AS Physics Errors and uncertainties Study Notes
AS Physics Errors and uncertainties Study Notes
AS Physics Errors and uncertainties Study Notes at IITian Academy focus on specific topic and type of questions asked in actual exam. Study Notes focus on AS Physics Study Notes syllabus with Candidates should be able to:
- understand and explain the effects of systematic errors (including zero errors) and random errors in measurements
- understand the distinction between precision and accuracy
- assess the uncertainty in a derived quantity by simple addition of absolute or percentage uncertainties
1.3.1—Understand and explain the effects of systematic errors (including zero errors) and random errors in measurements
- An error is the difference between the measured value and the actual value.
- In real experiments, it is difficult to find the true value of any physical quantity, thus, average of a large data set is taken.
- The deviation in each measured value from the true value of the quantity is said to be an error in measurement.
- Systematic Errors: These errors remain constant under different measurement conditions or change in a predictable manner.
- These can occur due to environmental, instrumental or observational errors.
- For example, wrong calibration of a weighing balance will lead to systematic errors or wrong reading due to parallax errors.
- Systematic errors affect the accuracy of the measurements.
- These can be reduced by improving the measurement techniques, better instrumental calibrations and removing personal bias.
- Random Errors: These errors occur in a random fashion and are not predictable.
- For example, temperature or voltage fluctuations.
- Random errors affect the precision of calculations and can be avoided by taking average of a large number of measurements.
- Zero Error: An instrument showing a reading when true reading should be zero is said to be having a zero error.
- For example, a weighing balance showing a weight of 500g when nothing is kept on it.
- Zero error should be adjusted with the measured values to get accurate results.
1.3.2—Understand the distinction between precision and accuracy
- Precision refers to how close are the individual measurements to each other, under same conditions.
- It is independent of accuracy.
- Accuracy refers to how close is the measured value to the accepted value.
- For example, while calculating the value of gravity, having value 9.7 is more accurate than 9.5.
- While values calculated as 9.25, 9.28, 9.31, 9.22 are not accurate but close to each other, therefore, precise.
1.3.3—Assess the uncertainty in a derived quantity by simple addition of absolute or percentage uncertainties
- Uncertainity in measurement represents the range of values that can have the actual value.
- Actual/Absolute uncertainity: It is represented as a fixed quantity (Δx).
- Fractional uncertainity: Represented as the fraction of measurement (Δx/x).
- Percentage uncertainity: Represented as the percentage of measurement ( (Δx/x) × 100%).
Rules for calculating uncertainity:
- For quantities added or subtracted, absolute uncertainities are added.
- In x ± y, uncertainty = Δx + Δy.
- For quantities being multiplied or divided, fractional uncertainities are added.
- Example, uncertainity in \(\frac{xy}{z}\), uncertainty is calculated as, \(\frac{Δx}{x}\)+\(\frac{Δy}{y}\)+\(\frac{Δz}{z}\).
- For quantities with indices, the uncertainty is multiplied with the power.
- Example, uncertainty in the term π\(r^{2}h\) will be 2\(\frac{Δr}{r}\)+\(\frac{Δh}{h}\).
Significant number
- Magnitudes of physical quantities are often quoted in terms of significant number.
- Can you tell how many sig. fig. in these numbers?
- 103, 100.0, 0.030, 0.4004, 200
- If you multiply 2.3 and 1.45, how many sf should you quote?
- 3.19, 3.335, 3.48
- 3.312, 3.335, $3 cdot 358$
The rules for identifying significant figures
- The rules for identifying significant figures when writing or interpreting numbers are as follows:-
- All non-zero digits are considered significant. For example, 91 has two significant figures ( 9 and 1 ), while 123.45 has five significant figures $(1,2,3,4$ and 5$)$.
- Zeros appearing anywhere between two non-zero digits are significant. Example: 101.1203 has seven significant figures: $1,0,1,1,2,0$ and 3.
- Leading zeros are not significant. For example, 0.00052 has two significant figures: 5 and 2.
The rules for identifying significant figures (cont)
- Trailing zeros in a number containing a decimal point are significant. For example, 12.2300 has six significant figures: $1,2,2,3,0$ and 0 . The number 0.000122300 still has only six significant figures (the zeros before the 1 are not significant). In addition, 120.00 has five significant figures since it has three trailing zeros.
- Often you will be asked to estimate some magnitudes of physical quantities around you.
- E.g. estimate the height of the ceiling, volume of an apple, mass of an apple, diameter of a strand of hair,
Estimates of physical quantities
• When making an estimate, it is only reasonable to give the figure to 1 or at most 2 significant figures since an estimate is not very precise.
$begin{array}{cc}text { Physical Quantity } & text { Reasonable Estimate } \ text { Mass of } 3 text { cans }(330 mathrm{ml}) text { of } & 1 mathrm{~kg} \ text { Pepsi } & 1000 mathrm{~kg} \ text { Mass of a medium-sized car } & 100 mathrm{~m} \ text { Length of a football field } & 0.2 mathrm{~s}end{array}$
- Occasionally, students are asked to estimate the area under a graph. The usual method of counting squares within the enclosed area is used.
Convention for labelling tables and graphs
• The symbol / unit is indicated at the italics as indicated in the data column left.
• Then fill in the data with pure numbers.
• Then plot the graph after labelling x axis and y axis
[Illustration with sample graph on left]
Prefixes
- For very large or very small numbers, we can use standard prefixes with the base units.
- The main prefixes that you need to know are shown in the table. (next slide)
Prefixes
- Prefixes simplify the writing of very large or very small quantities
Prefixes
- Alternative writing method
- Using standard form
- $N times 10^n$ where $1 leq N<10$ and $n$ is an integer
This galaxy is about $2.5 times 10^6$ light years from the Earth. The diameter of this atom is about $1 times 10^{-10} mathrm{~m}$.