Download First Year Laboratory Scripts MODULE I

Analysis of Results
In order to calculate the error in the final result, it is necessary to combine the errors from
your individual quantities in some way. You will learn more about this later in the
introductory module so for the moment we will simply quote the formula that you can use
to calculate the error (see laboratory handbook chapter 4).
You should have worked out the distance, d using the following equation:
d=
w
θ 
2 tan 
2
(5)
Before we can work out the error in d, we need to work out the error in θ/2. To measure
θ, you measured the difference between two angles which we will call θ1 and θ2 and you
should have errors in each quantity:
e.g. θ1 = 14.93 ± 0.01º
θ2 = 8.18 ± 0.01º
so ∆θ1 = 0.01º
so ∆θ2 = 0.01º
Therefore θ = θ1 - θ2 = 6.75º and the error in θ, ∆θ, is approximately 0.02º (by adding
together each error) so θ = 6.75±0.02º and therefore θ/2 = 3.38 ± 0.01º.
We can work out the error in d using the expression:
2
θ

 ∆ tan 
∆d
 ∆w  
2
where θ is in radians
(6)
= 
 +
d
 w   tan θ 



2 
The fractional error in tan(θ/2) is a rather complicated expression and must be calculated
using θ in radians:
2
θ
(
)
1 sec 2 θ ∆ θ
∆θ
2 = 2
2
2=
2
θ
θ
θ
tan
2 cos sin θ
tan
2
2
2
2
∆ tan
(7)
However there is an alternative way to calculate the error in tan(θ/2) and this is the
method we recommend that you use for this exercise:
1. Using equation 5, work out d using your measured values of w and θ/2.
e.g. using w = 86.7cm and θ/2 = 3.38º, d = 734 cm
2. Now work out d using your measured value of w and a value for θ/2 = measured
value of θ/2 + ∆θ/2:
e.g. using w = 86.7cm and θ/2 = 3.39º, d = 733 cm
3. So the error in d considering only the error in θ/2 = 734 - 733 = 1 cm.
4. Assume that the percentage error in d due to θ/2 equals the percentage error in
tan(θ/2) in equation 6.
e.g. % error in d =
1
× 100 % = 0.1% so % error in tan(θ/2) = 0.1%
734
We can now work out the error in d by combining the errors in w and θ/2 using equation
6:
7