Two Forces At Right Angles

Part of eval pr72 v4 branch rerun 2, agent version stated-pi-squared · 5f5be23

solved NewtonsSecondLaw

Q. A body of mass 5 kg $m = 5$ kg is pushed by two forces at right angles to each other $F_{net} = \sqrt{{F_{p1}}^{2} + {F_{p2}}^{2}}$ , $F_{p1,x} = F_{p1}$ , $F_{p1,y} = 0$ , $F_{p2,x} = 0$ , $F_{p2,y} = F_{p2}$ , 8 N $F_{p1} = 8$ N and 6 N $F_{p2} = 6$ N. How strongly does it accelerate, in m/s²?

Inputs

mass

m

5 kg

force1

F_{p1}

8 N

force2

F_{p2}

6 N

Outputs

acceleration

a

How it's solved

Solved with NewtonsSecondLaw

Given

$m = 5$ kg
$F_{p1} = 8$ N
$F_{p2} = 6$ N
$F_{net} = \sqrt{{F_{p1}}^{2} + {F_{p2}}^{2}}$ from “at right angles to each other”
$F_{p1,x} = F_{p1}$ from “at right angles to each other”
$F_{p1,y} = 0$ from “at right angles to each other”
$F_{p2,x} = 0$ from “at right angles to each other”
$F_{p2,y} = F_{p2}$ from “at right angles to each other”

Find $a$ .

Working

at right angles to each other (force on body by pusher1 during pushed along x) $F_{p1,x} = F_{p1}$
$F_{p1,x} = 8$
$F_{p1,x} = 8$ N
at right angles to each other (force on body by pusher2 during pushed along x) $F_{p2,x} = 0$
$F_{p2,x} = 0$ N
net force is the resultant(body, pushed, x) $F_{net,x} = F_{p1,x} + F_{p2,x} + 0$
$F_{net,x} = 8 + 0 + 0$
$F_{net,x} = 8$ N
second law(pushed, x) $F_{net,x} = m \cdot a_{x}$
$8 = 5 \cdot a_{x}$
$a_{x} = 1.6$ m/s²
at right angles to each other (force on body by pusher1 during pushed along y) $F_{p1,y} = 0$
$F_{p1,y} = 0$ N
at right angles to each other (force on body by pusher2 during pushed along y) $F_{p2,y} = F_{p2}$
$F_{p2,y} = 6$
$F_{p2,y} = 6$ N
net force is the resultant(body, pushed, y) $F_{net,y} = F_{p1,y} + F_{p2,y} + 0$
$F_{net,y} = 0 + 6 + 0$
$F_{net,y} = 6$ N
second law(pushed, y) $F_{net,y} = m \cdot a_{y}$
$6 = 5 \cdot a_{y}$
$a_{y} = 1.2$ m/s²
magnitude from components(acceleration of body during pushed) $a = \sqrt{{a_{x}}^{2} + {a_{y}}^{2}}$
$a = \sqrt{{1.6}^{2} + {1.2}^{2}}$
$a = 2$ m/s²

Conclusion

The two forces are perpendicular, so the net force is their resultant, and the acceleration follows from F = ma.

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, %{force1} and %{force2}. " \4            "%{acceleration}, in m/s²?"5  given mass: "5 kg"6  given force1: "8 N"7  given force2: "6 N"8  asks acceleration: "How strongly does it accelerate"9end

solution.rb

1solve :two_forces_at_right_angles, using: NewtonsSecondLaw do2  objects :body, :pusher1, :pusher23  axes :x, :y45  body.goes(:pushed)67  take mass: body.mass8  take force1: body.force(by: pusher1, during: pushed).magnitude9  take force2: body.force(by: pusher2, during: pushed).magnitude10  deduce_from "at right angles to each other", body.net_force(during: pushed).magnitude => sqrt(force1**2 + force2**2)11  deduce_from "at right angles to each other", body.force(by: pusher1, during: pushed).direction => x, body.force(by: pusher2, during: pushed).direction => y1213  acceleration = body.acceleration.magnitude14  answer acceleration:, in: "m/s2"1516  conclusion "The two forces are perpendicular, so the net force is their resultant, and the acceleration follows from F = ma."17end

