Introduction to Physics - CXC/CSEC Mathematics

Overview of Physics

Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, with its main goal being to understand how the universe behaves.

In the CXC/CSEC Mathematics curriculum, physics concepts are applied through mathematical principles and formulas to solve real-world problems. Understanding the basics of physics is essential for successfully tackling many mathematical applications.

Branches of Physics Relevant to CXC Mathematics

Mechanics - dealing with motion, forces, energy
Thermodynamics - dealing with heat and temperature
Electricity and Magnetism - dealing with electrical charges and magnetic fields
Waves - dealing with sound, light, and other wave phenomena

Fundamental Physical Quantities and Units

SI Base Units

The International System of Units (SI) defines seven base units from which all other units are derived:

Physical Quantity	SI Unit	Symbol
Length	Meter	m
Mass	Kilogram	kg
Time	Second	s
Electric Current	Ampere	A
Temperature	Kelvin	K
Amount of Substance	Mole	mol
Luminous Intensity	Candela	cd

Derived Units

Other physical quantities are expressed as combinations of the base units:

Physical Quantity	SI Unit	Symbol	Formula
Area	Square meter	m²	length × width
Volume	Cubic meter	m³	length × width × height
Velocity	Meter per second	m/s	displacement/time
Acceleration	Meter per second squared	m/s²	velocity change/time
Force	Newton	N	mass × acceleration
Energy/Work	Joule	J	force × distance
Power	Watt	W	energy/time
Pressure	Pascal	Pa	force/area

Motion in One Dimension

Scalars and Vectors

In physics, quantities are classified as either scalars or vectors:

Scalar quantities have only magnitude (size) - examples: time, mass, temperature, speed
Vector quantities have both magnitude and direction - examples: displacement, velocity, acceleration, force

Figure 1: Comparison of scalar (mass) and vector (force) quantities

Key Concepts in Motion

Position: The location of an object relative to a reference point
Distance: The total path length traveled by an object (scalar)
Displacement: The straight-line distance and direction from initial to final position (vector)
Speed: The rate of change of distance (scalar)
Velocity: The rate of change of displacement (vector)
Acceleration: The rate of change of velocity (vector)

Equations of Motion

For constant acceleration in a straight line, we can use the SUVAT equations:

v = u + at
s = ut + ½at²
v² = u² + 2as
s = ½(u + v)t

Where:

s = displacement
u = initial velocity
v = final velocity
a = acceleration
t = time

Important: When solving motion problems, always make sure to:

Identify what variables you know and what you need to find
Choose the appropriate equation
Be consistent with your units
Pay attention to signs (positive/negative) for direction

Practical Application: Distance-Time and Velocity-Time Graphs

Figure 2: Distance-Time graph showing constant velocity (left) and Velocity-Time graph showing constant velocity (right)

In a distance-time graph:

The slope represents the velocity
A straight line indicates constant velocity
A curve indicates changing velocity

In a velocity-time graph:

The slope represents the acceleration
The area under the curve represents the displacement

Forces and Newton's Laws of Motion

Newton's First Law (Law of Inertia)

An object will remain at rest or in uniform motion in a straight line unless acted upon by an external force.

Newton's Second Law (F = ma)

The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass:

F = ma

Where:

F = force (Newtons, N)
m = mass (kilograms, kg)
a = acceleration (meters per second squared, m/s²)

Newton's Third Law (Action-Reaction)

For every action, there is an equal and opposite reaction.

Figure 3: Newton's Third Law - action and reaction forces

Types of Forces

Gravitational Force: The attraction between masses
Normal Force: The perpendicular force exerted by a surface
Friction: The force that opposes motion between surfaces
Tension: The pulling force transmitted through a string, rope, or cable
Spring Force: The force exerted by a compressed or stretched spring

Weight vs. Mass: Mass is a measure of how much matter an object contains (kg), whereas weight is the gravitational force acting on that mass (N).

Weight = mass × gravitational acceleration (W = mg)

Work, Energy, and Power

Work

Work is done when a force causes displacement in the direction of the force:

W = F × d × cos(θ)

Where:

W = work (Joules, J)
F = force (Newtons, N)
d = displacement (meters, m)
θ = angle between force and displacement

Energy

Energy is the capacity to do work. The main forms of energy relevant to CXC Mathematics include:

Kinetic Energy: Energy of motion
KE = ½mv²
Potential Energy: Stored energy due to position or configuration
PE = mgh

Conservation of Energy

Energy cannot be created or destroyed, only transformed from one form to another.

