Measurements and Uncertainties

Units

7 Fundamental units

Measurement Unit Symbol
Time Second s
Displacement Metre m
Mass Kilogram Kg
Current Ampere A
Temperature Kelvin K
Amount of substance Mole mol
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Derived quantities

To convert between the derived and fundamental units, look at the formula and substitute the units

E.g Momentum has derived unit - p.

To find fundamental, substitute units into the formula. 

p = mass*velocity = kgms-1

 

Dimensional consistency

This is checking if the units are correct for measurement by converting it to fundamental units and seeing if it fits the equation. 

E.g are the units for velocity at2?

ms-1 ≠ ms-2s2  Therefore this is not dimensionally consistent.

 

Metric multipliers and si prefixes

Factor Name Symbol Factor Name Symbol
101 deca d 10-1 deci d
102 hecto h 10-2 centi c
103 kilo k 10-3 milli m
106 mega M 10-6 micro μ
109 giga G 10-9 nano n
1012 tera T 10-12 pico p
1015 peta P 10-15 femto f

 

Order of magnitude

The nearest power of 10.

6.5 x 105 has a magnitude of 106 because 6.5 rounds up to 10.

 

Significant figures

Your answer must be to least precise significant figures in the question you were asked.

 

Uncertainties

What is uncertainty

This is how much a value measured may vary from the actual value. 

Absolute uncertainty: the same units as the measurements

percentage uncertainty: Given as a percentage as measurement

 

Precise Vs Accurate

 

Uncertainties Practical

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Types of error

Type Description Effect on the graph
Random Changes every trial  The best fit line will not pass through all the error bars.
Systematic Caused by equipment, which is constant for every trial. Shifts graph vertically or/and horizontally. Error bars represent this error.

 

Rules to determine the uncertainty

Rounding: 1 sig fig on exam OR don't round for lab report

Averages: Range / 2

Analog Measurements: For reliable equipment use plus or minus half smallest division (rounded to 1sf) or the uncertainty listed on the equipment.

 

Combining Uncertainties

Added or subtracted

Add the absolute uncertainty

ΔU+ΔU\Delta U + \Delta U

 

Multiplied or divided

Add the percentage uncertainty

ΔU%+ΔU%\Delta U\% + \Delta U\%

 

Raised to a power

Times the percentage uncertainty by the power (X) of the equation

X×ΔU%X \times \Delta U\%

 

Determine the uncertainty of a series

Calculate the range of values using the min and max value (not the uncertainty).

VMax-VMin2=ΔU\frac{V_{Max} - V_{Min}}{2} = \Delta U

 

Determine the uncertainty of x number of oscillations

Divide the absolute uncertainty by the number of oscillations.

ΔUX=ΔU

 

Trigonometric Uncertainty

Calculate the range of values using the min and max value (normal plus and minus the uncertainty).

Cos(V+U)-Cos(V-U)2=ΔU

 

Exam Tip

Do not expend a great lot of effort in expressing uncertainties during an examination, unless specifically instructed. Assess the evidence, use common sense, and (generally) round-off absolute uncertainty to one significant figure. An uncertainty of ±0.389s can be rounded to ±0.4s to agree with the least significant digit position in the measured value.

 

Uncertainty graphs

Gradient uncertainty: (Maximum gradient - the minimum gradient)/2

Y-intercept: Maximum/minimum y-intercept - best fit y-intercept (largest difference determines maximum or minimum)

Error bars: Shows the uncertainty of the best line based on the systematic error.

 

Vectors

What are Vectors and Scalars?

A vector is a way of representing a quantity with both magnitude and direction. It can be represented in two ways. Either the magnitude is stated directly,  then it's direction is given as an angle from something. This is useful when working with a polar coordinate system.

Vector: Magnitude and direction

Scalar: ONLY magnitude

 

Examples of vectors and scalars

Vectors

Scalars

velocity

Mass

displacement

Time

Weight

Distance

Acceleration

Length

 

Difficult to remember Vector and scalars quantities

 

Ways of representing a vector

 

Horizontal and Vertical Components

Split each component into its horizontal and vertical components using trigonometry. Then for each axis add the components together. 

Vector addition and subtraction graphically:

To add two vectors, draw the first vector with its tail at the origin and then draw the second vector with its tail at the previous vectors head. Finally, draw a line from the origin to the head of the second vector. This line is the new vector, as shown below.

W= U+V

To subtract a vector, do the same as with addition but flip the vector which you are subtracting.

 

 

Practise Test

Here is a practise test I put together. It is 1hr long and made from V3 Past exam questions.

https://drive.google.com/file/d/1SxW4rSdmN5T4j08TqpJ9cBx3Py5JNOnI/view?usp=sharing

 

Editors

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