Measurement, scalars and vectors
Introduced| English | Meaning |
|---|---|
| measurement/ˈmeʒəmənt/ | Finding a quantity by comparing it with a standard unit. |
| scalar/ˈskeɪlə/ | A quantity described by magnitude alone, with no direction. |
| vector/ˈvektə/ | A quantity with both magnitude and direction. |
| parallax/ˈpærəlæks/ | The apparent change in an object's position when viewed from different locations or angles. |
| meniscus/ˈmenɪskəs/ | The curved upper surface of a liquid in a container. |
| period/ˈpɪərɪəd/ | The time taken for one complete oscillation or cycle. |
| resultant/rɪˈzʌltənt/ | The single vector equal to the sum of two or more vectors. |
Why measure carefully?
- Physics starts with measurement 测量 — turning the real world into good numbers.
- A careless reading can ruin a whole experiment.
- So first we learn to measure length and volume, handle tiny amounts, and tell scalars 标量 from vectors 矢量.
Length and volume
- Measure length with a ruler.
- Put your eye straight in front of the mark — looking from the side gives a wrong reading (a parallax 视差 error).
- Measure the volume of a liquid with a measuring cylinder.
- Read the scale at the bottom of the curved surface (the meniscus 弯月面), with your eye level with it.
Read a measuring cylinder at the bottom of the meniscus, with your eye level
Scalars & vectors
resultant = a + b
Add two vectors tip-to-tail to find the resultant.
You measure water in a measuring cylinder. How should you read the scale?
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Read at the bottom of the meniscus with your eye level with it — looking from above or below gives a parallax error (a wrong reading).
Measuring tiny amounts
- One thin page, or one quick swing, is too small to measure well.
- The trick: measure many, then divide.
- Stack 100 sheets of paper, measure the height, then $\div 100$ → the thickness of one sheet.
- Time 20 swings of a pendulum, then $\div 20$ → the period 周期 (the time for one swing).
- Measuring many at once makes the error much smaller.
Two vectors at right angles add to a resultant 合矢量, the diagonal of the rectangle
A stack of 100 sheets of paper is $12\ \text{mm}$ tall. How thick is one sheet, in mm?
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Measure many and divide: $12 \div 100 = 0.12\ \text{mm}$. Dividing makes the small measurement much more accurate.
20 swings of a pendulum take $30\ \text{s}$. What is the period (time for one swing), in s?
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The period is the time for one swing: $30 \div 20 = 1.5\ \text{s}$.
Scalars and vectors
- A scalar has size only.
- A vector has size and direction.
- Scalars: distance, speed, time, mass, energy, temperature.
- Vectors: force, weight, velocity, acceleration, momentum.
- "$30\ \text{m/s}$" is a speed (scalar). "$30\ \text{m/s}$ due north" is a velocity (vector).
Select all the quantities that are vectors.
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Vectors have size and direction: force, velocity, acceleration. Mass and temperature are scalars (size only).
Speed is a vector.
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Speed has size only, so it is a scalar. Velocity (speed in a stated direction) is the vector.
Adding vectors at right angles
- Two vectors at $90°$ add to a resultant — the diagonal of the rectangle they make.
- Find its size with Pythagoras:
- Find its direction with trigonometry (an angle measured from one side).
A force of $3.0\ \text{N}$ acts east and $4.0\ \text{N}$ acts north. What is the size of the resultant, in N?
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They are at right angles, so use Pythagoras: $R = \sqrt{3.0^2 + 4.0^2} = \sqrt{25} = 5.0\ \text{N}$.
Worked examples
1. A student times $20$ swings of a pendulum: $31.0\ \text{s}$. One swing takes $\dfrac{31.0}{20} = 1.55\ \text{s}$. Timing many swings makes the reaction-time error (about $0.2\ \text{s}$) small compared with the total.
2. You walk $30\ \text{m}$ east, then $40\ \text{m}$ north. Resultant displacement $= \sqrt{30^2 + 40^2} = 50\ \text{m}$, at an angle $\tan\theta = \dfrac{40}{30}$, so $\theta = 53°$ north of east.
Vectors do not add like numbers. In example 2 the distance walked is $70\ \text{m}$, but the displacement is only $50\ \text{m}$. Two forces of $3\ \text{N}$ and $4\ \text{N}$ at right angles give $5\ \text{N}$, not $7\ \text{N}$.
You've got it
- read a scale with your eye level with the mark; read a liquid at the meniscus
- for tiny amounts, measure many and divide
- scalar = size only; vector = size and direction
- two vectors at $90°$ → resultant $R = \sqrt{a^2 + b^2}$
Write from a sheet thickness of 0.40 mm
- A stack of 50 sheets is 20 mm tall. An 8 N force acts east and a 15 N force acts north.
- Find one sheet's thickness and the size of the resultant.
A stack of 50 sheets is 20 mm tall. An 8 N force acts east and a 15 N force acts north. What is the thickness of one sheet, and what is the size of the resultant?
20 divided by 50 is 0.40 mm. Pythagoras gives 17 N for the 8 N and 15 N forces.
How thick is one sheet?
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20 divided by 50 is 0.40 mm. Other options — 0.12 mm (the lesson example's result for 100 sheets in 12 mm); 17 mm (the resultant written with the wrong unit).