Total Energy = KE + PE = constant (in an isolated system)

Power

Power is the rate of doing work or transferring energy:

P = W/t = E/t

Where:

P = power (Watts, W)
W = work (Joules, J)
E = energy (Joules, J)
t = time (seconds, s)

Simple Machines and Mechanical Advantage

Types of Simple Machines

Lever
Wheel and axle
Pulley
Inclined plane
Wedge
Screw

Mechanical Advantage

Mechanical advantage (MA) is the ratio of output force to input force:

MA = F_output / F_input

For different machines:

Lever: MA = effort arm / load arm
Inclined Plane: MA = length / height
Pulley System: MA = number of supporting strands

Figure 4: First-class lever showing effort arm and load arm

Fluid Mechanics

Pressure

Pressure is defined as force per unit area:

P = F/A

Where:

P = pressure (Pascals, Pa)
F = force (Newtons, N)
A = area (square meters, m²)

Pressure in Liquids

Pressure in a liquid increases with depth:

P = ρgh

Where:

P = pressure (Pascals, Pa)
ρ (rho) = density of the liquid (kg/m³)
g = gravitational acceleration (9.8 m/s²)
h = depth (meters, m)

Archimedes' Principle

Any object wholly or partially immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced:

F_buoyant = ρ_fluid × g × V_displaced

Bernoulli's Principle

As the speed of a fluid increases, its pressure decreases.

Waves and Sound

Wave Characteristics

Wavelength (λ): Distance between consecutive identical points
Frequency (f): Number of complete waves per second
Amplitude: Maximum displacement from equilibrium
Period (T): Time taken for one complete cycle

Wave Speed

The speed of a wave can be calculated using:

v = f × λ = λ/T

Figure 5: Wave characteristics showing wavelength and amplitude

Types of Waves

Transverse Waves: Particles move perpendicular to the direction of wave propagation (e.g., light waves)
Longitudinal Waves: Particles move parallel to the direction of wave propagation (e.g., sound waves)

Electricity and Magnetism

Electric Current

Electric current is the rate of flow of electric charge:

I = Q/t

Where:

I = current (Amperes, A)
Q = charge (Coulombs, C)
t = time (seconds, s)

Ohm's Law

The current through a conductor is directly proportional to the voltage and inversely proportional to the resistance:

V = I × R

Where:

V = voltage (Volts, V)
I = current (Amperes, A)
R = resistance (Ohms, Ω)

Figure 6: Simple circuit illustrating Ohm's Law

Electric Power

Electric power is the rate at which electrical energy is transferred:

P = VI = I²R = V²/R

Electromagnetic Induction

A changing magnetic field induces an electromotive force (EMF) in a conductor:

EMF = -N × (Δϕ/Δt)

Where:

EMF = electromotive force (Volts, V)
N = number of turns in the coil
Δϕ = change in magnetic flux
Δt = time taken for the change

Simple Harmonic Motion

Characteristics of SHM

Simple Harmonic Motion (SHM) is a type of oscillatory motion where the restoring force is directly proportional to the displacement and acts in the opposite direction:

F = -kx

Where:

F = restoring force (Newtons, N)
k = spring constant (N/m)
x = displacement from equilibrium (meters, m)
The negative sign indicates that the force is in the opposite direction to the displacement

Period and Frequency in SHM

For a mass-spring system:

T = 2π√(m/k)
f = 1/T = (1/2π)√(k/m)

For a simple pendulum:

T = 2π√(L/g)
f = 1/T = (1/2π)√(g/L)

Where:

T = period (seconds, s)
f = frequency (Hertz, Hz)
m = mass (kilograms, kg)
k = spring constant (N/m)
L = length of pendulum (meters, m)
g = gravitational acceleration (9.8 m/s²)

Glossary of Physics Terms

Acceleration: The rate of change of velocity with respect to time.
Amplitude: The maximum displacement of a particle from its equilibrium position in oscillatory motion.
Archimedes' Principle: An object immersed in a fluid experiences an upward force equal to the weight of the fluid displaced.
Bernoulli's Principle: As the speed of a fluid increases, its pressure decreases.
Buoyancy: The upward force exerted by a fluid that opposes the weight of an immersed object.
Conservation of Energy: Energy cannot be created or destroyed, only transformed from one form to another.
Density: Mass per unit volume of a substance (kg/m³).
Displacement: The change in position of an object, including both distance and direction.
Efficiency: The ratio of useful output energy to input energy, usually expressed as a percentage.
Elasticity: The property of a material to return to its original shape after a deforming force is removed.
Electric Current: The flow of electric charge, measured in amperes (A).
Electromagnetic Induction: The process of generating an electric current by moving a conductor through a magnetic field.
Frequency: The number of complete cycles or oscillations per unit time, measured in hertz (Hz).
Friction: The force that opposes the relative motion or tendency of motion between two surfaces in contact.
Gravitational Potential Energy: The energy possessed by an object due to its position in a gravitational field.
Hooke's Law: The extension of a spring is directly proportional to the applied force (F = kx).
Inertia: The resistance of an object to changes in its state of motion.
Kinetic Energy: The energy possessed by an object due to its motion.
Magnetic Field: A region around a magnetic material or a moving electric charge where magnetic force can be detected.
Mass: A measure of the amount of matter in an object.
Mechanical Advantage: The ratio of output force to input force in a machine.
Moment: The turning effect of a force, calculated as force × perpendicular distance from the pivot.
Momentum: The product of an object's mass and velocity.
Newton's Laws of Motion: Three fundamental laws that describe the relationship between the motion of an object and the forces acting on it.
Ohm's Law: The current through a conductor is directly proportional to the voltage and inversely proportional to the resistance (V = IR).
Potential Energy: Stored energy that an object has due to its position or configuration.
Power: The rate of doing work or transferring energy, measured in watts (W).
Pressure: Force per unit area, measured in pascals (Pa).
Resistance: The opposition to the flow of electric current in a conductor, measured in ohms (Ω).
Scalar: A physical quantity that has magnitude but no direction.
Simple Harmonic Motion: Oscillatory motion where the restoring force is directly proportional to the displacement from equilibrium.
Speed: The rate of change of distance with respect to time.
Torque: A measure of the force that causes an object to rotate around an axis.
Vector: A physical quantity that has both magnitude and direction.
Velocity: The rate of change of displacement with respect to time (includes both speed and direction).
Wavelength: The distance between successive crests or troughs in a wave.
Weight: The gravitational force acting on an object, equal to mass × gravitational field strength.
Work: The transfer of energy when a force causes a displacement, measured in joules (J).

