Topic 1: Quantities and Measurement
Course: Prep4Uni Physics 1
Chapter 1: Quantities and Measurement
Chapter 2: Forces and Moments
Chapter 3: Motion and Forces
Chapter 4: Energy and Fields
Chapter 5: Projectile Motion
🚁Overview
This topic introduces the fundamental language of physics — the ability to describe, quantify, and measure physical phenomena. These foundational ideas underpin all topics in physics and engineering.
📖Contents
- Physical Quantities and SI Units
- Prefixes for Multiples and Submultiples
- Derived Units and Dimensional Homogeneity
- Errors and Uncertainties
- Scalars and Vectors
🎯 Learning Outcomes
By the end of this topic, you should be able to:
Recall and use SI base quantities and units
Convert units using SI prefixes
Use dimensional analysis to check equations
Identify types of measurement errors and estimate uncertainties
Distinguish between scalar and vector quantities
Add vectors using graphical and component methods
Table of Contents
🔢 1. Physical Quantities and SI Units
A physical quantity is something that can be measured — like length, time, mass, or temperature.
All physical quantities are expressed using a numerical value and a unit (e.g., 5.2 m, 3.0 s).
SI Base Quantities and Units
| Quantity | Unit Name | Symbol |
|---|---|---|
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Electric current | ampere | A |
| Temperature | kelvin | K |
| Amount of substance | mole | mol |
Example:
The height of a doorway is measured to be 2.10 m. This means the physical quantity “length” has a numerical value of 2.10 and a unit of metre (m).
✅ Learning Check: Which of the following is not a base SI quantity?
A. Speed B. Mass C. Time D. Temperature
(Answer: A — speed is a derived quantity: distance/time)
📏 2. Prefixes for Multiples and Submultiples
SI prefixes make it easier to express very large or small numbers.
| Prefix | Symbol | Factor |
| kilo | k | 10³ (1,000) |
| mega | M | 10⁶ |
| giga | G | 10⁹ |
| milli | m | 10⁻³ |
| micro | μ | 10⁻⁶ |
| nano | n | 10⁻⁹ |
Example 1:
A 5 km road = 5 × 10³ m = 5000 m
Example 2:
A computer processor operates at 3.2 GHz = 3.2 × 10⁹ Hz
✅ Learning Check: Convert the following:
1200 mm = ? m → Answer: 1.2 m
2.5 μs = ? s → Answer: 2.5 × 10⁻⁶ s
⚙️ 3. Derived Units and Dimensional Homogeneity
Derived units are combinations of base units. For example:
Speed = distance/time → m/s
Force = mass × acceleration → kg·m/s² = newton (N)
Dimensional homogeneity means that every term in an equation has the same unit structure.
Example: In the equation:
s = ut + ½at²
u is in m/s
a is in m/s²
t is in s
s must be in metres for the equation to be valid
📐 Learning Check: Check dimensional consistency of F = ma
Force → [F] = kg·m/s²
Mass = kg, Acceleration = m/s² → ✅ Equation is dimensionally consistent.
Learn how to instantly spot invalid physics equations using the principle of dimensional homogeneity! In this short tutorial, you’ll learn:
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✅ What dimensional homogeneity means and why it’s a must-have check
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🔍 Step-by-step how to compare units on both sides of any equation
-
⚠️ Common pitfalls to watch out for when you’re solving problems
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📚 Real-world examples that demonstrate why mismatched dimensions always signal an error
By the end of this video, you’ll be able to scan any physics or engineering formula and verify its validity in seconds—no calculator required!
⚖️ 4. Errors and Uncertainties
All measurements have some uncertainty due to limitations of instruments or human judgment.
Types of Errors
- Random errors: unpredictable variations (e.g., reading fluctuations)
- Systematic errors: consistent bias (e.g., zero error in a balance)
Expressing Uncertainty
- Write as: value ± uncertainty
- Example: L = 25.4 ± 0.2 cm
Propagation of Uncertainty
- When adding/subtracting: add absolute uncertainties
- When multiplying/dividing: add percentage uncertainties
Example:
- You measure time t = 2.00 ± 0.05 s and distance s = 10.0 ± 0.2 m
- Velocity v = s/t, so % uncertainty = (0.2/10.0 + 0.05/2.00) × 100% = 7%
✅ Learning Check: If you add 12.0 ± 0.2 cm and 8.0 ± 0.1 cm, what’s the total length and uncertainty?
