Damping is a fundamental concept in the study of oscillatory systems, playing a crucial role in various physical phenomena encountered in both academic and real-world contexts. Understanding the different types of damping—light, critical, and heavy—is essential for students pursuing AS & A Level Physics (9702), as it provides insights into how systems respond to disturbances and return to equilibrium. This article delves into the intricacies of damping, exploring its classifications, underlying principles, and applications.

Key Concepts

1. Definition of Damping

Damping refers to the process by which oscillating systems lose energy over time, typically due to resistive forces such as friction or air resistance. This energy loss causes the amplitude of oscillations to decrease progressively until the system eventually comes to rest. Mathematically, damping introduces a frictional term into the equations of motion, modifying the system's natural frequency and affecting its response to external forces.

2. The Damped Harmonic Oscillator

The damped harmonic oscillator is a model that describes systems experiencing both restoring and damping forces. Its equation of motion is given by:

$$m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0$$

where:

m is the mass of the oscillator
c is the damping coefficient
k is the spring constant
x is the displacement

The solution to this differential equation reveals the system's behavior over time, influenced by the damping coefficient.

3. Types of Damping

Damping is categorized into three main types based on the damping coefficient:

Light Damping: Occurs when the damping is weak, allowing the system to oscillate with gradually decreasing amplitude.
Critical Damping: Represents the threshold between oscillatory and non-oscillatory motion, enabling the system to return to equilibrium as quickly as possible without oscillating.
Heavy Damping: Involves strong damping, preventing oscillations and causing the system to return to equilibrium sluggishly.

4. Determining the Type of Damping

The classification of damping types depends on the discriminant in the solution of the damped harmonic oscillator equation. By introducing the damping ratio ζ, defined as:

$$\zeta = \frac{c}{2\sqrt{mk}}$$

We can determine the damping type based on the value of ζ:

ζ < 1: Light Damping
ζ = 1: Critical Damping
ζ > 1: Heavy Damping

5. Light Damping

In lightly damped systems (ζ < 1), the oscillations persist with amplitudes that decrease exponentially over time. The system exhibits underdamped motion, characterized by oscillatory behavior with a frequency slightly lower than the natural frequency.

The damped frequency ω_d is given by:

$$\omega_d = \omega_0 \sqrt{1 - \zeta^2}$$

where ω₀ is the natural frequency:

$$\omega_0 = \sqrt{\frac{k}{m}}$$

Example: A mass-spring system with a low friction coefficient will oscillate multiple times before coming to rest.

6. Critical Damping

Critical damping occurs when ζ = 1, representing the minimum damping required to prevent oscillations. The system returns to equilibrium without overshooting, doing so in the shortest possible time compared to other damping types.

The critical damping coefficient c_c is calculated as:

$$c_c = 2\sqrt{mk}$$

In critically damped systems, the solution to the motion equation is a repeated exponential decay without oscillations.

Example: The suspension systems in some cars are designed to be critically damped to ensure a smooth ride without excessive oscillations.

7. Heavy Damping

When ζ > 1, the system is heavily damped, leading to non-oscillatory motion where the system slowly returns to equilibrium. Heavy damping results in a sluggish response, taking longer to settle compared to critical damping.

Example: Door closers in some buildings are designed with heavy damping to prevent the door from slamming shut but still require significant time to close fully.

8. Energy Considerations in Damping

Damping affects the energy of the oscillatory system by converting mechanical energy into thermal energy. The energy dissipation rate depends on the damping coefficient and the velocity of the system:

$$\frac{dE}{dt} = -c\left(\frac{dx}{dt}\right)^2$$

This equation illustrates that higher damping coefficients result in faster energy loss, influencing how quickly the system's oscillations dampen.

9. Applications of Different Damping Types

Understanding the types of damping is essential for designing systems that require specific responses to disturbances:

Light Damping: Useful in systems where sustained oscillations are desired, such as in musical instruments or certain mechanical watches.
Critical Damping: Ideal for applications requiring quick stabilization without oscillations, like automotive suspension systems or door closers.
Heavy Damping: Employed in scenarios where oscillations must be suppressed entirely, such as in building designs to withstand seismic activity.

