A Force Along a Constrained Path

Q. A body constrained to move along the z-axis of a coordinate system is subject to a constant force F given by F = (-1 N) î + (2 N) ĵ + (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 along the z-axis, in joules? expects 12 J

What the draft establishes

Fills in once a draft is checked.

The web of concepts

given read from the words equation asked

The equations it used

None yet.

Check an answer

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?

The playground

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}

The solution

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 Ruby

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

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

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

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)