Answer: 20.0 ± 0.3 cm
🎥 Visual Lesson: Vector Addition
🔗
Watch on YouTube and subscribe for more Prep4Uni.Online lessons
🧭 5. Scalars and Vectors
A scalar has magnitude only (e.g., speed, time, energy)
A vector has both magnitude and direction (e.g., velocity, force, displacement)
Vector Representation
Use arrows: length = magnitude, direction = orientation
Notation: → examples are v or italic v or bold v
Vector Addition
Tip-to-tail method: Place one vector’s tail at the other’s tip
Vector Addition Tip to Tail Method Parallelogram method: Construct parallelogram to add vectors visually
Component method: Break vectors into x and y components and add separately
Example: A plane flies 100 km east, then 80 km north.
Resultant displacement = √(100² + 80²) = 128.1 km
Direction = tan⁻¹(80/100) = 38.7° north of east
✅ Learning Check: A force of 50 N acts at 30° to the horizontal. What are the horizontal and vertical components?
Horizontal: 50 cos(30°) = 43.3 N
Vertical: 50 sin(30°) = 25.0 N
| Concept | Definition / Formula |
|---|---|
| Physical Quantity | Something measurable, expressed as value + unit (e.g. 5.2 m, 3.0 s) |
| SI Base Units | Length (m), Mass (kg), Time (s), Current (A), Temperature (K), Amount (mol), Luminous intensity (cd) |
| SI Prefixes | kilo (k, 10³), mega (M, 10⁶), giga (G, 10⁹), milli (m, 10⁻³), micro (μ, 10⁻⁶), nano (n, 10⁻⁹) |
| Derived Units & Dimensional Homogeneity | Combine base units (e.g. N=kg·m/s²). Ensure each term in an equation has same dimensions. |
| Measurement Uncertainty | Express as value ± uncertainty. Add absolute errors for ± operations; add percentage errors for ×/÷ |
| Random vs Systematic Errors | Random: unpredictable scatter; Systematic: consistent bias (e.g. zero offset) |
| Scalars & Vectors | Scalar: magnitude only (e.g. time, energy). Vector: magnitude + direction (e.g. displacement, velocity) |
| Vector Addition | Graphical: Tip-to-tail or parallelogram; Component: add x and y separately |
Proceed to: Topic 2: Forces and Moments
Return to Prep4Uni Physics 1
📝EXERCISES
Questions and Answers (Try to answer before looking at the answer)
🔹 Section 1: Physical Quantities and SI Units
1. What are the seven SI base quantities?
Answer: Length, mass, time, electric current, thermodynamic temperature, amount of substance, luminous intensity.
2. State the SI base unit for electric current.
Answer: Ampere (A)
3. Which SI unit is used to measure temperature?
Answer: Kelvin (K)
4. Which physical quantity is measured in kilograms (kg)?
Answer: Mass
5. What is the SI unit for time?
Answer: Second (s)
🔹 Section 2: Prefixes for Multiples and Submultiples
6. What is the prefix for 10⁶?
Answer: Mega (M)
7. Convert 5 mm to metres.
Answer: 5 mm = 0.005 m
8. What does the prefix “nano” represent?
Answer: 10⁻⁹
9. Express 0.000001 s using a prefix.
Answer: 1 µs (microsecond)
10. How many metres are there in 2.5 km?
Answer: 2.5 km = 2500 m
🔹 Section 3: Derived Units and Dimensional Homogeneity
11. What is the SI unit for force?
Answer: Newton (N)
12. Show that the unit of force (N) is a derived unit.
Answer: N = kg·m/s²
13. Write the dimensions of kinetic energy.
Answer: [M][L]²[T]⁻²
14. Check if the equation
is dimensionally consistent.
Answer: Yes, all terms have the dimension of length [L].
15. What is meant by dimensional homogeneity?
Answer: All terms in a physically valid equation must have the same dimensions.
🔹 Section 4: Errors and Uncertainties
16. Name two types of systematic errors.
Answer: Zero error and calibration error.
17. What is the difference between random error and systematic error?
Answer: Random errors vary unpredictably; systematic errors are consistent and repeatable.
18. A length is measured as 15.2 ± 0.1 cm. What is the percentage uncertainty?
Answer: (0.1 / 15.2) × 100% ≈ 0.66%
19. If two quantities A = 5.0 ± 0.2 and B = 3.0 ± 0.1 are added, what is the uncertainty in A + B?
Answer: ±(0.2 + 0.1) = ±0.3
20. What is the best way to reduce random errors?
Answer: Take repeated measurements and average the results.
🔹 Section 5: Scalars and Vectors
21. Define scalar quantity and give an example.
Answer: A scalar has magnitude only. Example: temperature.