10. Mathematical Derivation of Damping Ratios

Starting with the standard form of the damped harmonic oscillator equation:

$$m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0$$

We divide through by m to obtain:

$$\frac{d^2x}{dt^2} + 2\zeta\omega_0\frac{dx}{dt} + \omega_0^2 x = 0$$

Here, ω₀ is the natural frequency, and ζ is the damping ratio. The characteristic equation associated with this differential equation is:

$$r^2 + 2\zeta\omega_0 r + \omega_0^2 = 0$$

Solving for r yields the roots:

$$r = -\zeta\omega_0 \pm \omega_0\sqrt{\zeta^2 - 1}$$

The nature of these roots determines the damping type:

ζ < 1: Complex conjugate roots, indicating oscillatory (lightly damped) motion.
ζ = 1: Repeated real roots, indicative of critically damped motion.
ζ > 1: Distinct real roots, corresponding to heavily damped motion.

11. Phase Space Analysis

Phase space plots (displacement vs. velocity) provide a visual representation of damping types:

Light Damping: Trajectories spiral inward, reflecting oscillatory decay.
Critical Damping: Trajectories move directly toward equilibrium without oscillating.
Heavy Damping: Trajectories approach equilibrium more slowly, without oscillations.

These plots help in understanding the system's dynamic behavior under different damping conditions.

12. Real-World Examples and Case Studies

Examining practical applications reinforces the theoretical understanding of damping:

Automotive Suspension Systems: Designed to be critically damped to balance comfort and responsiveness.
Seismology: Heavy damping is crucial in building structures to absorb and dissipate energy from earthquakes.
Mechanical Clocks: Light damping ensures smooth oscillations for accurate timekeeping.

Advanced Concepts

1. Mathematical Oscillation Solutions

For a damped harmonic oscillator, the solutions to the equation of motion differ based on the damping ratio ζ:

Underdamped (ζ < 1):
The solution is:
$$x(t) = e^{-\zeta\omega_0 t}\left( A\cos(\omega_d t) + B\sin(\omega_d t) \right)$$
where A and B are constants determined by initial conditions.
Critically Damped (ζ = 1):
The solution simplifies to:
$$x(t) = (A + Bt)e^{-\omega_0 t}$$
Overdamped (ζ > 1):
The solution involves two distinct exponential terms:
$$x(t) = A e^{r_1 t} + B e^{r_2 t}$$
where r₁ and r₂ are the roots of the characteristic equation.

2. Energy Dissipation in Damped Systems

Analyzing energy dissipation involves calculating the work done by the damping force over time:

$$W = \int F_d \cdot dx = \int -c \frac{dx}{dt} \cdot dx$$

This integral quantifies the energy converted from mechanical to thermal forms, highlighting how different damping types affect the system's energy over time.

3. Nonlinear Damping

While linear damping assumes the damping force is proportional to velocity, real-world systems may exhibit nonlinear damping where the relationship is more complex: $$F_d = -c |v|^n \cdot v$$

Nonlinear damping can lead to phenomena such as amplitude-dependent oscillation frequencies and complex dynamic behavior, requiring advanced mathematical techniques for analysis.

4. Damping in Multi-Degree-of-Freedom Systems

Extending damping analysis to systems with multiple degrees of freedom involves examining mode shapes and damping ratios for each mode. Techniques such as matrix algebra and modal analysis become essential to solve for the system's dynamic response.

5. Resonance and Damping

Damping significantly influences resonance phenomena, where systems exhibit maximum amplitude at specific driving frequencies. Appropriate damping can prevent destructive resonance by reducing peak amplitudes, essential in engineering applications like bridge design and electronics.

6. Forced Oscillations with Damping

Incorporating external driving forces into damped systems introduces complexity in the response. The steady-state solution involves a balance between the driving force and damping, leading to phenomena such as phase shifts between displacement and driving force.

The amplitude of the steady-state oscillations is given by:

$$x_{max} = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + (2\zeta\omega\omega_0)^2}}$$

where F₀ is the driving force amplitude and ω is the driving frequency.

