{av,y} = \frac{Δ x_{y}}{Δ t}

displacement a time(accelerating, y)

c_{av} = \frac{d}{Δ t}

path a time(accelerating, x)

c_{av} = \frac{d}{Δ t}

path a time(accelerating, y)

Δ v_{x} = v_{1,x} - v_{0,x}

velocity gained(accelerating, x)

Δ v_{y} = v_{1,y} - v_{0,y}

velocity gained(accelerating, y)

a_{av,x} = \frac{Δ v_{x}}{Δ t}

velocity gained a time(accelerating, x)

a_{av,y} = \frac{Δ v_{y}}{Δ t}

velocity gained a time(accelerating, y)

v_{1,x} = v_{0,x} + a_{x} \cdot Δ t

velocity after a time(accelerating, x)

v_{1,y} = v_{0,y} + a_{y} \cdot Δ t

velocity after a time(accelerating, y)

x_{1,x} = x_{0,x} + v_{0,x} \cdot Δ t + \frac{a_{x} \cdot {Δ t}^{2}}{2}

position after a time(accelerating, x)

x_{1,y} = x_{0,y} + v_{0,y} \cdot Δ t + \frac{a_{y} \cdot {Δ t}^{2}}{2}

position after a time(accelerating, y)

{v_{1,x}}^{2} = {v_{0,x}}^{2} + 2 \cdot a_{x} \cdot (x_{1,x} - x_{0,x})

velocity after a distance(accelerating, x)

{v_{1,y}}^{2} = {v_{0,y}}^{2} + 2 \cdot a_{y} \cdot (x_{1,y} - x_{0,y})

velocity after a distance(accelerating, y)

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

mean of the velocities(accelerating, x)

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

mean of the velocities(accelerating, y)

F_{net,x} = m \cdot a_{x}

second law(accelerating, x)

F_{net,y} = m \cdot a_{y}

second law(accelerating, y)

{F_{net}}^{2} = {F_{pa}}^{2} + {F_{pb}}^{2} + 2 \cdot F_{pa} \cdot F_{pb} \cdot \cos θ_{pa;pb}

net force is the resultant(body, accelerating, 1)

F_{net,x} = F_{pa,x} + F_{pb,x} + 0

net force is the resultant(body, accelerating, x)

F_{net,y} = F_{pa,y} + F_{pb,y} + 0

net force is the resultant(body, accelerating, y)

F_{net} = m \cdot a

second law in size(body, accelerating)

Δ x = \sqrt{{Δ x_{x}}^{2} + {Δ x_{y}}^{2}}

magnitude from components(displacement of body during accelerating)

x_{1} = \sqrt{{x_{1,x}}^{2} + {x_{1,y}}^{2}}

magnitude from components(position of body at end of accelerating)

x_{0} = \sqrt{{x_{0,x}}^{2} + {x_{0,y}}^{2}}

magnitude from components(position of body at start of accelerating)

v_{av} = \sqrt{{v_{av,x}}^{2} + {v_{av,y}}^{2}}

magnitude from components(average velocity of body during accelerating)

Δ v = \sqrt{{Δ v_{x}}^{2} + {Δ v_{y}}^{2}}

magnitude from components(change of velocity of body during accelerating)

v_{1} = \sqrt{{v_{1,x}}^{2} + {v_{1,y}}^{2}}

magnitude from components(velocity of body at end of accelerating from ground)

v_{0} = \sqrt{{v_{0,x}}^{2} + {v_{0,y}}^{2}}

magnitude from components(velocity of body at start of accelerating from ground)

a_{av} = \sqrt{{a_{av,x}}^{2} + {a_{av,y}}^{2}}

magnitude from components(average acceleration of body during accelerating)

a = \sqrt{{a_{x}}^{2} + {a_{y}}^{2}}

magnitude from components(acceleration of body during accelerating)

F_{net} = \sqrt{{F_{net,x}}^{2} + {F_{net,y}}^{2}}

magnitude from components(net force on body during accelerating)

F_{pa} = \sqrt{{F_{pa,x}}^{2} + {F_{pa,y}}^{2}}

magnitude from components(force on body by pusher a during accelerating)

F_{pb} = \sqrt{{F_{pb,x}}^{2} + {F_{pb,y}}^{2}}

magnitude from components(force on body by pusher b during accelerating)

m \neq 0

has mass?(body)

m_{pa} \neq 0

has mass?(pusher a)

m_{pb} \neq 0

has mass?(pusher b)

F_{net,x} = 0

free of forces?(body, accelerating, x)

F_{net,y} = 0

free of forces?(body, accelerating, y)