Two Forces At Right Angles

Part of eval multi-llm branch, agent version multi-llm-servers · 2b11b97

Q. A body of 5 kg is pushed by two forces at right angles to each other, 8 N and 6 N. How strongly does it accelerate, in m/s²? expects 2 m/s²

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draft 1: agreed

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question.rb1 question :two_forces_at_right_angles do2   kind :numerical3   describes "A body of %{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"9 end

solution.rb1 solve :two_forces_at_right_angles, using: NewtonsSecondLaw do2   objects :body3   axes :x, :y4 5   body.goes(:pushed)6 7   take mass: body.mass8   take force1: body.force(by: :pusher1, during: pushed).magnitude9   take force2: body.force(by: :pusher2, during: pushed).magnitude10 11   deduce_from "at right angles to each other", body.force(by: :pusher1, during: pushed).direction => x,12               body.force(by: :pusher2, during: pushed).direction => y13 14   acceleration = body.acceleration.magnitude15   answer acceleration:, in: "m/s2", figures: 116 17   conclusion "The two forces are perpendicular, so the resultant is the hypotenuse of the right triangle they form; " \18              "Newton's second law gives the acceleration from the resultant force and the mass."19 end

round 1

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

The playground

Inputs

mass

m

5 kg

force1

F_{p}

8 N

force2

F_{p}

6 N

Outputs

acceleration

a

The solution

Solved with NewtonsSecondLaw

Given

$m = 5$ kg
$F_{p} = 8$ N
$F_{p} = 6$ N
$F_{p,x} = F_{p}$ from “at right angles to each other”
$F_{p,y} = 0$ from “at right angles to each other”
$F_{p,x} = 0$ from “at right angles to each other”
$F_{p,y} = F_{p}$ 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) $F_{p,x} = F_{p}$
$F_{p,x} = 8$
$F_{p,x} = 8$ N
at right angles to each other (force on body by pusher2 during pushed along x) $F_{p,x} = 0$
$F_{p,x} = 0$ N
net force is the resultant(body, pushed, x) $F_{net,x} = F_{p,x} + F_{p,x} + 0$
$F_{net,x} = 8 + 0 + 0$
$F_{net,x} = 8$ 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) $F_{p,y} = 0$
$F_{p,y} = 0$ N
at right angles to each other (force on body by pusher2 during pushed along y) $F_{p,y} = F_{p}$
$F_{p,y} = 6$
$F_{p,y} = 6$ N
net force is the resultant(body, pushed, y) $F_{net,y} = F_{p,y} + F_{p,y} + 0$
$F_{net,y} = 0 + 6 + 0$
$F_{net,y} = 6$ 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
$a \approx 2$ m/s²to 1 significant figure

Conclusion

The two forces are perpendicular, so the resultant is the hypotenuse of the right triangle they form; Newton's second law gives the acceleration from the resultant force and the mass.

The Ruby

1question :two_forces_at_right_angles do2  kind :numerical3  describes "A body of %{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

1solve :two_forces_at_right_angles, using: NewtonsSecondLaw do2  objects :body3  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).magnitude1011  deduce_from "at right angles to each other", body.force(by: :pusher1, during: pushed).direction => x,12              body.force(by: :pusher2, during: pushed).direction => y1314  acceleration = body.acceleration.magnitude15  answer acceleration:, in: "m/s2", figures: 11617  conclusion "The two forces are perpendicular, so the resultant is the hypotenuse of the right triangle they form; " \18             "Newton's second law gives the acceleration from the resultant force and the mass."19end

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

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 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	—
weight on ground	$W_{g}$	N	—
weight on body	$W$	N	—
force on body by pusher1 during pushed	$F_{p}$	N	—
force on body by pusher2 during pushed	$F_{p}$	N	—
force on body by pusher1 during pushed along x	$F_{p,x}$	N	—
force on body by pusher2 during pushed along x	$F_{p,x}$	N	—
force on body by pusher1 during pushed along y	$F_{p,y}$	N	—
force on body by pusher2 during pushed along y	$F_{p,y}$	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_{p,x} + F_{p,x} + 0

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

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

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

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

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

F_{p,x} = F_{p}

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

F_{p,y} = 0

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

F_{p,x} = 0

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

F_{p,y} = F_{p}

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

m \neq 0

has mass?(body)

F_{net,x} = 0

free of forces?(body, pushed, x)

F_{net,y} = 0

free of forces?(body, pushed, y)