Welcome to the Foundation of Physics!
Before we can calculate the speed of a galaxy or the force of an atom, we need to know how to measure things properly. This chapter is all about the "rules of the game." You will learn how to handle units, make smart guesses (estimations), and deal with the fact that no measurement is ever 100% perfect. Don't worry if the math seems a bit picky at first—once you master these basics, the rest of Physics becomes much easier!
1. Physical Quantities and Units
In Physics, a number by itself is useless. If I tell you a car is traveling at "50," you don't know if that's 50 miles per hour, 50 meters per second, or 50 centimeters per year!
What makes a Physical Quantity?
Every physical quantity consists of two vital parts:
1. A numerical value (the "how much").
2. A unit (the "what").
The SI Base Units
Scientists across the world use the Système Internationale (S.I.) to make sure everyone is speaking the same language. There are six base units you must memorize for this course:
• Mass: kilogram (kg)
• Length: meter (m)
• Time: second (s)
• Current: ampere (A)
• Temperature: kelvin (K)
• Amount of substance: mole (mol)
Derived Units
Most other units are "Derived Units"—they are built by combining the base units.
Example: Density is \(\text{mass} / \text{volume}\). Its derived unit is \(kg \cdot m^{-3}\).
Example: Momentum is \(\text{mass} \times \text{velocity}\). Its derived unit is \(kg \cdot m \cdot s^{-1}\).
Homogeneity: The "Golden Rule"
For a Physics equation to be correct, it must be homogeneous. This simply means the units on the left-hand side must be exactly the same as the units on the right-hand side. We use this to check if an equation is "possible" or if we’ve made a mistake in our algebra.
Quick Review: The Base Six
Mnemonic: Many Lovely Tigers Can Teatime Always (Mass, Length, Time, Current, Temperature, Amount). Okay, it's a bit silly, but it helps!
Key Takeaway: Always include your units! If an equation's units don't match on both sides, the equation is wrong.
2. Prefixes and Estimation
Physics deals with the very big (stars) and the very small (atoms). We use prefixes to avoid writing too many zeros.
Common Prefixes to Memorize
• Tera (T): \(10^{12}\)
• Giga (G): \(10^{9}\)
• Mega (M): \(10^{6}\)
• Kilo (k): \(10^{3}\)
• Deci (d): \(10^{-1}\)
• Centi (c): \(10^{-2}\)
• Milli (m): \(10^{-3}\)
• Micro (\(\mu\)): \(10^{-6}\)
• Nano (n): \(10^{-9}\)
• Pico (p): \(10^{-12}\)
Making Estimates
Sometimes you don't need an exact answer; you just need to know if your answer "makes sense." You should be able to estimate common values:
• Mass of an adult: \(\approx 70\text{ kg}\)
• Height of a door: \(\approx 2\text{ m}\)
• Mass of an apple: \(\approx 100\text{ g} (0.1\text{ kg})\)
• Walking speed: \(\approx 1\text{ m}\cdot\text{s}^{-1}\)
Key Takeaway: Use prefixes to keep your numbers manageable, and always check your final answer against a "common sense" estimate.
3. Errors, Accuracy, and Precision
No measurement is perfect. We categorize these imperfections into two types of errors.
Random vs. Systematic Errors
Random Errors: These cause measurements to be scattered around the true value. They happen because of unpredictable things like air currents or human reaction time.
How to fix: Take repeat readings and calculate a mean.
Systematic Errors: These cause measurements to be constantly "off" by the same amount in the same direction. A common type is a zero error (where a scale reads "0.1g" before you even put anything on it).
How to fix: Re-calibrate your equipment or subtract the error from every reading.
Accuracy vs. Precision
These two words mean very different things in Physics!
• Accuracy: How close your measurement (or the mean of your measurements) is to the true value.
• Precision: How close your repeated measurements are to each other. (If your results are all very similar, they are precise, even if they are all wrong!)
Analogy: Think of a dartboard. If all your darts hit the bullseye, you are accurate and precise. If they all hit the same spot in the far corner, you are precise but not accurate.
Key Takeaway: Repeating readings helps reduce random errors and improves precision, but it won't fix a systematic error.
4. Uncertainties
An uncertainty is a range of values within which the true value is expected to lie. We usually write it as \((\text{Value} \pm \text{Uncertainty})\).
Types of Uncertainty
1. Absolute Uncertainty: The actual range (e.g., \(\pm 0.1\text{ cm}\)).
2. Percentage Uncertainty: The uncertainty expressed as a percentage of the value.
Formula: \(\text{Percentage Uncertainty} = \frac{\text{Absolute Uncertainty}}{\text{Measured Value}} \times 100\)
Combining Uncertainties (The Rules)
When you use measurements in a calculation, the uncertainties "add up." Follow these steps:
• Adding or Subtracting: Add the Absolute uncertainties.
• Multiplying or Dividing: Add the Percentage uncertainties.
• Raising to a Power (e.g., \(x^2\)): Multiply the percentage uncertainty by the power (e.g., multiply by 2).
Don't worry if this seems tricky! Just remember: percentage uncertainties are your friends when multiplying or dividing.
Key Takeaway: As you do more with your data (like squaring it or multiplying it), the "doubt" or uncertainty always increases.
5. Data and Graphs
In your practical work (PAGs), you will often plot graphs to find relationships between variables.
Labelling Conventions
Always label table columns and graph axes with the Quantity / Unit.
Example: Speed / \(m \cdot s^{-1}\) or Time / \(s\).
The forward slash (/) means "Quantity divided by unit," which leaves just the pure number on the axis.
Error Bars and Lines of Best Fit
• Error Bars: These are little "I" shapes drawn on each data point to show the uncertainty in that measurement.
• Line of Best Fit (LOBF): A smooth line that passes as close to as many points as possible.
• Worst Acceptable Line (WAL): The steepest or shallowest possible line that still passes through all your error bars.
Finding Uncertainty in the Gradient
To find out how certain you are of your graph's results, use this simple trick:
\(\text{Uncertainty in gradient} = |\text{gradient of LOBF} - \text{gradient of WAL}|\)
Percentage Difference: This compares your experimental result to the accepted "textbook" value:
\(\text{Percentage Difference} = \frac{|\text{your value} - \text{accepted value}|}{\text{accepted value}} \times 100\)
Key Takeaway: Graphs aren't just for looking at trends; they allow us to calculate the uncertainty in our final results by comparing the best and worst possible lines.
Quick Review Checklist:
• Did I check for zero errors? (Systematic)
• Did I take repeats? (Random)
• Are my graph axes labelled with Quantity / Unit?
• Did I add percentage uncertainties when multiplying?