A Force Along a Constrained Path

solved Work

Q. A body constrained to move along the z-axis $Δ x_{x} = 0$ m, $Δ x_{y} = 0$ m of a coordinate system is subject to a constant force F given by F = (-1 N $F_{a,x} = -1$ N) î + (2 N $F_{a,y} = 2$ N) ĵ + (3 N $F_{a,z} = 3$ N) k̂, where î, ĵ, k̂ are unit vectors along the x-, y- and z-axis of the system respectively. What is the work done by this force in moving the body a distance of 4 m $Δ x_{z} = 4$ m along the z-axis, in joules?

Inputs

F_{a,x}

-1 N

F_{a,y}

2 N

F_{a,z}

3 N

distance

Δ x_{z}

4.00 m

Outputs

work

W_{F,a}

How it's solved

Solved with Work

Given

$F_{a,x} = -1$ N
$F_{a,y} = 2$ N
$F_{a,z} = 3$ N
$Δ x_{z} = 4$ m
$Δ x_{x} = 0$ m from “constrained to move along the z-axis”
$Δ x_{y} = 0$ m from “constrained to move along the z-axis”

Find $W_{F,a}$ .

Working

work of a steady force(body, moving, 1) $W_{F,a} = F_{a,x} \cdot Δ x_{x} + F_{a,y} \cdot Δ x_{y} + F_{a,z} \cdot Δ x_{z}$
$W_{F,a} = (−1) \cdot 0 + 2 \cdot 0 + 3 \cdot 4$
$W_{F,a} = 12$ J

Conclusion

Only the part of the force along the way the body moves does work; the parts across the z-axis push against the constraint and do none.

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 :a_force_along_a_constrained_path do4  kind :numerical5  describes "A body constrained to move along the z-axis of a coordinate system is subject to a constant force F " \6            "given by F = (%{fx}) î + (%{fy}) ĵ + (%{fz}) k̂, where î, ĵ, k̂ are unit vectors along the x-, y- and " \7            "z-axis of the system respectively. %{work} in moving the body a distance of %{distance} along the " \8            "z-axis, in joules?"9  given fx: "-1 N"10  given fy: "2 N"11  given fz: "3 N"12  given distance: "4 m"13  asks work: "What is the work done by this force"14end

solution.rb

1require_relative "../questions/a_force_along_a_constrained_path"2require_relative "../science/work_energy_and_power/work"34solve :a_force_along_a_constrained_path, using: Work do5  objects :body, :agent6  axes :x, :y, :z7  body.goes(:moving)89  take fx: body.force(by: agent, along: x)10  take fy: body.force(by: agent, along: y)11  take fz: body.force(by: agent, along: z)12  take distance: body.displacement(along: z)1314  deduce_from "constrained to move along the z-axis", body.displacement(along: x) => "0 m",15              body.displacement(along: y) => "0 m"1617  work = body.work(by: agent)18  answer work:, in: "J"1920  conclusion "Only the part of the force along the way the body moves does work; the parts across the z-axis " \21             "push against the constraint and do none."22end

playground.rb

1require_relative "../solutions/a_force_along_a_constrained_path"23play_with :a_force_along_a_constrained_path do4  title "A Force Along a Constrained Path"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	$W_{F}$	J	—	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
work by gravity	$W_{g}$	J	—	Work
net work	$W_{net}$	J	—	Work
displacement of body during moving along x	$Δ x_{x}$	m	—
position of body at end of moving along x	$x_{1,x}$	m	—
position of body at start of moving along x	$x_{0,x}$	m	—
displacement of body during moving along y	$Δ x_{y}$	m	—
position of body at end of moving along y	$x_{1,y}$	m	—
position of body at start of moving along y	$x_{0,y}$	m	—
displacement of body during moving along z	$Δ x_{z}$	m	—
position of body at end of moving along z	$x_{1,z}$	m	—
position of body at start of moving along z	$x_{0,z}$	m	—
average velocity of body during moving along x	$v_{av,x}$	m/s	—
time taken of moving	$Δ t$	s	0 … 1e+05
average velocity of body during moving along y	$v_{av,y}$	m/s	—
average velocity of body during moving along z	$v_{av,z}$	m/s	—
average speed of body during moving	$c_{av}$	m/s	—
path length of body during moving	$d$	m	—
change of velocity of body during moving along x	$Δ v_{x}$	m/s	—
velocity of body at end of moving from ground along x	$v_{1,x}$	m/s	—
velocity of body at start of moving from ground along x	$v_{0,x}$	m/s	—
change of velocity of body during moving along y	$Δ v_{y}$	m/s	—
velocity of body at end of moving from ground along y	$v_{1,y}$	m/s	—
velocity of body at start of moving from ground along y	$v_{0,y}$	m/s	—
change of velocity of body during moving along z	$Δ v_{z}$	m/s	—
velocity of body at end of moving from ground along z	$v_{1,z}$	m/s	—
velocity of body at start of moving from ground along z	$v_{0,z}$	m/s	—
average acceleration of body during moving along x	$a_{av,x}$	m/s²	—
average acceleration of body during moving along y	$a_{av,y}$	m/s²	—
average acceleration of body during moving along z	$a_{av,z}$	m/s²	—
acceleration of body during moving along x	$a_{x}$	m/s²	—
acceleration of body during moving along y	$a_{y}$	m/s²	—
acceleration of body during moving along z	$a_{z}$	m/s²	—
net force on body during moving along x	$F_{net,x}$	N	—
mass of body	$m$	kg	0 … ∞
net force on body during moving along y	$F_{net,y}$	N	—
net force on body during moving along z	$F_{net,z}$	N	—
mass of agent	$m_{a}$	kg	0 … ∞
weight on ground	$W_{g}$	N	—
weight on body	$W$	N	—
weight on agent	$W_{a}$	N	—
force on body by agent during moving	$F_{a}$	N	—
force on body by agent during moving along x	$F_{a,x}$	N	—
force on body by agent during moving along y	$F_{a,y}$	N	—
force on body by agent during moving along z	$F_{a,z}$	N	—
work on body by agent during moving	$W_{F,a}$	J	—
net work on body during moving	$W_{net}$	J	—
displacement of body during moving	$Δ x$	m	—
position of body at end of moving	$x_{1}$	m	—
position of body at start of moving	$x_{0}$	m	—
average velocity of body during moving	$v_{av}$	m/s	—
change of velocity of body during moving	$Δ v$	m/s	—
velocity of body at end of moving from ground	$v_{1}$	m/s	—
velocity of body at start of moving from ground	$v_{0}$	m/s	—
average acceleration of body during moving	$a_{av}$	m/s²	—
acceleration of body during moving	$a$	m/s²	—
net force on body during moving	$F_{net}$	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(moving, x)

