TOPIC 01
Further Mechanics
Unit 4 extends mechanics into impulse, two-dimensional momentum and circular motion.
Impulse and Change of Momentum
Impulse is the product of force and the time for which the force acts. It is equal to the change in momentum.
A large force acting for a short time can therefore produce the same change in momentum as a smaller force acting for a longer time.
Conservation of Momentum in Two Dimensions
In an isolated system, the total momentum before an interaction equals the total momentum after the interaction.
Momentum is a vector, so horizontal and vertical components must be considered separately in two-dimensional collisions.
Elastic and Inelastic Collisions
Momentum is conserved in both elastic and inelastic collisions when external resultant forces are negligible.
- Elastic collision — kinetic energy is also conserved.
- Inelastic collision — kinetic energy is not conserved.
Kinetic Energy and Momentum
For a non-relativistic particle, kinetic energy can be expressed in terms of momentum.
Angular Displacement
Angular displacement may be measured in radians or degrees.
Angular Velocity
Angular velocity describes the rate at which angular displacement changes.
Centripetal Acceleration
An object moving in a circle at constant speed is continuously changing direction. Therefore it has an acceleration directed towards the centre of the circle.
Centripetal Force
The resultant force directed towards the centre of the circular path is called the centripetal force.
Centripetal force is not a new type of force. It is the name given to the resultant force acting towards the centre of circular motion.
Worked Example
A 2.0 kg object travels in a horizontal circle of radius 0.50 m at a speed of 4.0 m s⁻¹. Find the centripetal force.
F = mv² / r
F = (2.0 × 4.0²) / 0.50
F = 64 N
TOPIC 02
Electric & Magnetic Fields
Electric and magnetic fields describe interactions between charged particles, conductors and electromagnetic systems.
Electric Field
An electric field is a region in which a charged particle experiences a force.
Electric Field Strength
Electric field strength is measured in N C⁻¹.
Force Between Two Point Charges
Coulomb's law describes the force between two point charges.
The force is attractive for opposite charges and repulsive for charges of the same sign.
Electric Field of a Point Charge
Electric Potential
Electric potential is the work done per unit charge in bringing a positive test charge from infinity to a point.
Electric field strength is related to the rate of change of electric potential with distance.
For a radial electric field:
Electric Field Lines and Equipotentials
- Field lines show the direction of the electric field.
- Field lines point away from positive charges and towards negative charges.
- Equipotential lines represent points of equal electric potential.
- Electric field lines are perpendicular to equipotential lines.
Capacitance
Capacitance is the charge stored per unit potential difference.
Energy Stored in a Capacitor
Charging and Discharging Capacitors
A resistor-capacitor circuit has a time constant given by:
During discharge, charge, current and potential difference decrease exponentially.
Magnetic Flux Density
Magnetic flux density describes the strength of a magnetic field.
Magnetic Flux
Magnetic flux is represented by φ, while flux linkage is represented by Nφ.
Force on a Moving Charge
The direction of the force can be determined using Fleming's left-hand rule.
Force on a Current-Carrying Conductor
Electromagnetic Induction
An electromotive force can be induced when there is a change in magnetic flux linkage.
Faraday's law and Lenz's law give:
The negative sign represents Lenz's law: the induced effect opposes the change that produced it.
TOPIC 03
Nuclear & Particle Physics
This section explores atomic structure, particle accelerators, particle detectors and the Standard Model's quark-lepton classification.
Nucleon Number and Proton Number
The proton number is the number of protons in a nucleus. The nucleon number is the total number of protons and neutrons.
Alpha Scattering and the Nuclear Model
Large-angle alpha-particle scattering provided evidence that atoms contain a small, dense, positively charged nucleus.
Most alpha particles pass through the atom with little deflection, while a small number are strongly deflected. This demonstrates that most of the atom is empty space.
Thermionic Emission
Thermionic emission is the release of electrons from a heated material.
Electric and magnetic fields can then be used to accelerate and control the motion of these electrons.
Particle Accelerators
Particle accelerators use electric fields to increase the energy of charged particles and magnetic fields to control their paths.