7. Damping in Quantum Systems

At the quantum level, damping influences particle behavior in potential wells and affects phenomena like quantum tunneling. Understanding damping in quantum systems requires the integration of concepts from quantum mechanics and statistical physics.

8. Numerical Methods for Damped Systems

Analytical solutions for complex damped systems are often intractable, necessitating numerical approaches. Techniques such as the Runge-Kutta method allow for approximate solutions by discretizing time and iteratively solving the motion equations.

9. Experimental Measurement of Damping Ratios

Determining damping ratios experimentally involves analyzing oscillation data to extract parameters like amplitude decay rates and frequencies. Methods include logarithmic decrement and frequency response analysis using tools such as oscilloscopes and data acquisition systems.

10. Damping in Electrical Circuits

Analogous to mechanical systems, electrical circuits can exhibit damping behavior. In RLC circuits, resistance R provides the damping effect, with the damping ratio defined as:

$$\zeta = \frac{R}{2}\sqrt{\frac{C}{L}}$$

Understanding damping in electrical circuits is essential for designing stable electronic systems without unwanted oscillations.

11. Advanced Modal Analysis

Modal analysis extends damping concepts to systems with multiple interacting modes, requiring the computation of natural frequencies, damping ratios, and mode shapes for each independent mode. This analysis is critical in fields like structural engineering and aerospace.

12. Stability of Damped Systems

Evaluating the stability of damped systems involves assessing whether perturbations grow or decay over time. Critical and heavy damping typically ensure stability by preventing unbounded oscillations, while improper damping can lead to instability in certain configurations.

13. Coupled Oscillators with Damping

In systems of coupled oscillators, damping affects the energy transfer between oscillators and the collective behavior of the system. Analyzing such systems requires solving coupled differential equations, often leading to complex dynamic patterns.

14. Damping in Biological Systems

Biological organisms exhibit damping in various physiological processes, such as muscle oscillations and neural signal processing. Understanding damping in biological contexts aids in developing medical devices and understanding bodily responses to stimuli.

Comparison Table

Aspect	Light Damping	Critical Damping	Heavy Damping
Damping Ratio (ζ)	< 1	= 1	> 1
Oscillatory Behavior	Yes, with decreasing amplitude	No oscillations	No oscillations
Time to Equilibrium	Longer due to oscillations	Shortest possible without oscillating	Slower due to strong damping
Energy Dissipation	Gradual	Balanced for optimal decay	Rapid
Applications	Musical instruments, certain mechanical watches	Automotive suspensions, door closers	Seismic dampers, heavy machinery

Summary and Key Takeaways

Damping reduces oscillation amplitude over time through energy dissipation.
There are three types of damping: light, critical, and heavy, each defined by the damping ratio ζ.
Critical damping offers the quickest return to equilibrium without oscillation.
Understanding damping is essential for designing stable and efficient mechanical and electrical systems.

Examiner Tip

Tips

To remember the types of damping based on ζ: "Light ζ for Zigzag motions, Critical ζ for Zero oscillations, and Heavy ζ for Halted swings."

When solving problems, always first calculate the damping ratio ζ to accurately determine the damping type before proceeding with further analysis.

Use dimensional analysis to ensure that all terms in your equations are consistent, helping to avoid common calculation errors.

Did You Know

1. The concept of damping isn't limited to mechanical systems; it also plays a crucial role in electrical engineering, particularly in preventing unwanted oscillations in circuits.

2. The famous Swiss watchmaker Rolex employs critical damping in their timepiece movements to ensure precise and smooth operation without oscillatory errors.

3. Earthquake-resistant buildings utilize heavy damping techniques to absorb and dissipate seismic energy, significantly reducing structural damage during earthquakes.

Common Mistakes

Mistake 1: Confusing the damping ratio ζ with the natural frequency ω₀. While ζ determines the type of damping, ω₀ defines the system's inherent oscillation rate.

Mistake 2: Assuming that higher damping always leads to quicker stabilization. In reality, excessive damping (heavy damping) can slow down the return to equilibrium compared to critical damping.

