Year 13 Physics Uncertainties and Graphing

These units are: Physics involves measuring physical quantities such as the length of a spring the charge of an electron. Units are used to help us know what we are measuring. There are 7 quantities that make up the fundamental units which are called the Systeme International d’Unites’or SI Units.

Physical Quantities & their Units QuantitySI Unit NameSymbol(s)NameSymbol Length Mass Time Electric Current Temperature Luminous Intensity Amount of substance l, x, y,d, M, m T, t I T I N Metre Kilogram Second Ampere Kelvin Candela Mole m kg s A K cd mol

Prefix multipliers can be added to each of the units in both numerically and in the form of names. You are expected to be able to convert from name to numbers and back again. You already use some of the these prefixes: Kilogram = 1000grams OR 1 x 10 3 g The standard SI prefixes are as follows:

Standard Prefix Multipliers PowerPrefixSymbol GMkmnpGMkmnp

Solution a.43 g = kg b.137 mA = A c.2.8 km = 2800 m d.38 cm 2 = m 2 e.6.8 x 10 4 cm 3 = 6.8 x m 3 a. Convert 43 g to kg. b. Convert 137mA to A c. Convert 2.8km to m d. Convert 38cm ² to m² e. Convert 6.8 x 10 4 cm 3 to m 3.

These give us an indication as to how accurate a measurement can be. For instance: 23.0cm is 3 sig. figs 23.01cm is 4 sig. figs and is more accurate. Our measurement tool can measure to 0.01cm. In Physics we use sig. figs to show the level of accuracy we are measuring to. In a lot of cases we measure to 2 significant figures. When converting to specific sig. figs we count from the first digit, not zero.

Exercise: Exercise: A metre ruler was used to make distance measurements. The following results were recorded by different students. a. 43cmb. 43.0cmc cm Comment on each of these measurements and explain which of the three measurements are valid? Measurement A is given to the nearest cm. A metre ruler can measure to the nearest mm, and so the measurement should have been recorded as 43.0cm. Measurement C implies it is possible to accurately read to one tenth of a millimetre. This is not possible using a metre ruler, so the measurement should have been recorded as 43.0cm. Measurement B correctly expresses the measurement to the nearest mm and so is valid.

the measurement should be the least number of significant figures of the data values used. Multiplication or Division the measurement should be rounded to the least number of decimal places of the data values used. Addition or Subtraction

SOLUTION a.Perimeter of the plate: = = = 32.7 cm (to 1 decimal point) b.Area of the plate: = 15.4 x 0.94 = = 14 cm 2 (to 2 sf) A metal plate has the following dimensions: Length 15.4cm. Width 0.94cm. Calculate : Perimeter of the plate Area of the plate

Measurement Measuring necessarily involves some element of inaccuracy. This may be due to such things as: Limits to the precision which a scale may be read. Reaction times when using a stopwatch. An incorrectly calibrated meter. A zero error on a measuring device. This means that any measurement taken must be recorded with its absolute uncertainty. eg. t = 5.3s + 0.1s The +0.1s is the ‘uncertainty’ or ‘error’. Systematic Uncertainties These are errors arising due to such things as: An incorrectly zeroed meter. An inaccurate timing device. Using an approximate theory. (ie. Ignoring friction in a pulley) etc. Random Uncertainties These errors tend to arise due to limitations on our ability to measure or judge. For example: Timing with a stopwatch. Estimating the sharpest image formed by a lens. Marking the amplitude on a swinging pendulum. etc

Measuring necessarily involves some element of inaccuracy. This may be due to such things as: Limits to the precision which a scale may be read. Reaction times when using a stopwatch. An incorrectly calibrated meter. A zero error on a measuring device. This means that any measurement taken must be recorded with its absolute uncertainty. eg. t = 5.3s + 0.1s The +0.1s is the ‘uncertainty’ or ‘error’. Systematic Uncertainties These are errors arising due to such things as: An incorrectly zeroed meter. An inaccurate timing device. Using an approximate theory. (ie. Ignoring friction in a pulley) etc. Random Uncertainties These errors tend to arise due to limitations on our ability to measure or judge. For example: Timing with a stopwatch. Estimating the sharpest image formed by a lens. Marking the amplitude on a swinging pendulum. etc

1. Limitations in the accuracy of the measuring instrument. 2. Limitations in the skill of the experimenter. 3. Fluctuations in the physical quantity being measured. 4. External influences on the measurement procedure. Uncertainties are not caused by mistakes rather they are due to such things as: If the measurements are used to draw a graph, there will be uncertainty in the gradient of the graph line. This will cause uncertainty in the value of any physical quantity calculated from the gradient.

