Converging mirrors

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

2. 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

3. 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

4. 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

6. 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.

7. Experiment Diagram

8. 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.

9. Part 1 Data

10. 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

14. Mathematical Analysis 1/di 1/Image distance 1/do 1/object distance y = mx + b 1/di = k (1/do) + b k = Δ(1/di)/Δ(1/do) k = -1.05 1/cm / 1/cm k = -1.05 b = 0.0419 1/cm 1/di = -1.05 (1/do) + 0.0419 1/cm All slopes and values by LoggerPro

15. Error analysis: k Accepted Value: 1 Experimental 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

16. Error analysis: b Accepted Value: 1/focal length = 1/26 cm = 0.0385 1/cm Experimental Value (Source: y-intercept of 1/di vs. 1/dograph): 0.0419 1/cm ABS error: ǀAccepted value-Experimental valueǀ ǀ 0.0385 1/cm – 0.0419 1/cm ǀ 0.0034 1/cm Relative error: ABS/accepted value (0.0034 1/cm)/ (0.0385 1/cm) 0.0883 8.8% error

17. 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.

18. Results 1/di = b- k(1/do) k = 1 b = 1/f 1/di = 1/f - (1/do) 1/di + 1/do = 1/f

19. 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.

20. DIAGRAM

21. Data

23. Math analyses part 2 hiImage height ho Object height hi∝ ho y = mx hi = k (ho) k = Δhi/ Δho k = 1.89 cm/cm k = 1.89 hi = 1.89 ho All slopes and values by LoggerPro

24. Error part 2 Accepted Value: f/(do – f) = 26 cm / [1.5 (26cm) – 26 cm] = 2 Experimental Value (Source: magnitude of hivs. hograph slope): 1.89 ABS error: ǀAccepted value-Experimental valueǀ ǀ 2 – 1.89 ǀ 0.11 Relative error: ABS/accepted value 0.11/2 0.055 5.5% error

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

26. Results part 2 hi = k(ho) k = f/(do – f) hi = [f/(do – f)](ho)

27. Part 3 data

28. Graph part 3

29. Math analysis part 3 hi Image height ho Object height hi ∝ ho y = mx hi = k (ho) k = Δhi/ Δho k = 0.593 cm/cm k = 0.593 hi = 0.593 ho All slopes and values by LoggerPro

30. Error part 3 Accepted Value: f/(do – f) = 26 cm / [2.5 (26cm) – 26 cm] = 0.667 Experimental Value (Source: magnitude of hi vs. ho graph slope): 0.593 ABS error: ǀAccepted value-Experimental valueǀ ǀ 0.667 – 0.593 ǀ 0.074 Relative error: ABS/accepted value 0.074/0.667 0.111 11% error

31. 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.

32. Results part 3 [This might possibly look familiar.] hi = k(ho) k = f/(do – f) hi = [f/(do – f)](ho)

33. Let’s put it all together! From experiment 1: 1/di + 1/do = 1/f So: 1/di = 1/f- 1/do 1/di = do/fdo- f/fdo 1/di = (do- f)/fdo do/di = (do- f)/f di/do = f/(do- f) From experiments 2 and 3: hi = [f/(do – f)](ho) hi / ho= f/(do – f) hi / ho= di/do 

34. Ray diagram: Object outside C

35. Ray diagram: at C

36. Ray diagram: between C and F

37. Ray diagram: inside F

38. Graphical representation:

39. THE END