1. Tsokos1-5The Realm of physics

2. IB Assessment Statements • Measurements and Uncertainties • Uncertainties in Graphs • 1.2.12 Identify uncertainties as error bars in graphs. • 1.2.13State random uncertainty as an uncertainty range (±) and represent it graphically as an ‘error bar’’ • 1.2.14Determine the uncertainties in the gradient and intercepts of a straight-line graph.

3. IB Assessment Statements • Measurements and Uncertainties • Uncertainties in Calculated Results • 1.2.10 State uncertainties as absolute, fractional and percentage uncertainties. • 1.2.11Determine the uncertainties in results.

4. Objectives • Deal with logarithmic functions, semi- logarithmic and logarithmic plots • Find the error in a calculated quantity in terms of the errors of the dependent quantities • Find the error in the slope and intercept of a straight-line graph

5. Logarithmic Functions a) b) Radioactive decay, , where λ is the decay constant i) Negative exponent = decay ii) Positive exponent = growth

7. c) You can make a straight line graph of this function by taking the log of each side: d) The slope in this case would be - λ which is the decay constant e) The intercept would be , , call it i) If

8. f) This is a semi-logarithmic plot since we take the log of only one of the variables

9. Logarithmic Plots a) b) A straight-line plot of this data can be made by taking the log of both sides c) The slope in this case would be n d) The intercept would be

10. Propagation of Error • Addition and Subtraction • i) The absolute uncertainty of the result is the sum of the absolute uncertainty of the parts • ii) Assumes all uncertainties are in the same units • iii) Large errors can sometimes result when subtracting two numbers that are very close

11. Propagation of Error Multiplication, division, powers and roots • i) The fractional uncertainty of the result is the sum of the fractional uncertainties of the parts.

12. Propagation of Error Other functions i) For complex equations such as trigonometric functions, surface area equations, etc., it may be easier to find the largest and smallest values of Q and deduce that the area is half of the difference between the two.

13. Propagation of Error Example: Find the surface area of a sphere with a radius of 4.0±0.5mm.

14. Uncertainties in Slope and Intercept • a) Graph should include error bars showing the uncertainty of each data point • b) To find the uncertainty in the slope, draw two extreme lines and find the slope of each • i) The lines should pass through a point halfway in the range of x • ii) One should have the largest slope possible while remaining within the error bars • iii) The other should have the smallest slope possible while remaining within the error bars • iv) Find the slopes of these two lines and the differences between these and the best fit line to find the error for each • v) Take the average of these two to find the error in the slope

16. Uncertainties in Slope and Intercept • c) To find the uncertainty in the intercept, average the absolute values of these two extreme values to find the uncertainty of the intercept

17. Summary Review • Are you able to: • Deal with logarithmic functions, semi- logarithmic and logarithmic plots • Find the error in a calculated quantity in terms of the errors of the dependent quantities • Find the error in the slope and intercept of a straight-line graph

18. Self-Assessment • Can you identify uncertainties as error bars in graphs. • Can you state random uncertainty as an uncertainty range (±) and represent it graphically as an ‘error bar’’ • Can you determine the uncertainties in the gradient and intercepts of a straight-line graph. • Can you state uncertainties as absolute, fractional and percentage uncertainties. • Can you determine the uncertainties in results.

19. Homework • 1-9 and 11-18