Converging mirrors

Converging mirrors(JL)^2 [aka Jackie and Jocelyn]

Purpose

Determine the relationship between object distance and image distance for real images in a concave spherical mirror

Determine the relationship between object height and image height for a set object distance

Hypotheses

The inverse of image distance will be linearly related to the inverse of object distance

The object height will be directly proportional to image height

Focal Length

Focal point – where rays parallel to the principal axis come to a focus

Focal length – distance between focal point and center of the mirror

Determining Focal Length Set curved mirror on track facing an

open window. Set screen on track between window

and mirror. Adjust distance between screen and

mirror until focused image (far away object through window, like a power pole) appears on the screen. Measure this distance with meter stick.

Experiment Diagram

Procedure – Part 1 Place object 15 focal lengths from mirror using cart tracks. Place the screen between mirror and object/light source

(see diagram). Move the screen until a focused image of the object

appears on the screen. Measure and record the image height that appears on the

screen and the distance between the screen and the mirror (image distance).

Decrease object distance by 1 focal length and repeat process until 3 focal lengths away from mirror.

Change distance increments to 5 cm. Continue process until 2 focal lengths away.

Reduce distance increments to 1 cm. Continue process until 1 cm less than 1 focal length away.

Part 1 Data

Sample Calcs

1/object distance = (1/do)

Example:1/do = 1/390 cm 1/do = 0.003 1/cm

1/image distance = (1/di)

Example:1/di = 25 cm 1/di = 0.040 1/cm

Mathematical Analysis

1/di 1/Image distance1/do 1/object distance

y = mx + b1/di = k (1/do) + bk = Δ(1/di)/Δ(1/do)

k = -1.05 1/cm / 1/cmk = -1.05

b = 0.0419 1/cm1/di = -1.05 (1/do) + 0.0419 1/cm

All slopes and values by LoggerPro

Error analysis: k

Accepted Value: 1Experimental Value (Source: magnitude of 1/di vs. 1/do graph slope): 1.05 ABS error: ǀAccepted value-Experimental valueǀ

ǀ 1 – 1.05 ǀ 0.05

Relative error: ABS/accepted value(0.05)/ (1)0.05 5% error

Error analysis: b

Accepted Value: 1/focal length = 1/26 cm = 0.0385 1/cmExperimental Value (Source: y-intercept of 1/di vs. 1/do graph): 0.0419 1/cm ABS error: ǀAccepted value-Experimental valueǀ ǀ 0.0385 1/cm – 0.0419 1/cm ǀ 0.0034 1/cmRelative error: ABS/accepted value(0.0034 1/cm)/ (0.0385 1/cm)0.0883 8.8% error

Error discussion

The method for taking data for this part largely relied on eyeballing distances and heights, so that was probably the largest source of error.

Also, for distance, we used a ruler to measure distances, which we had to constantly reset—so distances may not have been consistent, esp. with room darkness.

Results

1/di = b - k(1/do)k = 1

b = 1/f1/di = 1/f - (1/do)1/di + 1/do = 1/f

Part 2/3 procedure Place mirror on stand, at head of track. Use large lamp as light source. Place smallest arrow

card in front holder. Move light source 1.5 focal lengths from mirror.

Place screen between light source and mirror. Do not block light source. Adjust mirror so that image appears on screen.

Move screen until image focuses. Measure image height and object height with ruler. Repeat process for all arrow cards, recording the

height of actual arrow as well. Move light source until it is 2.5 focal lengths from the

mirror and repeat entire process.

DIAGRAM

Data

Math analyses part 2

hi Image heightho Object height

hi ∝ ho y = mx

hi = k (ho)k = Δhi / Δho

k = 1.89 cm/cmk = 1.89

hi = 1.89 ho All slopes and values by LoggerPro

Error part 2

Accepted Value: f/(do – f) = 26 cm / [1.5 (26cm) – 26 cm] = 2Experimental Value (Source: magnitude of hi vs. ho graph slope): 1.89 ABS error: ǀAccepted value-Experimental valueǀ ǀ 2 – 1.89 ǀ 0.11Relative error: ABS/accepted value0.11/20.055 5.5% error

Error discussion

Same as in part 1: eyeballing distances was the largest source of error, as well as estimating where the image focused exactly.

Results part 2

hi = k(ho)

k = f/(do – f)

hi = [f/(do – f)](ho)

Part 3 data

Graph part 3

Math analysis part 3

hi Image heightho Object height

hi ∝ ho y = mx

hi = k (ho)k = Δhi / Δho

k = 0.593 cm/cmk = 0.593

hi = 0.593 ho All slopes and values by LoggerPro

Error part 3

Accepted Value: f/(do – f) = 26 cm / [2.5 (26cm) – 26 cm] = 0.667Experimental Value (Source: magnitude of hi vs. ho graph slope): 0.593 ABS error: ǀAccepted value-Experimental valueǀ ǀ 0.667 – 0.593 ǀ 0.074Relative error: ABS/accepted value0.074/0.6670.111 11% error

Error discussion

Same as in parts 1 and 2: eyeballing distances was the largest source of error, as well as estimating where the image focused exactly.

Results part 3

[This might possibly look familiar.]

hi = k(ho)

k = f/(do – f)

hi = [f/(do – f)](ho)

Let’s put it all together!From experiments 2 and 3:hi = [f/(do – f)](ho)hi / ho= f/(do – f)

From experiment 1: 1/di + 1/do = 1/f

So:1/di = 1/f- 1/do1/di = do/fdo- f/fdo1/di = (do- f)/fdodo/di = (do- f)/fdi/do = f/(do- f)

hi / ho= di/do

Ray diagram: Object outside C

Ray diagram: at C

Ray diagram: between C and F

Ray diagram: inside F

Graphical representation:

THE END