Self-Assessment Questions

A car accelerates uniformly from rest to 20 m/s in 5 seconds. Calculate:

a) The acceleration of the car

b) The distance traveled during this time

a) Acceleration = (final velocity - initial velocity) / time

a = (20 m/s - 0 m/s) / 5s = 4 m/s²

b) Distance = initial velocity × time + ½ × acceleration × time²

s = 0 × 5 + ½ × 4 × 5² = 50 meters
A force of 50 N is applied to a 10 kg object. What is the acceleration of the object?

Using Newton's Second Law: F = ma

a = F/m = 50 N / 10 kg = 5 m/s²
Calculate the work done when a force of 20 N moves an object 5 meters in the direction of the force.

Work = Force × Distance × cos(θ)

W = 20 N × 5 m × cos(0°) = 20 N × 5 m × 1 = 100 J
A 2 kg object is moving at 5 m/s. Calculate its kinetic energy.

Kinetic Energy = ½ × mass × velocity²

KE = ½ × 2 kg × (5 m/s)² = ½ × 2 kg × 25 m²/s² = 25 J
A lever has an effort arm of 2 meters and a load arm of 0.5 meters. What is the mechanical advantage of this lever?

Mechanical Advantage = effort arm / load arm

MA = 2 m / 0.5 m = 4

This means the lever multiplies the input force by a factor of 4.
A wave has a frequency of 50 Hz and a wavelength of 0.5 meters. Calculate the speed of the wave.

Wave Speed = Frequency × Wavelength

v = f × λ = 50 Hz × 0.5 m = 25 m/s
A 6 kg object is submerged in water (density = 1000 kg/m³). If the object has a volume of 0.008 m³, calculate:

a) The buoyant force acting on the object

b) Whether the object will sink or float

a) Buoyant Force = ρ × g × V = 1000 kg/m³ × 9.8 m/s² × 0.008 m³ = 78.4 N

b) Weight of object = mg = 6 kg × 9.8 m/s² = 58.8 N

Since the buoyant force (78.4 N) is greater than the weight (58.8 N), the object will float.
A circuit has a resistance of 5 Ω and a current of 3 A flowing through it. Calculate:

a) The voltage across the circuit

b) The power dissipated in the circuit

a) Using Ohm's Law: V = IR = 3 A × 5 Ω = 15 V

b) Power = VI = 15 V × 3 A = 45 W

Alternatively: Power = I²R = (3 A)² × 5 Ω = 9 × 5 = 45 W
A simple pendulum has a length of 0.4 meters. Calculate its period on Earth (g = 9.8 m/s²).

Period of a simple pendulum: T = 2π√(L/g)

T = 2π√(0.4/9.8) = 2π√0.0408 = 2π × 0.202 = 1.27 seconds
Identify whether each of the following is a scalar or vector quantity:

a) Speed

b) Displacement

c) Mass

d) Acceleration

e) Temperature

a) Speed - Scalar (magnitude only)

b) Displacement - Vector (has magnitude and direction)

c) Mass - Scalar (magnitude only)

d) Acceleration - Vector (has magnitude and direction)

e) Temperature - Scalar (magnitude only)

Summary of Key Physics Concepts for CXC Mathematics

In this introduction to physics, we have covered:

Basic physical quantities and their units
Motion in one dimension (distance, displacement, speed, velocity, acceleration)
Newton's laws of motion and their applications
Work, energy, and power
Simple machines and mechanical advantage
Fluid mechanics including pressure and buoyancy
Wave characteristics and properties
Basic electricity and magnetism concepts
Simple harmonic motion

Understanding these physics concepts provides a strong foundation for tackling applied mathematics problems in the CXC/CSEC syllabus. Remember that physics and mathematics are deeply connected - physics provides the real-world context for many mathematical principles, while mathematics gives us the tools to describe and analyze physical phenomena.

Examination Tips:

Always include appropriate units in your final answers
Pay attention to significant figures in calculations
Draw clear diagrams when solving mechanics problems
Remember to convert units if necessary (e.g., km to m, hours to seconds)
When solving problems, identify the known quantities and what you need to find, then select the appropriate formula