Mistake 3: Neglecting the units when calculating damping coefficients, leading to incorrect damping ratios and misclassification of damping types.

FAQ

What is the physical significance of the damping ratio ζ?

The damping ratio ζ determines the nature of the system's response to perturbations, indicating whether the system will oscillate, return to equilibrium quickly, or do so sluggishly.

How does damping affect the natural frequency of an oscillator?

Damping reduces the natural frequency of an oscillator. The damped frequency ω_d is lower than the undamped natural frequency ω₀, especially noticeable in lightly damped systems.

Why is critical damping preferred in automotive suspension systems?

Critical damping allows the suspension to return to equilibrium quickly without oscillating, providing a smooth ride while maintaining vehicle stability and comfort.

Can a system switch from one type of damping to another?

Yes, by altering the damping coefficient c or other system parameters like mass m and spring constant k, a system can transition between light, critical, and heavy damping.

How is damping implemented in electronic circuits?

In electronic circuits, damping is achieved using resistors. For example, in RLC circuits, resistance provides the damping effect that controls oscillations.

1. Capacitance

1.1 Energy Stored in a Capacitor

1.1.1 Determine electric potential energy stored in a capacitor from the area under the potential–charge g

1.1.2 Recall and use W = ½QV = ½CV²

1.2 Discharging a Capacitor

1.2.1 Analyze graphs of potential difference, charge, and current during capacitor discharge

1.2.2 Recall and use τ = RC for the time constant during discharge

1.2.3 Use equations of the form x = x₀e^(-t / RC) for current, charge, or potential during discharge

1.3 Capacitors and Capacitance

1.3.1 Define capacitance for isolated spherical conductors and parallel plate capacitors

1.3.2 Recall and use C = Q / V

1.3.3 Derive formulae for combined capacitance of capacitors in series and parallel

1.3.4 Use capacitance formulae for capacitors in series and parallel

2. Nuclear Physics

2.1 Mass Defect and Nuclear Binding Energy

2.1.1 Sketch the variation of binding energy per nucleon with nucleon number

2.1.2 Explain nuclear fusion and nuclear fission

2.1.3 Calculate the energy released in nuclear reactions using E = c²∆m

2.1.4 Understand the equivalence between energy and mass (E = mc²)

2.1.5 Define and use the terms mass defect and binding energy

2.2 Radioactive Decay

2.2.1 Understand that fluctuations in count rate provide evidence for random decay

2.2.2 Understand that radioactive decay is spontaneous and random

2.2.3 Define activity and decay constant, and recall A = λN

2.2.4 Define half-life

2.2.5 Use λ = 0.693 / t₁/₂ for the half-life relationship

2.2.6 Understand exponential decay and use the relationship x = x₀e^(-λt)

3. Kinematics

3.1 Equations of Motion

3.1.1 Define and use distance, displacement, speed, velocity, acceleration

3.1.2 Use graphical methods to represent motion

3.1.3 Determine displacement from velocity–time graph

3.1.4 Determine velocity using gradient of displacement–time graph

3.1.5 Determine acceleration using gradient of velocity–time graph

3.1.6 Derive equations for uniformly accelerated motion

3.1.7 Solve problems of uniformly accelerated motion

3.1.8 Describe experiment to determine acceleration due to gravity

3.1.9 Explain motion with uniform velocity and perpendicular acceleration

4. Deformation of Solids

4.1 Stress and Strain

4.1.1 Understand and use the terms load, extension, compression, and limit of proportionality

4.1.2 Recall and use Hooke’s law

4.1.3 Recall and use the formula for spring constant k = F / x

4.1.4 Define and use terms stress, strain, and Young's modulus

4.1.5 Describe an experiment to determine Young’s modulus

4.1.6 Understand deformation caused by tensile or compressive forces

4.2 Elastic and Plastic Behaviour

4.2.1 Understand elastic and plastic deformation and elastic limit

