An Object Before a Concave Mirror

solved SphericalMirror

Q. An object is placed 10 cm $d_{o} = 10$ cm in front of a concave $R = - r_{size}$ mirror of radius of curvature 15 cm $r_{size} = 15$ cm. Where is the image, in cm, measured along the incoming light from the mirror?

Inputs

distance

d_{o}

10 cm

radius

r_{size}

15 cm

Outputs

where

v

How it's solved

Solved with SphericalMirror

Given

$d_{o} = 10$ cm
$r_{size} = 15$ cm
$R = - r_{size}$ from “concave”

Find $v$ .

Working

$d_{o} = 10$ cm $= 0.1$ m
$r_{size} = 15$ cm $= 0.15$ m
an object in front is against the light(mirror) $u = - d_{o}$
$u = - 0.1$
$u = −0.1$ m
concave $R = - r_{size}$
$R = - 0.15$
$R = −0.15$ m
focus halfway to the centre(mirror) $f = \frac{R}{2}$
$f = \frac{(−0.15)}{2}$
$f = −0.075$ m
mirror equation(mirror) $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$
$\frac{1}{v} + \frac{1}{(−0.1)} = \frac{1}{(−0.075)}$
$v = −30$ cm
$v = −0.3$ m $= −30$ cm
$v \approx −30$ cmto 2 significant figures

Conclusion

Distances in front of the mirror are negative; the image at minus thirty centimetres is real, in front of the mirror, inverted and three times the size.

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 :an_object_before_a_concave_mirror do4  kind :numerical5  describes "An object is placed %{distance} in front of a concave mirror of radius of curvature %{radius}. " \6            "%{where}, in cm, measured along the incoming light from the mirror?"7  given distance: "10 cm"8  given radius: "15 cm"9  asks where: "Where is the image"10end

solution.rb

1require_relative "../questions/an_object_before_a_concave_mirror"2require_relative "../science/ray_optics_and_optical_instruments/spherical_mirror"34solve :an_object_before_a_concave_mirror, using: SphericalMirror do5  objects :mirror67  take distance: mirror.distance_in_front8  take radius: mirror.size_of_the_radius910  deduce_from "concave", mirror.radius_of_curvature => -mirror.size_of_the_radius1112  where = mirror.image_distance13  answer where:, in: "cm", figures: 21415  conclusion "Distances in front of the mirror are negative; the image at minus thirty centimetres is real, " \16             "in front of the mirror, inverted and three times the size."17end

playground.rb

1require_relative "../solutions/an_object_before_a_concave_mirror"23play_with :an_object_before_a_concave_mirror do4  title "An Object Before a Concave Mirror"5end

The laws it drew on

quantity	written	units	within	from
radius of curvature	$R$	m	—	SphericalMirror
size of the radius	$r_{size}$	m	—	SphericalMirror
distance in front	$d_{o}$	m	—	SphericalMirror
focal length	$f$	m	—	SphericalMirror
size of the focal length	$f_{size}$	m	—	SphericalMirror
object distance	$u$	m	—	SphericalMirror
image distance	$v$	m	—	SphericalMirror
magnification	$m$	—	—	SphericalMirror
object height	$h_{o}$	m	—	SphericalMirror
image height	$h_{i}$	m	—	SphericalMirror
object distance of ground	$u_{g}$	m	—
distance in front of ground	$d_{o,g}$	m	—
object distance of mirror	$u$	m	—
distance in front of mirror	$d_{o}$	m	—
focal length of ground	$f_{g}$	m	—
radius of curvature of ground	$R_{g}$	m	—
focal length of mirror	$f$	m	—
radius of curvature of mirror	$R$	m	—
image distance of ground	$v_{g}$	m	—
image distance of mirror	$v$	m	—
magnification of ground	$m_{g}$	—	—
magnification of mirror	$m$	—	—
image height of ground	$h_{i,g}$	m	—
object height of ground	$h_{o,g}$	m	—
image height of mirror	$h_{i}$	m	—
object height of mirror	$h_{o}$	m	—
size of the radius of mirror	$r_{size}$	m	—
size of the focal length of mirror	$f_{size}$	m	—

u_{m} = - d_{o,m}

an object in front is against the light

f_{m} = \frac{R_{m}}{2}

focus halfway to the centre

\frac{1}{v_{m}} + \frac{1}{u_{m}} = \frac{1}{f_{m}}

mirror equation

m_{m} = \frac{- v_{m}}{u_{m}}

magnification of a mirror

m_{m} = \frac{h_{i,m}}{h_{o,m}}

magnification compares the heights

u_{g} = - d_{o,g}

an object in front is against the light(ground)

u = - d_{o}

an object in front is against the light(mirror)

f_{g} = \frac{R_{g}}{2}

focus halfway to the centre(ground)

f = \frac{R}{2}

focus halfway to the centre(mirror)

\frac{1}{v_{g}} + \frac{1}{u_{g}} = \frac{1}{f_{g}}

mirror equation(ground)

\frac{1}{v} + \frac{1}{u} = \frac{1}{f}

mirror equation(mirror)

m_{g} = \frac{- v_{g}}{u_{g}}

magnification of a mirror(ground)

m = \frac{- v}{u}

magnification of a mirror(mirror)

m_{g} = \frac{h_{i,g}}{h_{o,g}}

magnification compares the heights(ground)

m = \frac{h_{i}}{h_{o}}

magnification compares the heights(mirror)

R = - r_{size}

concave