Physics notes, chapter 1

Measurement

Textbook Physics Form 4 KSSM (DLP), pages 2–22

What this chapter covers

Identify physical quantities and state their magnitude and S.I. unit (1.1.1)
List the seven base quantities with their S.I. units and symbols (1.1.2)
Express derived quantities and units in terms of base quantities and base units (1.1.3)
Distinguish scalar quantities from vector quantities (1.1.4)
Interpret the shapes of graphs and the relationship between two quantities (1.2.1)
Analyse a graph using gradient, area under the graph, interpolation and extrapolation (1.2.2)
Plan and report a scientific investigation, using a simple pendulum as the example (1.2.3)

1.1Physical quantities

Measurement is a method to determine the value of a physical quantity. A physical quantity is always stated as magnitude + unit, for example length = 100 m.

Metric units (centimetre, kilopascal) are used in science. Imperial units (inch, mile, psi) are seldom used. Mixing the two caused the Mars Climate Orbiter to be lost in 1999. Scientists everywhere use the International System of Units, the S.I., agreed in 1960.

Base quantities

A base quantity cannot be derived from another physical quantity. There are seven.

Base quantity	Symbol	S.I. unit	Unit symbol
Length	l	metre	m
Mass	m	kilogram	kg
Time	t	second	s
Thermodynamic temperature	T	kelvin	K
Electric current	I	ampere	A
Luminous intensity	I_v	candela	cd
Amount of substance	n	mole	mol

Remember the S.I. unit of mass is the kilogram, not the gram, so always convert before using a formula.

Derived quantities

A derived quantity is built from base quantities through a formula. Its unit is built from the base units in the same way.

Derived quantity	Formula	In base quantities	S.I. unit
Area	A = l²	l × l	m²
Volume	V = l³	l × l × l	m³
Density	ρ = m / V	m / l³	kg m⁻³
Velocity	v = l / t	l / t	m s⁻¹
Acceleration	a = v / t	l / t²	m s⁻²
Force	F = m × a	ml / t²	kg m s⁻² = newton (N)
Momentum	p = m × v	ml / t	kg m s⁻¹
Pressure	P = F / A	m / (l t²)	kg m⁻¹ s⁻² = pascal (Pa)
Work or energy	W = F × l	ml² / t²	kg m² s⁻² = joule (J)
Charge	Q = I × t	I × t	A s = coulomb (C)

Worked example: unit of power

Power = work ÷ time, and work = force × length.
Unit of work = kg m s⁻² × m = kg m² s⁻².
Unit of power = kg m² s⁻² ÷ s = kg m² s⁻³ (the watt, W).

Writing the newton as a base unit. N, Pa, J and C are named derived units; they are never base units.

Derive the unit of pressure in base units.

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P = F / A = (kg m s⁻²) / m² = kg m⁻¹ s⁻².

Scalar and vector quantities

A scalar has magnitude only. A vector has magnitude and direction.

Scalar	Vector
Distance, length, area, volume, time, speed, work, energy, temperature, density	Displacement, velocity, acceleration, force, momentum

Example

"Run 50 m" gives a distance (scalar). "Run 50 m towards that tree" gives a displacement (vector), because a direction is stated.

Speed is the scalar partner of velocity. If a question gives a direction (north, upward, towards), the quantity is a vector.

1.2Scientific investigation

Shapes of graphs

Six graph shapes: straight line through origin, straight line with positive intercept, straight line with negative gradient, upward curve, downward curve, inverse curve — Six common shapes of y against x. (Drawn after textbook Figure 1.8, pp. 10–11.)

Graph	Meaning	Equation
1 Straight line through origin	y is directly proportional to x	y = mx
2 Straight line, positive gradient, intercept c	y increases linearly with x	y = mx + c
3 Straight line, negative gradient	y decreases linearly with x	y = mx + c (m < 0)
4 Curve rising from origin	y increases with x	no simple equation
5 Curve falling	y decreases with x	no simple equation
6 Curve that never touches either axis	y is inversely proportional to x; the graph of y against 1/x is a straight line through the origin	y = k / x

Saying "y increases with x" is not the same as "directly proportional". Directly proportional means a straight line that passes through the origin.

Five things to analyse in a graph

Relationship between the variables: read the shape.
Gradient: m = Δy / Δx = (y₂ − y₁) / (x₂ − x₁). Use a large triangle and points on the line, not table values.
Area under the graph: rectangle = base × height, triangle = ½ × base × height.
Interpolation: reading a value between plotted points.
Extrapolation: extending the line beyond the data to predict a value.

The gradient and area carry units. The unit of the gradient is (unit of y) ÷ (unit of x); the unit of the area is (unit of y) × (unit of x).

Worked example: spring (textbook Activity 1.3)

Graph of force against extension of a spring, a straight line through the origin up to 4.0 N at 6.4 cm — Force F against extension x. Data: F = 0.5 N to 4.0 N, x = 0.8 cm to 6.4 cm.

(a) Straight line through the origin: F is directly proportional to x.
(b) Gradient k = (4.0 − 0.5) / (6.4 − 0.8) = 3.5 / 5.6 = 0.625 N cm⁻¹ = 62.5 N m⁻¹.
(c) Work done = area under the graph = ½ F x. At x = 5 cm, F = 0.625 × 5 = 3.125 N, so W = ½ × 3.125 × 0.05 = 0.078 J (convert 5 cm to 0.05 m).
(d) Interpolation: at x = 3.5 cm, F = 0.625 × 3.5 = 2.2 N (read between plotted points).
(e) Extrapolation: F = 5.0 N gives x = 5.0 / 0.625 = 8.0 cm.

Data and set-up: Physics Form 4 (DLP) textbook, Activity 1.3, p. 13.

A straight line passes through (0, 10) and (15, 40). State its gradient and equation.

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Gradient = (40 − 10) / (15 − 0) = 2.0. Intercept = 10, so y = 2x + 10.

Plotting a graph

Put the manipulated variable on the x-axis and the responding variable on the y-axis.
Label each axis with the quantity and its unit, for example l / cm.
Choose even scales so the points fill at least half of the graph paper.
Mark each point clearly and draw one smooth best-fit line or curve, with points on both sides of it.

Simple pendulum investigation (Experiment 1.1)

Part	In this experiment
Inference	The period of a pendulum depends on its length
Hypothesis	The longer the pendulum, the longer its period
Aim	To investigate the relationship between length l and period T
Manipulated variable	Length of pendulum, l
Responding variable	Period of oscillation, T
Constant variable	Mass of the bob

Apparatus: retort stand, thread, pendulum bob, stopwatch, metre rule, protractor and G-clamp.
Change l (20.0 cm to 70.0 cm). For each length, displace the bob through less than 10° and time 20 complete oscillations, twice. Then t_average = (t₁ + t₂) / 2 and T = t_average / 20.
Timing 20 oscillations reduces the error caused by the reaction time of the person using the stopwatch; repeating the timing and averaging reduces random error.
Plot T² against l: a straight line through the origin. Since T² = 4π²l / g, the gradient m = 4π² / g, so g = 4π² / m.

Worked example

20 oscillations take 28.0 s. Period T = 28.0 / 20 = 1.40 s. With g = 10 m s⁻², l = gT² / 4π² = 10 × 1.96 / 39.5 = 0.50 m.

Taking T as the time for one oscillation measured once. A single oscillation is too short to time accurately; always use 20 and divide.

The same pendulum is taken to the Moon. Will its gradient of T² against l change?

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Yes. The gradient is 4π² / g and g on the Moon is smaller, so the gradient is larger.