 An absolute uncertainty is always given to 1 sf only.  A measurement must not be expressed to a greater accuracy that the absolute uncertainty.  A processed measurement must not be expressed to a greater number of sf than the raw data from which it was found. Significant figures and uncertainties The number of sig. figs (s.f) used when giving a measurement should reflect the amount of uncertainty involved. The following principles should be applied whenever decisions about s.f are made. Significant figures and uncertainties The number of sig. figs (s.f) used when giving a measurement should reflect the amount of uncertainty involved. The following principles should be applied whenever decisions about s.f are made.

Calculating Uncertainty from data:  The absolute uncertainty “  X” for a number of data entries can be calculated by finding the range. This is done as follows: Calculating Uncertainty from data:  The absolute uncertainty “  X” for a number of data entries can be calculated by finding the range. This is done as follows: Absolute Uncertainty = Difference between highest and lowest entry 2 Written as (measurement) X ±  X (uncertainty) Absolute Uncertainty = Difference between highest and lowest entry 2 Written as (measurement) X ±  X (uncertainty) The relative uncertainty is a percentage and is used when processing uncertainties. The relative uncertainty can be calculated by: Relative Uncertainty =  X x 100 X The relative uncertainty is a percentage and is used when processing uncertainties. The relative uncertainty can be calculated by: Relative Uncertainty =  X x 100 X

Rules For Manipulating Uncertainties 1. When adding or subtracting quantities – ADD the absolute errors. 2. When multiplying or dividing quantities – ADD the relative errors. Rules For Manipulating Uncertainties 1. When adding or subtracting quantities – ADD the absolute errors. 2. When multiplying or dividing quantities – ADD the relative errors. From these rules we can infer that: When raising to a power, multiply the relative error by that power. Note: That includes fractional powers. When multiplying a quantity by a constant, multiply the absolute error by that constant. Examples: a = , b = Find: From these rules we can infer that: When raising to a power, multiply the relative error by that power. Note: That includes fractional powers. When multiplying a quantity by a constant, multiply the absolute error by that constant. Examples: a = , b = Find:

Express the errors as percentages.a = (3.8%) b = (6.7%) (in case they are needed) The error is Note that the final result should be given to no more than 2 sig. fig. The error must be given to only 1 significant figure. Adding percentage errors means finding 10.5% of 39 The error is 7.6% (double 3.8%) The percentage error is 2.2% (one third of 6.7%) 2.2% of 1.96 is 0.04 which is best rounded up to be on the safe side. The final result can still be only 2 sig. fig.

(v) Working should be presented as follows: 3a + b = (4.8%) (3 x ) 2a – b = (31%) 3a + b= (36%) 2a – b = 8 + 3NB. Only 1 s.f for the error means that in this case there will only be 1s.f. in the result.

Find the time period of a mass on a spring under simple harmonic motion including uncertainties using the following formula: T = 2  (m/k) Where m = 0.5 ± 5% Kg k = 30.0 ± 0.2 Nm -1

T = 0.81 ± 0.02 s

Using these skills complete exercises from page 4-12 Rutter

GRAPHING SKILLS......