4.2.2 Understand area under the force–extension graph as work done

4.2.3 Calculate elastic potential energy using EP = ½kx²

5. Gravitational Fields

5.1 Gravitational Field of a Point Mass

5.1.1 Understand that g is approximately constant for small height changes near Earth's surface

5.1.2 Recall and use g = GM / r²

5.1.3 Derive and use g = GM / r² for gravitational field strength

5.2 Gravitational Potential

5.2.1 Define gravitational potential as work done per unit mass in bringing a test mass from infinity

5.2.2 Use ϕ = -GM / r for gravitational potential due to a point mass

5.2.3 Use EP = -GMm / r for gravitational potential energy between two point masses

5.3 Gravitational Field

5.3.1 Define gravitational field as force per unit mass

5.3.2 Represent a gravitational field using field lines

5.4 Gravitational Force Between Point Masses

5.4.1 For point outside uniform sphere, treat mass as a point mass at its center

5.4.2 Analyze circular orbits in gravitational fields

5.4.3 Recall and use Newton's law of gravitation F = Gm₁m₂ / r²

6. Electricity

6.1 Resistance and Resistivity

6.1.1 Understand that the resistance of a thermistor decreases as temperature increases

6.1.2 Define resistance and recall and use V = IR

6.1.3 Sketch I–V characteristics for metallic conductor, semiconductor diode, and filament lamp

6.1.4 Explain that the resistance of a filament lamp increases as current increases

6.1.5 State Ohm’s law and recall and use R = ρL / A

6.1.6 Understand that the resistance of an LDR decreases as light intensity increases

6.2 Electric Current

6.2.1 Understand that an electric current is a flow of charge carriers

6.2.2 Understand that the charge on charge carriers is quantised

6.2.3 Recall and use Q = It

6.2.4 Use I = Anvq for current-carrying conductors

6.3 Potential Difference and Power

6.3.1 Define potential difference as energy transferred per unit charge

6.3.2 Recall and use V = W / Q

6.3.3 Recall and use P = VI, P = I²R, P = V² / R

7. D.C. Circuits

7.1 Practical Circuits

7.1.1 Recall and use the circuit symbols in the syllabus

7.1.2 Draw and interpret circuit diagrams using circuit symbols

7.1.3 Define electromotive force (e.m.f.) of a source as energy transferred per unit charge

7.1.4 Distinguish between e.m.f. and potential difference (p.d.)

7.1.5 Understand the effects of internal resistance on terminal potential difference

7.2 Kirchhoff’s Laws

7.2.1 Recall Kirchhoff’s first law: conservation of charge

7.2.2 Recall Kirchhoff’s second law: conservation of energy

7.2.3 Derive the formula for combined resistance of resistors in series using Kirchhoff’s laws

7.2.4 Use the formula for combined resistance of resistors in series

7.2.5 Derive the formula for combined resistance of resistors in parallel using Kirchhoff’s laws

7.2.6 Use the formula for combined resistance of resistors in parallel

7.2.7 Use Kirchhoff’s laws to solve simple circuit problems

7.3 Potential Dividers

7.3.1 Understand the principle of a potential divider circuit

7.3.2 Recall and use the principle of the potentiometer for comparing potential differences

7.3.3 Understand the use of a galvanometer in null methods

7.3.4 Explain the use of thermistors and LDRs in potential dividers to provide a potential dependent on te

8. Thermal Equilibrium

8.1 Thermal Energy Transfer

8.1.1 Understand that thermal energy transfers from higher to lower temperature

8.1.2 Understand that regions of equal temperature are in thermal equilibrium

9. Temperature Scales

9.1 Temperature Measurement

9.1.1 Understand physical properties that vary with temperature (e.g., density, gas volume, resistance)

9.1.2 Convert temperatures between Kelvin and Celsius, and recall T(K) = θ(°C) + 273.15

10. Magnetic Fields

10.1 Concept of a Magnetic Field

10.1.1 Understand that a magnetic field is a field of force produced by moving charges or permanent magnets

10.1.2 Represent a magnetic field using field lines

10.2 Force on a Current-Carrying Conductor

10.2.1 Understand that a force acts on a current-carrying conductor in a magnetic field

