Topic 1: Measurements and Uncertainties

1.1 Measurements in Physics

Fundamental and derived SI units

The fundamental units are the basic units of measurement of the International System of Units (SI), which combined make up the derived units, e.g. newtons (N).

Fundamental units:

  • Length: metre (m)

  • Mass: kilogram (kg)

  • Time: second (s)

  • Electrical current: ampere (A

  • Temperature: kelvin (K)

  • Amount of substance: mole (mol)

  • Luminous intensity: candela (cd) (Not required for the IB!)

 

Metric multipliers and scientific notation

Metric multipliers: correspond to a power of ten, e.g. kilo (k) - 10³.

  • All available on the Physics Data Booklet! (page 5)

Site Title

Scientific notation: convenient way of expressing numbers that are too small or too big.​

  • Notation: m x 10^n, where 1 ≤ m < 10 and n is an integer (positive or negative).

  • Example: 213 000 000 = 2.13 x 10^8

 

Order of magnitude: approximation of a number to the nearest power of ten.

 

Significant figures (s.f.) 

The number of digits that should be used to express a certain number, which shows how precise the information is. In any calculation or experimentation, the final answer should be expressed with the same number of s.f. as the value with least s.f. used.

 

Rules to count the number of significant figures:

  • Non-zero numbers are always significant. Example: 1234 - four s.f.​

  • "Sandwiched" zeros are always significant. Example: 5403 - four s.f.

  •  ​Zeros to the left are never significant. Example: 0.0004578 - four s.f.​

  • Zeros to the right are only significant if​ there is a point. Example: 1403.00000 - nine s.f.

 

Rules to round a number:

  • If the number following the last significant digit is less than five, the digit remains equal, e.g. 678.4 (4 s.f.) rounded to 3 s.f. = 678.

  • If the number following the last significant digit is greater than five, the digit rounds up (i.e. +1), e.g. 678.6 (4 s.f.) rounded to 3 s.f. = 679.

  • If the number following the last significant digit is five and it is only followed by zeros:

    • The last significant digit (number before five) remains equal ​if it is even, e.g. 3.2500 (5 s.f.) rounded to 2 s.f. = 3.2.

    • The last significant digit (number before five) rounds up ​if it is odd, e.g. 3.3500 (5 s.f.) rounded to 2 s.f. = 3.4.

1.2 Uncertainties and Errors

Errors (or uncertainties) in experimentation

All measurements are an estimate of the real value, since they are always subject to errors:

  • Systematic error: biases measurements in the same direction, e.g. always +0.1 cm.

    • Cause (e.g.): Not adequately calibrated equipment.​

    • Cause (e.g.): Ignoring the effects of friction (given that it is constant).

  • Random error: biases measurements in all directions,​ yielding a wide spread of values.

    • Cause (e.g.): Using a stopwatch manually - some measurements (of time) will be above the real time and some measurements will be below the real time.

    • Cause (e.g.): Changing external circumstances, e.g. alternating atmospheric conditions.

    • Solution: Gathering a wide range of values and then taking the average.

 

Accuracy​ and precision:

  • Accurate measurement: Low systematic error - average close to real value.

  • Precise measurement: Low random error - values close to each other.

Errors (or uncertainties) in measurements

Estimation of random errors ​in instruments: +- uncertainty. The uncertainty should always have the same number of decimal places as the value measured, and normally only 1 s.f.​​

  • Digital instrument (e.g. stopwatch)Smallest possible width of graduation, i.e. smallest division that the instrument can read.

    • Example: Image to the right (stopwatch)

      • Smallest width of graduation:​ 0.01 s

      • Uncertainty: +- 0.01 s

      • Value:  8.78 +- 0.01 s

  • Analogical instrument (e.g. a ruler): Half the smallest possible width of graduation

  • Ruler A: 

    • Smallest width of graduation:​ 1 cm

    • Uncertainty: +- 0.5 cm

    • Value: 12.5 +- 0.5 cm

  • Ruler B: more precise than Ruler A.

    • Smallest width of graduation:​ 0.1 cm

    • Uncertainty: +- 0.05 cm

    • Value:12.50 +- 0.05 cm

 

Errors (or uncertainties) in calculations

Consider the following value: L1: 8.3 +- 0.1 cm.

  • Absolute uncertainty (∆x): has the same units as the value, e.g. for L1: +- 0.1 cm

  • Fractional uncertainty: division between the absolute uncertainty and the value itself, e.g. for L1: 0.1/8.3 = 0.012 

  • Percentage uncertainty: the product of the fractional uncertainty by 100%, e.g. for L1: 0.012 x 100% = 1.2%

Now consider the following value as well: L2: 7.4 +- 0.5 cm.

 

Propagation of uncertainties:

  • Addition or Subtraction: Addition or subtraction of the values and addition of the absolute uncertainties.

    • Example: L1 - L2:​ (8.3 +- 0.1) - (7.4+- 0.5) = 0.9 +- 0.6 cm.

  • Multiplication or division: Multiplication or division of the values and addition of the fractional uncertainties or percentage uncertainties.

    • Example: L1 x L2: (8.3 +- 1.2%) x (7.4 +- 6.8%) = 61 +- 8.0% cm².

  • Power and roots:​ Value raised to a certain power and multiplication of the fractional uncertainty or percentage uncertainty by the value of the power.

    • Example: L1³:​ (8.3 +- 1.2%)^3 = 570 +- 3.6% cm³.

 

Errors (or uncertainties) in graphs

Error box:

  • Uncertainties of one value in a graph is commonly represented by error bars.

  • Error of y-value may differ from the error of x-value.

 

Best-fit line: line that goes through all error bars (it does not have to be a straight line!)

  • Direct proportionality (in the form y = ax) only if best-fit line is a straight line that passes through the origin (0,0).​​

  • More than 2 points are needed to confirm a relationship​ between two variables (e.g. x and y)

  • Gradient: found by using two points at least half the line's length away from each other: gradient = rise/run = ∆x/∆y

  • Uncertainty in the gradient:

    ∆gradient = gradientMAX - gradientMIN/2

  • Uncertainty in the y-intercept:                         

    ∆y - intercept = y - interceptMAX - y - interceptMIN/2

1.3 Vectors and Scalars

 

Representation of a vector

vector is represented by a line with an arrow at its end or at its middle, as shown by the two equal vectors a to the right.

  • The arrow indicates the direction.

  • The length indicates the magnitude.

Components of a vector

vector may be decomposed into one vertical and one horizontal component, as follows:

  • vector's magnitude may be found from its vertical and horizontal components, through the formula (known as Pythagoras Theorem): A^2 = Ax^2 = Ay^2 .

  • vector's direction (angle with the horizontal) may be calculated by means of the following formula: θ = tan^-1 Ay/Ax

 

Vector manipulation

When two or more vectors are added or subtracted, a resultant vector is formed.

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