In order to produce an equation from data the following steps should be observed: 1. Tabulate your raw data and process data including uncertainties. 2.Sketch measurement data onto a graph. 3. Decide what relationship has been produced i.e. y  x; y  1/x; y  x2 e.t.c. 4. Do the function to measurement and uncertainty i.e. if y  x2 then do x2 to all the x values and multiply relative uncertainty by two. 5. Plot y versus x2 and this should produce a straight line if you have drawn it correctly. 6. Error bars need to be drawn on to show uncertainty above & below point. 7. Produce a line of best fit and then worst fit (using error bars)

8. Calculate the gradient of the line of best fit (m) and worst fit (m’) using the rise/run method. 9. Calculate uncertainty for line of best fit (m) by subtracting worst fit (m’) from best fit (m).  m = m’ – m 10. Substitute m ±  m into the ‘y = mx + c’ equation. 11. Calculate ‘c’, from y intercept of line of best fit and its uncertainty (  c) from subtracting y intercept for the line of best fit (c) from y intercept for the line of worst fit (c’).  c = c’ – c. 12. Rewrite the equation y = mx +c, with the function of ‘x ±  x’, the new gradient (m ±  m ) and the y intercept if there is one (c ±  c).

 Experimental data will contain uncertainties  These uncertainties must be considered when trying to determine the relationship between variables  Error bars are used to show the uncertainty in individual data points  Lines of best & worst fit are used to find the overall uncertainty in the relationship

Uncertainties on Graphs When plotting measured, or calculated points, error bars must be shown. The length of the bar will depend on the size of the uncertainty. In most cases only the dependent variable errors are plotted as the independent only show equipment errors. Uncertainties on Graphs When plotting measured, or calculated points, error bars must be shown. The length of the bar will depend on the size of the uncertainty. In most cases only the dependent variable errors are plotted as the independent only show equipment errors. m m’ The ‘ line of best fit ’ is drawn and its gradient m calculated. Another line (either a maximum or a minimum) error line is drawn. This line should still be within the bounds of the error bars. It’s gradient m’ is calculated. The error in the gradient is |m – m’|. ie. The absolute value of the difference. The ‘ line of best fit ’ is drawn and its gradient m calculated. Another line (either a maximum or a minimum) error line is drawn. This line should still be within the bounds of the error bars. It’s gradient m’ is calculated. The error in the gradient is |m – m’|. ie. The absolute value of the difference.

Graphing: Your graph must show: Graphing: Your graph must show: Simple Pendulum (heading) Period vs Length (a statement of what is being plotted) Dependent variable Independent variable T(s) (symbols with appropriate SI unit) l (m) Suitable scales.

A m l θ Graphing and Data Collection Consider the experiment to find the relationship between The period and length of a simple pendulum Independent variable Usually the variable that you have control over. eg: The length of a pendulum. (You are physically able to adjust and measure that.) Dependent Variable This is the quantity that you have to measure and is affected by how you alter the independent variable. eg. The period of the pendulum. Other variables : If the aim is to discover a relationship between the period and length of a pendulum, then other variables that may affect the period must be identified and kept constant. eg: the mass of the pendulum bob, the amplitude or the size of the semi-vertical angle.

Time (s)Speed (ms -1 ) Uncertainty in time values is ±0.2 s and speed values are ±10% Calculate the value of each uncertainty Calculate the value of each uncertainty

Time (s)  Time Speed (ms -1 )  Speed Uncertainty in time values is ±0.2 s and speed values are ±10% Graph the data

Add error bars

+ 0.2 s s Total length of error bar is 0.4 s Each bar shows the possible values for data point

Draw line of BEST fit

Draw line of WORST fit

LABEL the lines Line of worst fit Line of best fit Calculate gradients & record intercepts for BOTH lines

Line of worst fit Line of best fit Line of worst fit: Intercept = 2.0 ms -1 Gradient = 7.6/3.4 = Calculate the overall error

Line of best fit: Intercept = 1.4 ms -1 Gradient = 10/3.9 = Line of worst fit: Intercept = 2.0 ms -1 Gradient = 7.6/3.4 = Final Errors: Intercept = (2.0 – 1.4) = ± 0.6 ms -1 Gradient = (2.235 – 2.564) = ± ms -2 State the relationship between the variables

This statement shows linear relationship between speed and time error in the GRADIENT error in the INTERCEPT units

Using these skills complete exercises from Rutter Page 13-27

LE 06 Page 29 Rutter. Follow instructions and draw up a table similar to the one below. Hwk: complete the write-up.m mmmmx xxxxF50g 100g 150g 500g