22. Define vector quantity and give an example.
Answer: A vector has both magnitude and direction. Example: velocity.
23. Is mass a scalar or a vector?
Answer: Scalar
24. How can vectors be added graphically?
Answer: By using the tip-to-tail method or the parallelogram method.
25. A displacement of 4 m east is added to 3 m north. What is the resultant magnitude?
Answer: √(4² + 3²) = √25 = 5 m
Problems and Solutions 1 (Try to answer before looking at the solution)
Problem 1: A car travels 240 km in 3 hours. What is its average speed in m/s?
Solution:
240 km = 240,000 m
3 hours = 3 × 3600 = 10,800 s
Average speed = distance ÷ time = 240,000 ÷ 10,800 ≈ 22.2 m/s
Problem 2: Convert 5.6 × 10⁻³ m into a unit with a prefix.
Solution:
5.6 × 10⁻³ m = 5.6 mm (millimetres)
Problem 3: The unit of energy is the joule (J). Show that it is dimensionally consistent with the formula E = ½mv².
Solution:
½mv² → dimensions are [M] × [L²][T⁻²] = [M][L²][T⁻²]
This is the dimension of energy, which matches 1 J = 1 kg·m²/s²
Problem 4: A stopwatch shows a time of 3.52 s. The least count is 0.01 s. What is the absolute and percentage uncertainty?
Solution:
Absolute uncertainty = ±0.01 s
Percentage uncertainty = (0.01 ÷ 3.52) × 100 ≈ 0.28%
Problem 5: Two masses are measured as A = 5.0 ± 0.2 kg and B = 3.0 ± 0.1 kg. What is the total mass and its uncertainty?
Solution:
Total mass = A + B = 5.0 + 3.0 = 8.0 kg
Uncertainty = 0.2 + 0.1 = ±0.3 kg
Answer: 8.0 ± 0.3 kg
Problem 6: Determine whether the equation s = vt² is dimensionally consistent.
Solution:
RHS: vt² → [L][T²] = [L][T²] = [L·T²]
LHS: s → [L]
Since LHS ≠ RHS, the equation is not dimensionally consistent.
Problem 7: Express 1.2 × 10⁹ Hz using a suitable prefix.
Solution:
1.2 × 10⁹ Hz = 1.2 GHz (gigahertz)
Problem 8: A displacement vector 6 m at 30° is added to another vector 4 m at 120°. Find the magnitude of the resultant using components.
Solution:
Break into components:
Vector A (6 m at 30°):
x₁ = 6 cos(30°) ≈ 5.20, y₁ = 6 sin(30°) = 3.00
Vector B (4 m at 120°):
x₂ = 4 cos(120°) ≈ -2.00, y₂ = 4 sin(120°) ≈ 3.46
Total x = 5.20 + (–2.00) = 3.20
Total y = 3.00 + 3.46 = 6.46
Resultant = √(3.20² + 6.46²) ≈ √52.1 ≈ 7.22 m
Problem 9: A reading of 1.55 m is taken with a ruler marked in mm. What is the uncertainty and its percentage?
Solution:
Least count = 1 mm = 0.001 m
Uncertainty = ±0.001 m
Percentage = (0.001 ÷ 1.55) × 100 ≈ 0.065%
Problem 10: The unit of pressure is the pascal (Pa). Express it in terms of SI base units.
Solution:
Pressure = Force ÷ Area = N/m²
1 N = kg·m/s²
So, 1 Pa = kg·m⁻¹·s⁻²
Answer: [M][L⁻¹][T⁻²]
Problems and Solutions 2 (Try to answer before looking at the solution)
1. Vector Addition (tip-to-tail method – travel context)
Problem: A hiker walks 4 km east, then 3 km north. What is her total displacement from the starting point?
Solution:
This forms a right triangle.
Displacement = √(4² + 3²) = √(16 + 9) = √25 = 5 km northeast
2. Vector Addition (components – flight context)
Problem: A plane flies 200 km due north, then 150 km due east. What is the resultant displacement?
Solution:
x = 150 km, y = 200 km
Resultant = √(150² + 200²) = √(62500) = 250 km
Direction = tan⁻¹(200/150) ≈ 53.1° east of north
3. Vector Subtraction (geographical displacement)
Problem: A ship sails 80 km north and then 50 km south. What is its final displacement?
Solution:
Net displacement = 80 – 50 = 30 km north
4. Vector Subtraction (flight path correction)
Problem: A drone is intended to fly 100 m east but is blown 60 m west by wind. What is its actual displacement relative to the starting point?
