Page 261 - Physics Coursebook 2015 (A level)

P1: Practical skills at AS level
WORKED EXAMPLES
2 A student investigates the depth D of a crater made when ball-bearings of different diameters d are dropped into sand. He drops two ball bearings from the same height and measures the depth of the craters using a 30 cm ruler. The results are shown in Table P1.3.
Table P1.3 For Worked example 2.
It is suggested that the depth D of the crater is directly
proportional to the diameter d of the ball-bearing, that is: D = k d o r Dd = k
Do the readings support this hypothesis?
Step 1 Calculate the values of k = Dd . These values are
shown in the third column in Table P1.3, although they should only be given to 2 significant figures as values of
D are given to 2 significant figures and values of d to 3 significant figures. The more precise values for k are to be used in the next step.
Step2 Calculatethepercentagedifferenceinthek values. The percentage difference is:
0.45 × 100% = 7.2% 6.19
So the k values differ by 7% of the smaller value. Step3 Stateacriterionandcheckit.
‘My criterion is that, if the hypothesis is true, then the percentage difference in the k values will be less than the percentage uncertainty in D. I chose D as it obviously has the higher percentage uncertainty.’
QUESTIONS
3
The uncertainty in the smaller measurement of D can be calculated as:
uncertaintyinD= 2 ×100%=9.5% 21
The percentage difference in the k values is less than the uncertainty in the experimental results; therefore the experiment is consistent with the hypothesis.
Of course we cannot say for sure that the hypothesis is correct. To do that we would need to greatly reduce the percentage uncertainties.
A student obtains data shown in Table P1.4.
Table P1.4 For Worked example 3.
The first reading of x was found to have an uncertainty
of ± 0.1. Do the results show that d is proportional to x? Step 1 Calculate the ratio of dx in both cases:
d =1.50 d =2.29 x1 x2
Step2 Calculatehowclosetoeachotherthetworatios are:
Diameter of ball bearing d/mm
Depth of the crater D/mm
D/d
5.42 ± 0.01
36±2
6.64
3.39 ± 0.01
21±2
6.19
x/cm
d/cm
2.0
3.0
3.5
8.0
2.29 − 1.50 = 0.79
So the two values of x are 1.5 = 53% different.
15
A student obtains the following data for two variables T and m (Table P1.5).
Table P1.5 Data for Question 15.
The first value of T has an uncertainty of ± 0.2 s. Do the results show that T is proportional to m?
16
The uncertainty in the first value of x is 5% and, since the percentage difference between the ratios is 53% is much greater, the evidence does not support the suggested relationship.
A student obtains the following values of two variables r and t (Table P1.6).
Table P1.6 Data for Question 16.
The first value of r has an uncertainty of ± 0.2 cm, which is much greater than the percentage uncertainty in t. Do the results show that t 2 is proportional to r?
d 0.79
Step3 Comparethevaluesandwriteaconclusion.
T/s
m/kg
4.6
0.90
6.3
1.20
r/cm
t/s
6.2
4.6
12.0
6.0
249