Exam Study Notes: Units, Errors, and Vectors

Units are important for problem-solving.
Unless specifically asked for a unit (e.g., centimeters), it's not mandatory to write the unit at every step or at the end of the result.
Final results will be graded based on the inclusion of the correct unit.
Velocity is measured in meters per second (m/s); this unit doesn't have a special name.

The constant $k$ in an equation can be derived by solving for it and tracking the units.
If the units are kept consistent during the calculation, the resulting unit for $k$ will be correct.
Dimensional analysis is a method to ensure the units of the final result are correct.

By plugging a constant back into an equation, one can verify if the unit of force comes out to be in Newtons, which would validate the calculation.

Einstein's famous equation: $E = mc^2$
- $E$ is energy.
- $m$ is mass (kilograms).
- $c$ is the speed of light (meters per second).
The unit of energy is kilogram-meter squared per second squared (kg
vert m^2/s^2).

Calculations on your units: The system is lenient with extra significant figures but strict if you provide too few.
Determine the number of significant figures from the question itself. It may not be explicitly stated.
Zeros from powers of 10 do not count as significant figures. E.g. 123 x $10^{-7}$ contains 3 significant figures.

When measuring with error, the precision of the measurement should match the precision of the error.
- Correct Example: The measurement and the error both stop at the same decimal place.
- Incorrect Example: The error implies a higher precision than the measurement.

When numbers have scientific notation, you have to correct them for errors.
Example:
- Original numbers: $4.87 \times 10^{-5}$ and $0.234 \times 10^{-5}$
Steps:
- Write both numbers using the same power of scientific notation.
- Factor out the scientific notation: $(4.87 \pm 0.234) \times 10^{-5}$
- Adjust the numbers based on significant figures.
- The error cannot have more significant figures (digits after the decimal) than the main result.

If after rewriting with same power of 10, you have
- $(4.87 \pm 1.234) \times 10^{-5}$ , then round to $(4.87 \pm 1.23) \times 10^{-5}$
  - Drop the $4$ from $1.234$ because the main result only has two decimal places.
If the error term was 1.8, it would be rounded up to 2.
It's about decimal places, not overall significant figures, in the error.
What If the calculated error is too small?
- Always account for a minimum error possible, which is 1.
- If the error margin is too small it will be counted wrong if you round it too much.
- There is a 5% or 1% margin.

Basic trigonometric ratios:
- $\tan(\theta) = \frac{opposite}{adjacent}$
- Sine is opposite over hypotenuse.
- Cosine is adjacent over hypotenuse.
Where:
- $\theta$ is the angle.
- Opposite is the side opposite to the angle.
- Adjacent is the side adjacent to the angle.
- Hypotenuse is the longest side of the right triangle.
Radian vs Degrees

$\pi \text{ radians} = 180 \text{ degrees}$
$2\pi \text{ radians} = 360 \text{ degrees}$
Calculators should be in degree mode for most calculations.
The inverse function in calculators is not one divided by, it is called 'inverse function'.

The position of the angle is important to know which side is opposite, adjacent, and not be held blindly.

Vectors can be represented as components in a coordinate system.
A vector ( \vec{a} ) in 2D can be written as $\vec{a} = ax \hat{x} + ay \hat{y}$

The magnitude (or length) of a vector is calculated using the Pythagorean theorem.
For a vector $\vec{c} = 2 \hat{x} + 3 \hat{y}$ , the magnitude is
- $|\vec{c}| = \sqrt{2^2 + 3^2} = \sqrt{13}$