Solution:
100 m east – 60 m west = 40 m east
5. Vector Addition (components with angles)
Problem: A vector of 10 m acts at 30°, and another of 8 m at 120°. Find the resultant using components.
Solution:
First vector:
x₁ = 10 cos(30°) ≈ 8.66, y₁ = 10 sin(30°) = 5.00
Second vector:
x₂ = 8 cos(120°) ≈ –4.00, y₂ = 8 sin(120°) ≈ 6.93
Total x = 8.66 – 4.00 = 4.66
Total y = 5.00 + 6.93 = 11.93
Resultant = √(4.66² + 11.93²) ≈ √162.1 ≈ 12.73 m
Direction = tan⁻¹(11.93 / 4.66) ≈ 68.5° from x-axis
6. Navigation (vector resolution)
Problem: A boat heads north at 5 m/s, but a current pushes it east at 3 m/s. What is the resultant velocity?
Solution:
Resultant = √(5² + 3²) = √34 ≈ 5.83 m/s northeast
Direction = tan⁻¹(3/5) ≈ 31.0° east of north
7. Vector Subtraction (position change)
Problem: An object moves from point A (3, 2) to point B (7, 6). What is the displacement vector?
Solution:
Displacement = (7 – 3, 6 – 2) = (4, 4)
Magnitude = √(4² + 4²) = √32 = 5.66 units
8. Walking Path (component addition)
Problem: A person walks 6 km northeast, then 6 km southeast. Find the total displacement.
Solution:
Northeast = (6 cos 45°, 6 sin 45°) = (4.24, 4.24)
Southeast = (4.24, –4.24)
x total = 4.24 + 4.24 = 8.48
y total = 4.24 – 4.24 = 0
Resultant displacement = 8.48 km due east
9. Vector Addition (airplane with wind)
Problem: An airplane flies at 300 km/h north. A crosswind of 100 km/h blows from the west. What is the actual ground speed and direction?
Solution:
x = 100 km/h, y = 300 km/h
Resultant speed = √(100² + 300²) = √100000 = 316.2 km/h
Direction = tan⁻¹(100 / 300) ≈ 18.4° east of north
10. Round Trip Displacement (urban navigation)
Problem: A taxi goes 5 blocks east, 3 blocks north, 5 blocks west, and 3 blocks south. What is its final displacement?
Solution:
Net east-west = 5 – 5 = 0
Net north-south = 3 – 3 = 0
Displacement = 0 blocks (returns to start)
📝MORE EXERCISES
25 Learning‐Check Questions & Answers (Try to answer before looking at the answer)
(5 from each of the 5 subsections in “Forces & Moments”)
1. Types of Forces
Q: What contact force acts perpendicular to a surface?
A: Normal force.Q: Which force always opposes relative motion between two surfaces?
A: Frictional force.Q: Name the force that pulls on a mass hanging from a string.
A: Tension.Q: What non-contact force acts downward on all masses near Earth’s surface?
A: Gravitational force (weight).Q: Identify a viscous force example.
A: Air drag on a moving car.
2. Hooke’s Law and Elastic Forces
Q: State Hooke’s Law.
A: F= k xQ: A spring stretches 0.10 m under a 5 N load. What is ?
A: k=5/0.10=50 N/m.
Q: True or false: Hooke’s Law holds for any extension of a spring.
A: False—only within the elastic limit.Q: If N/m, how much force to compress the spring by 0.02 m?
A: N.
Q: A spring’s constant is 150 N/m. How far does it extend under 3 N?
A: x =3/150 = 0.02 m
3. Moment and Torque
Q: Define moment of a force.
A: (force × perpendicular distance to pivot).Q: What is the unit of moment?
A: Newton-metre (Nm).Q: A 12 N force applied 0.25 m from a hinge produces what moment?
A: Nm.Q: If you need a 10 Nm torque and apply 20 N, what lever arm?
A: m.Q: True or false: torque and moment are the same concept.
A: True (torque is a rotational moment).
4. Couples & Principle of Moments
Q: What is a pure couple?
A: Two equal and opposite forces whose lines of action are separate, causing rotation without net force.Q: Moment of a couple of two 15 N forces 0.4 m apart?
A: Nm.Q: State the principle of moments for a body in rotational equilibrium.
A: Sum of clockwise moments = sum of anticlockwise moments.Q: A 20 N weight at 1.5 m must be balanced by what force at 1 m?
A: NmN.
Q: True or false: a couple produces translation of its body.
A: False—only pure rotation.
5. Equilibrium Conditions
Q: What are the two equilibrium conditions for a rigid body?
A: ∑F = 0 and ∑M = 0.Q: A box at rest on a table: what is ∑F_y?
