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?
Difficulty 1/5 · 100 coins + a small bonus on first completion today
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?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
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?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
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.
Difficulty 2/5 · 140 coins + a small bonus on first completion today
Vectors have size and direction: force, velocity, acceleration. Mass and temperature are scalars (size only).
Speed is a vector.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
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?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
They are at right angles, so use Pythagoras: $R = \sqrt{3.0^2 + 4.0^2} = \sqrt{25} = 5.0\ \text{N}$.
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 $R = \sqrt{a^2 + b^2}$