Question

After plotting my graph of q versus p for the lens with the longer focal length, how do I determine from the graph what the asymptotes are, and then use them to determine the two values for f (focal length) on the graph? Please be as detailed as possible and explain every step!

Table of original data

Position of Lens (cm)	Position of Screen (cm)
23.8	70.0
45.6	70.0
23.1	72.0
48.6	72.0
22.3	75.0
52.2	75.0
21.9	77.0
54.7	77.0
21.7	79.0
56.9	79.0

Answer 1

I have to re arrange the given data as follows.

Position of lens(cm)	Position of screen(cm)
23.8	70
23.1	72
22.3	75
21.9	77
21.7	79
45.6	70
48.6	72
52.2	75
54.7	77
56.9	79

Plotted the graph as 2 data seris

80 79 78 77 76 75 Series1 74 73 -Series2 72 71 70 69 0 10 20 30 40 50 60

Both the curves are part of parabola

y = ax² + bx+c

We don't know the value of a b or c.

Since we know the x and y values

I have formulated 3 equations for each curve

70= a 23.8²+b 23.8 + 1*c 70 = 566.4 a + 23.8 b + 1c

72 = a 23.1² + b 23.1 + 1 c 72= 533.61 a + 23.1 b + 1c

75 = a 22.3² + b 22.3 + 1*c 75= 497.29 a+ 22.3 b + 1c

3 un knowns 3 equations we can solve the same

I used an online solver to solve the same

The a, b, c values are given below.

a	0.618812
b	-31.8441
c	477.3936

Similarly i have found the data for the second group of data

70= a 45.6²+b 45.6 + 1*c 70 = 2079.36 a + 45.6 b + 1c

72 = a 48.6² + b 48.6 + 1 c 72= 2361.96 a + 48.6 b + 1c

75 = a 52.2² + b 52.2 + 1*c 75= 2724.84 a+ 52.2 b + 1c

Solving we get

a	0.025253
b	-1.71212
c	95.56364

Calculated the y values for 2 data as shown below

Position of lens(cm)	Position of screen(cm)
29	74.33663366
28	70.90841584
27	68.71782178
26	67.76485149
25	68.04950495
24	69.57178218
23.8	70
23.1	72
22.3	75
21.9	77
21.7	79
20	71.42222222
25	68.54343434
30	66.92727273
36	66.65454545
37	66.78585859
38	66.96767677
39	67.2
40	67.48282828
41	67.81616162
42	68.2
43	68.63434343
44	69.11919192
45.6	70
48.6	72
52.2	75
54.7	77
56.9	79

Ploting the graph yields

80 78 76 74 Series1 72. Series2 70 68 66 0 10 20 30 40 50 60

I have carefully joined the data

The point just before crossing for the first data set

24	69.57178218

For second data set

25	68.54343434

Rearranging the data gives the following details

Position of lens(cm)	Position of screen(cm)
21.7	79
21.9	77
22.3	75
23.1	72
23.8	70
24	69.57
25	68.54
30	66.93
36	66.65
37	66.79
38	66.97
39	67.20
40	67.48
41	67.82
42	68.20
43	68.63
44	69.12
45.6	70
48.6	72
52.2	75
54.7	77
56.9	79

Position of screen(cm) 80 78 76 74 Position of screen(cm) 72 70 68 66 T 0 10 20 30 40 50 60

focal length can be calculated by the below formulae for any data point u, v

u= position from lens

v = position from screen

f= u*v / (u+v)

for example

for the values

40

67.48

f= 40*67.48 / (40+67.48) = 25.11 cm

Hope this helps