A: Zero (normal = weight).Q: Name the diagram listing all forces on an object.
A: Free-body diagram.Q: True or false: in static equilibrium, objects may rotate with constant angular speed.
A: False—static means no rotation (constant speed only if dynamic equilibrium).Q: A beam is at equilibrium under three forces; how many equations can you write?
A: Three in 2D: ∑F_x=0, ∑F_y=0, ∑M=0.
20 Problems & Detailed Solutions (Try to answer before looking at the solution)
(Covering all five sections of Forces & Moments)
Identify Forces on a Hanging Mass
N downward; Tension 49 N upward.
Problem: A 5 kg mass hangs from a ceiling by a rope. List and draw the forces acting on it.
Solution: WeightFriction on a Sliding Block
Problem: A 2 kg block slides on a horizontal surface with μ_k=0.3. What frictional force acts?
Solution:N →
N backward.
Spring Extension
Problem: A spring of N/m supports a 4 kg mass. How far does it extend?Solution:
N →
m.
Stokes’ Drag (Viscous Force)
Problem: A small sphere falls in oil with terminal speed 0.1 m/s and radius 0.01 m. Given η=0.5 Pa·s, find drag.
Solution:N.
Moment of a Force
Problem: 15 N perpendicular applied at 0.2 m from pivot. Moment?
Solution:Nm.
Torque Vector
Problem: ForceN applied at position m; compute torque about origin (into page).
Solution: τ = xF_y − yF_x = 0.5×20−0=10 Nm (into page).Balancing a Couple
Problem: Two equal opposite forces of 10 N are 0.5 m apart. Couple moment?
Solution:Nm.
Lever Balance
Problem: A 50 N weight 2 m left of pivot balanced by a force 1 m right. Find F.
Solution:→ F=100 N.
Support Reactions (Statics)
Problem: A uniform 4 m beam weighs 200 N rests on two supports at its ends. Find reactions at each support.
Solution: Symmetric → R₁=R₂=100 N.String Tension on Pulley
Problem: Two masses, 3 kg and 5 kg, hang over a frictionless pulley. Find tensions and acceleration.
Solution: a = (m₂−m₁)g/(m₁+m₂)=2×9.8/8=2.45 m/s²; T = m₁(g+a)=3×(9.8+2.45)=36.75 N.Spring & Weight Equilibrium
Problem: A 1 kg mass hung on spring with k=200 N/m. Find extension.
Solution: mg = kx → x = 9.8/200 = 0.049 m.Max Friction on Incline
Problem: μ_s=0.4, find max θ before sliding.
Solution: tanθ = μ → θ ≈ 21.8°.Free-Body & Equations
Problem: Block on 30° incline, μ=0.2, m=2 kg. Will it slide?
Solution: F_||=19.6×sin30=9.8 N; f_max=0.2×19.6×cos30≈3.39 N → 9.8>3.39 → yes slides.Distributed Forces to CG
Problem: Three point masses on x-axis: 2 kg at 0 m, 3 kg at 2 m, 5 kg at 5 m. Find CG.
Solution: x_CG = (2×0+3×2+5×5)/(2+3+5) = (0+6+25)/10 = 3.1 m.Rotational Equilibrium of Beam
Problem: Simply supported beam length 6 m with 300 N at 2 m. Find support reactions.
Solution: Moments about A: R_B×6 =300×2 → R_B=100 N; R_A=300−100=200 N.Torque with Angle
Problem: 50 N force at 30° to lever arm length 0.4 m. Compute τ.
Solution: τ=rF sinθ=0.4×50×sin30=0.4×50×0.5=10 Nm.Couple vs Single Force
Problem: Show two 20 N forces at ±1 m apart form a couple. Compute moment.
Solution: M=20×2=40 Nm.Buoyant Force
Problem: Cube 0.1 m side fully submerged in water (ρ=1000 kg/m³). Find buoyancy.
Solution: V=0.001 m³ → F_b =ρgV=1000×9.8×0.001=9.8 N.Magnetic Force
Problem: 0.5 A wire length 0.2 m in B=0.3 T perpendicular. Force?
Solution: F=ILB=0.5×0.2×0.3=0.03 N.Equilibrium of Concurrent Forces
Problem: Forces 30 N at 0° and 40 N at 90°. Find third force for equilibrium.
Solution: Sum = (30,0)+(0,40)=(30,40); R=−(30,40), magnitude=50 N, direction=216.9°.
Notation convention:
We sometimes use the underscore character (“_”) to indicate subscripts. For example,v_x → vx and s_total → stotal.
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