## Topic 1: Measurements and Uncertainties

### 1.1 Measurements in Physics

SI Units, scientific notation, significant figures

- SI system, mks
- Always convert all quantities to meters, kilograms, seconds. Do not work in millimetres, hours, grams and so on – that will probably be rather confusing.

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*base units*. Mass (kg), length (m), time (s), electric current (A), temperature (K), intensity of light (cd), amount of substance (mol) - All other units can be expressed in terms of these base units. They are called
*derived units*. For example, energy (J = kg m^{2}s^{-2}), acceleration (m s^{-2})- To find out how to express derived units in terms of base units, use a formula which relates the quantity in question to the base quantities.
- For example, E = F*d = m*a*d. Hence, J = (kg)(ms
^{-2})(m) = kgm^{2}s^{-2}

- Dimensional analysis. Only quantities of the
*same units*can be added or subtracted from*each other*. When two quantities are multiplied or divided, their units are likewise multiplied or divided- Checking whether the units are correct is a good way of making sure your workings are not invalid.
- Limits of dimensional analysis: Unable to determine dimensionless numerical prefactor (for example, v
^{2}= u^{2}+ 2as, the ‘2’ has to be determined from physical principles)

### 1.2 Errors and Uncertainty

Outline: Systematic errors. Determining errors in gradients and intercepts.

- Addition or subtraction:
**add***absolute uncertainties*- For example, consider l
_{total}= l_{1}+ l_{2}. If Δl_{1}= 0.1m and Δl_{2}= 0.2m, then Δl_{3}= Δl_{1}+ Δl_{2}= 0.3m

- For example, consider l
- Multiplication or subtraction:
**add***fractional uncertainties*- Hence, whenever there is an exponent, the fractional uncertainty is multiplied by the exponent. For example, consider E = 0.5 * k * x
^{2}. ΔE / E = Δk/k + 2 Δx/x. Another example, ω = sqrt(l/g) hence Δ ω / ω = 0.5*Δl/l + 0.5* Δg/g

- Hence, whenever there is an exponent, the fractional uncertainty is multiplied by the exponent. For example, consider E = 0.5 * k * x
- When dealing with functions like sin or cos, bear in mind that the above rules will not work. Instead, find the maximum and minimum values, and use that to determine the uncertainty.
- For example, y = sin x. y = (y
_{max}+ y_{min})/2, Δy = (y_{max}– y_{min})/2

- For example, y = sin x. y = (y
**Never**subtract uncertainties.- When writing the answer, give the uncertainty to 1 significant figure, and make sure that the value has the same number of decimal places as the uncertainty.
- Don’t forget to check that units make sense

### 1.3 Vectors and Scalars

You can move vectors around, as long as you don’t change their magnitude or direction. When adding vectors, place the tail of one vector at the head of the other. The resultant vector *starts* from the tail of the first vector and *points* to the head of the second. When subtracting vectors, reverse the direction of the vector where the negative sign is applied, and then add the resulting vectors together.

Resolving vectors. We can add two vectors to form one vector. Similar, we can take one vector and ‘split’ it into two or more vectors. We do this to make the problem more easy. For instance, we can ‘split’ (resolve) the vector into a vector in the x direction and another vector in the y direction. More generally, we resolve the vector such that the component vectors are in convenient directions.

Out of the IB syllabus: Scalar and vector products