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
displacement of body during pushed along x	$Δ x_{x}$	m	—
position of body at end of pushed along x	$x_{1,x}$	m	—
position of body at start of pushed along x	$x_{0,x}$	m	—
displacement of body during pushed along y	$Δ x_{y}$	m	—
position of body at end of pushed along y	$x_{1,y}$	m	—
position of body at start of pushed along y	$x_{0,y}$	m	—
average velocity of body during pushed along x	$v_{av,x}$	m/s	—
time taken of pushed	$Δ t$	s	0 … 1e+05
average velocity of body during pushed along y	$v_{av,y}$	m/s	—
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 along x	$Δ v_{x}$	m/s	—
velocity of body at end of pushed from ground along x	$v_{1,x}$	m/s	—
velocity of body at start of pushed from ground along x	$v_{0,x}$	m/s	—
change of velocity of body during pushed along y	$Δ v_{y}$	m/s	—
velocity of body at end of pushed from ground along y	$v_{1,y}$	m/s	—
velocity of body at start of pushed from ground along y	$v_{0,y}$	m/s	—
average acceleration of body during pushed along x	$a_{av,x}$	m/s²	—
average acceleration of body during pushed along y	$a_{av,y}$	m/s²	—
acceleration of body during pushed along x	$a_{x}$	m/s²	—
acceleration of body during pushed along y	$a_{y}$	m/s²	—
net force on body during pushed along x	$F_{net,x}$	N	—
mass of body	$m$	kg	0 … ∞
net force on body during pushed along y	$F_{net,y}$	N	—
mass of pusher1	$m_{p1}$	kg	0 … ∞
mass of pusher2	$m_{p2}$	kg	0 … ∞
weight on ground	$W_{g}$	N	—
weight on body	$W$	N	—
weight on pusher1	$W_{p1}$	N	—
weight on pusher2	$W_{p2}$	N	—
force on body by pusher1 during pushed	$F_{p1}$	N	—
force on body by pusher2 during pushed	$F_{p2}$	N	—
force on body by pusher1 during pushed along x	$F_{p1,x}$	N	—
force on body by pusher2 during pushed along x	$F_{p2,x}$	N	—
force on body by pusher1 during pushed along y	$F_{p1,y}$	N	—
force on body by pusher2 during pushed along y	$F_{p2,y}$	N	—
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	—
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	—

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(pushed, x)

Δ x_{y} = x_{1,y} - x_{0,y}

change of position(pushed, y)

v_{av,x} = \frac{Δ x_{x}}{Δ t}

displacement a time(pushed, x)

v_{av,y} = \frac{Δ x_{y}}{Δ t}

displacement a time(pushed, y)

c_{av} = \frac{d}{Δ t}

path a time(pushed, x)

c_{av} = \frac{d}{Δ t}

path a time(pushed, y)

Δ v_{x} = v_{1,x} - v_{0,x}

velocity gained(pushed, x)

Δ v_{y} = v_{1,y} - v_{0,y}

velocity gained(pushed, y)

a_{av,x} = \frac{Δ v_{x}}{Δ t}

velocity gained a time(pushed, x)

a_{av,y} = \frac{Δ v_{y}}{Δ t}

velocity gained a time(pushed, y)

v_{1,x} = v_{0,x} + a_{x} \cdot Δ t

velocity after a time(pushed, x)

v_{1,y} = v_{0,y} + a_{y} \cdot Δ t

velocity after a time(pushed, 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(pushed, 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(pushed, y)

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

velocity after a distance(pushed, x)

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

velocity after a distance(pushed, y)

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

mean of the velocities(pushed, x)

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

mean of the velocities(pushed, y)

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

second law(pushed, x)

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

second law(pushed, y)

F_{net,x} = F_{p1,x} + F_{p2,x} + 0

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

F_{net,y} = F_{p1,y} + F_{p2,y} + 0

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

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

magnitude from components(displacement of body during pushed)

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

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

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

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

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

magnitude from components(average velocity of body during pushed)

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

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

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

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

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

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

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

magnitude from components(average acceleration of body during pushed)

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

magnitude from components(acceleration of body during pushed)

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

magnitude from components(net force on body during pushed)

F_{p1} = \sqrt{{F_{p1,x}}^{2} + {F_{p1,y}}^{2}}

magnitude from components(force on body by pusher1 during pushed)

F_{p2} = \sqrt{{F_{p2,x}}^{2} + {F_{p2,y}}^{2}}

magnitude from components(force on body by pusher2 during pushed)

F_{net} = \sqrt{{F_{p1}}^{2} + {F_{p2}}^{2}}

at right angles to each other (net force on body during pushed)

F_{p1,x} = F_{p1}

at right angles to each other (force on body by pusher1 during pushed along x)

F_{p1,y} = 0

at right angles to each other (force on body by pusher1 during pushed along y)

F_{p2,x} = 0

at right angles to each other (force on body by pusher2 during pushed along x)

F_{p2,y} = F_{p2}

at right angles to each other (force on body by pusher2 during pushed along y)

m \neq 0

has mass?(body)

m_{p1} \neq 0

has mass?(pusher1)

m_{p2} \neq 0

has mass?(pusher2)

F_{net,x} = 0

free of forces?(body, pushed, x)

F_{net,y} = 0

free of forces?(body, pushed, y)