Δ x_{y} = x_{1,y} - x_{0,y}

change of position(moving, y)

Δ x_{z} = x_{1,z} - x_{0,z}

change of position(moving, z)

v_{av,x} = \frac{Δ x_{x}}{Δ t}

displacement a time(moving, x)

v_{av,y} = \frac{Δ x_{y}}{Δ t}

displacement a time(moving, y)

v_{av,z} = \frac{Δ x_{z}}{Δ t}

displacement a time(moving, z)

c_{av} = \frac{d}{Δ t}

path a time(moving, x)

c_{av} = \frac{d}{Δ t}

path a time(moving, y)

c_{av} = \frac{d}{Δ t}

path a time(moving, z)

Δ v_{x} = v_{1,x} - v_{0,x}

velocity gained(moving, x)

Δ v_{y} = v_{1,y} - v_{0,y}

velocity gained(moving, y)

Δ v_{z} = v_{1,z} - v_{0,z}

velocity gained(moving, z)

a_{av,x} = \frac{Δ v_{x}}{Δ t}

velocity gained a time(moving, x)

a_{av,y} = \frac{Δ v_{y}}{Δ t}

velocity gained a time(moving, y)

a_{av,z} = \frac{Δ v_{z}}{Δ t}

velocity gained a time(moving, z)

v_{1,x} = v_{0,x} + a_{x} \cdot Δ t

velocity after a time(moving, x)

v_{1,y} = v_{0,y} + a_{y} \cdot Δ t

velocity after a time(moving, y)

v_{1,z} = v_{0,z} + a_{z} \cdot Δ t

velocity after a time(moving, z)

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

position after a time(moving, 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(moving, y)

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

position after a time(moving, z)

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

velocity after a distance(moving, x)

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

velocity after a distance(moving, y)

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

velocity after a distance(moving, z)

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

mean of the velocities(moving, x)

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

mean of the velocities(moving, y)

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

mean of the velocities(moving, z)

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

second law(moving, x)

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

second law(moving, y)

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

second law(moving, z)

F_{net,x} = F_{a,x} + 0

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

F_{net,y} = F_{a,y} + 0

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

F_{net,z} = F_{a,z} + 0

net force is the resultant(body, moving, z)

W_{F,a} = F_{a,x} \cdot Δ x_{x} + F_{a,y} \cdot Δ x_{y} + F_{a,z} \cdot Δ x_{z}

work of a steady force(body, moving, 1)

W_{net} = F_{net,x} \cdot Δ x_{x} + F_{net,y} \cdot Δ x_{y} + F_{net,z} \cdot Δ x_{z}

work of the net force(body, moving)

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

magnitude from components(displacement of body during moving)

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

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

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

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

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

magnitude from components(average velocity of body during moving)

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

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

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

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

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

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

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

magnitude from components(average acceleration of body during moving)

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

magnitude from components(acceleration of body during moving)

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

magnitude from components(net force on body during moving)

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

magnitude from components(force on body by agent during moving)

m \neq 0

has mass?(body)

m_{a} \neq 0

has mass?(agent)

F_{net,x} = 0

free of forces?(body, moving, x)

F_{net,y} = 0

free of forces?(body, moving, y)

F_{net,z} = 0

free of forces?(body, moving, z)