Physical quantities and measurement techniques
Learning Objective 1: Describe the use of rulers and measuring cylinders.
To measure length accurately with a metre rule:
- Place the object against the ruler so it is parallel to the scale.
- Align the zero mark (or a known starting mark) with one end of the object.
- Read the scale at the other end with your eye directly above the mark to avoid parallax error.
To measure volume with a measuring cylinder:
- Place the cylinder on a flat, horizontal surface.
- Pour the liquid in and wait for it to settle.
- Read the scale at the bottom of the meniscus (the curved surface of the liquid).
- Position your eye at eye level with the meniscus to avoid parallax error.
Correction: Always read the bottom of the concave meniscus for water. Ensure your eye is exactly level with the liquid surface.
Use a stopwatch (analogue or digital) for short intervals. For longer intervals, use a clock.
Learning Objective 3: Determine an average value for a short interval by measuring multiples.
Human reaction time introduces significant error when timing short events (e.g., a ball falling 1m). To reduce this:
- Measure the total time t for many repetitions (e.g., 20 oscillations or 50 falls).
- Divide the total time by the number of repetitions N to find the average time for one event.
Formula: \text{Average time} = \frac{t}{N}
where t is the total time measured and N is the number of events.
Note: This method applies to any short interval where reaction time is significant, not just pendulums. For example, timing 50 drops of water can give a more accurate average drop interval than timing one drop.
When to use: When asked how to improve the accuracy of measuring the period T or length d.
Why examiners accept this: It directly addresses sources of random error (reaction time and parallax).
Correct phrasing examples:
- "Measure the time for 20 oscillations and divide by 20 to reduce the effect of reaction time."
- "Use a fiducial marker (e.g., a vertical line) at the equilibrium position (lowest point) because the bob moves fastest there, making it easier to judge the exact moment it passes."
- "Place a set-square horizontally against the bob to measure the length d from the pivot to the center of the bob accurately."
T = \frac{34.6}{20} = 1.73 \text{ s}
The period is 1.73 s.
- A scalar quantity has magnitude (size) only. It is described by a number and a unit.
- A vector quantity has both magnitude and direction. It requires a number, a unit, and a direction to be fully defined.
Learning Objectives 5 & 6: Classify quantities.
| Scalar Quantities | Vector Quantities |
|---|---|
| Distance | Force |
| Speed | Weight |
| Time | Velocity |
| Mass | Acceleration |
| Energy | Momentum |
| Temperature | Electric Field Strength |
| Gravitational Field Strength |
Note: Distance and displacement are different. Distance is scalar (total path). Displacement is vector (straight line from start to finish). Similarly, speed is scalar, while velocity is vector.
Learning Objective 7: Determine the resultant of two vectors at right angles.
Method 1: Calculation (Pythagoras and Trigonometry)
If two vectors A and B are perpendicular:
- Magnitude (R): Use Pythagoras' theorem.
R = \sqrt{A^2 + B^2} - Direction (\theta): Use trigonometry to find the angle relative to one of the vectors.
\tan(\theta) = \frac{B}{A} \implies \theta = \tan^{-1}\left(\frac{B}{A}\right)
Method 2: Graphical (Scale Drawing)
- Draw vector A to scale.
- From the tip of A, draw vector B perpendicular to it, to scale.
- The resultant is the line from the start of A to the end of B.
- Measure the length (magnitude) and angle (direction).
Solution:
Let v_1 = 3.0 \text{ m/s} (boat's speed) and v_2 = 4.0 \text{ m/s} (river's flow). These are magnitudes of perpendicular velocity vectors.
Magnitude:
R = \sqrt{3.0^2 + 4.0^2} = \sqrt{9 + 16} = \sqrt{25} = 5.0 \text{ m/s}Direction:
Let \theta be the angle relative to the boat's direction.
\tan(\theta) = \frac{4.0}{3.0}
\theta = \tan^{-1}(1.33) \approx 53^\circ
The resultant velocity is 5.0 \text{ m/s} at 53^\circ to the boat's direction.
Correction: Speed is just how fast (scalar). Velocity includes direction (vector). Similarly, distance is scalar, displacement is vector. Mass is scalar, weight is a force (vector).
When to use: When asked to 'draw' or 'describe' how to find a resultant.
Why examiners accept this: It demonstrates understanding of the 'tip-to-tail' method.
Correct phrasing:
- "Draw the first vector to scale. From the tip (arrow end) of the first vector, draw the second vector perpendicular to it. The resultant is the line from the tail of the first to the tip of the second."
- "Use a ruler to measure the length and a protractor to measure the angle."
R = \sqrt{30^2 + 40^2} = \sqrt{900 + 1600} = \sqrt{2500} = 50 \text{ N}
The magnitude is 50 N.