Linear Accelerators
In a linear accelerator, particles are accelerated through a series of electric fields along a straight path.
Cyclotrons
A cyclotron uses alternating electric fields for acceleration and magnetic fields to bend the path of charged particles.
Charged Particles in Magnetic Fields
A charged particle moving perpendicular to a magnetic field experiences a force that can produce circular motion.
Therefore:
Particle Detectors
Particle detectors provide evidence about the properties and interactions of high-energy particles.
Ionisation and deflection in magnetic fields can be used to identify and study charged particles.
Conservation Laws
Particle interactions must obey conservation laws.
- Conservation of charge
- Conservation of energy
- Conservation of momentum
- Conservation of baryon number
- Conservation of lepton number
Mass-Energy Equivalence
Matter and antimatter can be created or annihilated when the relevant conservation laws are satisfied.
Particle Energy and Mass Units
Particle physics commonly uses MeV and GeV for energy and MeV/c² and GeV/c² for mass.
- 1 GeV = 1000 MeV
- 1 MeV = 10⁶ eV
Relativistic Particle Lifetime
Fast-moving unstable particles can have significantly longer observed lifetimes than when measured in their rest frame. This is a consequence of time dilation.
The Standard Quark-Lepton Model
The Standard Model classifies particles into several groups.
Quarks
The six quarks are:
- Up
- Down
- Strange
- Charm
- Top
- Bottom
Baryons
Baryons are made from three quarks. Protons and neutrons are examples.
Mesons
Mesons are made from a quark and an antiquark. Pions are examples of mesons.
Leptons
Leptons are fundamental particles. Examples include electrons and neutrinos.
Photons
Photons are fundamental particles associated with the electromagnetic interaction.
Antiparticles
Every particle has a corresponding antiparticle. The antiparticle has the same mass as its particle but opposite values for relevant conserved charges.
Particle Equations
Particle equations can be tested by applying conservation laws to both sides of an interaction.
When checking whether an interaction is possible, systematically compare charge, baryon number and lepton number before and after the interaction.
PRACTICAL SKILLS
Unit 4 Practical Work
Practical knowledge is important because Unit 4 examination questions can assess understanding developed through practical activities.
Core Practical 9
Investigate the relationship between the force exerted on an object and its change of momentum.
Core Practical 10
Use ICT to analyse collisions between small spheres, such as ball bearings on a table.
Core Practical 11
Use an oscilloscope or data logger to display and analyse the potential difference across a capacitor as it charges and discharges through a resistor.
Be prepared to discuss variables, apparatus, measurements, graphs, uncertainties, sources of error and improvements to experimental methods.
PRACTICE
Unit 4 Mixed Practice
Show answer
Take east as positive.
Δp = m(v − u)
Δp = 0.50(−4.0 − 8.0) = −6.0 kg m s⁻¹
F = Δp / Δt = −6.0 / 0.20
Magnitude = 30 N
Show answer
F = mv² / r
F = (1200 × 10²) / 50
F = 2400 N
Show answer
E = Q / (4πε₀r²)
Show answer
W = ½CV²
W = ½ × 220 × 10⁻⁶ × 12²
W = 0.0158 J
Show answer
F = BQv sin θ
Since θ = 90°, sin θ = 1:
F = BQv
Show answer
The magnetic force acts perpendicular to the particle's velocity and therefore changes the direction of its velocity without changing its speed. The force acts towards the centre of the circular path and provides the centripetal force.
Show answer
Most alpha particles passed through the atom with little deflection, showing that most of the atom is empty space. A small fraction were strongly deflected, showing that positive charge and most of the mass are concentrated in a small region: the nucleus.
Show answer
Proton = uud
Neutron = udd
Show answer
The relevant quantities must have the same total value before and after the interaction. These include charge, energy, momentum, baryon number and lepton number. If a required conservation law is violated, the proposed interaction is not possible.
Show answer
p = BQr
r = p / BQ
r = (3.0 × 10⁻²⁰) / (0.50 × 1.60 × 10⁻¹⁹)
r = 0.375 m
QUICK REVIEW