10.2.2 Recall and use F = BIL sin θ for the force on a current-carrying conductor in a magnetic field

10.2.3 Define magnetic flux density as the force per unit current per unit length on a wire at right angles

10.3 Force on a Moving Charge

10.3.1 Determine the direction of the force on a charge moving in a magnetic field

10.3.2 Recall and use F = BQv sin θ for the force on a moving charge in a magnetic field

10.3.3 Understand the origin of the Hall voltage and use the expression VH = BI / (ntq) for the Hall voltag

10.3.4 Understand the use of a Hall probe to measure magnetic flux density

10.4 Magnetic Fields Due to Currents

10.4.1 Sketch magnetic field patterns due to currents in a straight wire, flat coil, and solenoid

10.4.2 Understand that magnetic field in a solenoid is increased by a ferrous core

10.4.3 Explain the origin of forces between current-carrying conductors and determine the direction of the

10.5 Electromagnetic Induction

10.5.1 Define magnetic flux as the product of magnetic flux density and area perpendicular to flux directio

10.5.2 Recall and use Φ = BA for magnetic flux

10.5.3 Understand and use the concept of magnetic flux linkage

10.5.4 Explain experiments that show changing magnetic flux induces an e.m.f.

10.5.5 Recall and use Faraday’s and Lenz’s laws of electromagnetic induction

11. Specific Heat Capacity and Latent Heat

11.1 Specific Heat Capacity

11.1.1 Define and use specific heat capacity

11.2 Latent Heat

11.2.1 Define and use specific latent heat; distinguish between fusion and vaporisation

12. Dynamics

12.1 Momentum and Newton’s Laws of Motion

12.1.1 Mass resists change in motion

12.1.2 Recall F = ma and solve related problems

12.1.3 Define and use linear momentum

12.1.4 Force as rate of change of momentum

12.1.5 State and apply Newton’s laws

12.1.6 Describe weight as effect of gravitational field

12.1.7 Weight of an object is mass × gravitational acceleration

12.2 Non-uniform Motion

12.2.1 Understand frictional and drag forces

12.2.2 Describe motion in uniform gravitational field with air resistance

12.2.3 Objects moving against resistive force may reach terminal velocity

12.3 Linear Momentum and its Conservation

12.3.1 State conservation of momentum

12.3.2 Apply conservation of momentum to solve problems

12.3.3 For elastic collision, kinetic energy and relative speed are conserved

12.3.4 Momentum is conserved, but some kinetic energy may change

13. Medical Physics

13.1 Production and Use of Ultrasound

13.1.1 Understand that a piezo-electric crystal changes shape when a p.d. is applied and generates an e.m.f

13.1.2 Understand how ultrasound waves are generated and detected by a piezoelectric transducer

13.1.3 Understand how ultrasound reflection at boundaries provides diagnostic information about internal st

13.1.4 Define specific acoustic impedance as Z = ρc

13.1.5 Use the equation IR / I₀ = (Z₁ – Z₂)² / (Z₁ + Z₂)² for the intensity reflection coefficient

13.2 Production and Use of X-rays

13.2.1 Explain that X-rays are produced by electron bombardment of a metal target and calculate the minimum

13.2.2 Understand the use of X-rays in imaging internal body structures and the term 'contrast' in X-ray im

13.2.3 Recall and use I = I₀e^(-μx) for the attenuation of X-rays in matter

13.2.4 Understand how computed tomography (CT) produces 3D images by combining multiple 2D X-ray images

13.3 PET Scanning

13.3.1 Understand that a tracer contains radioactive nuclei and is introduced into the body for imaging

13.3.2 Understand that β+ decay is used in positron emission tomography (PET)

13.3.3 Understand that annihilation occurs when a particle interacts with its antiparticle, conserving mass

13.3.4 Explain how gamma-ray photons from annihilation are detected to create an image of tracer concentrat

14. Waves

14.1 Progressive Waves

14.1.1 Describe wave motion in ropes, springs, and ripple tanks

14.1.2 Understand and use the terms displacement, amplitude, phase difference, period, frequency, wavelengt

14.1.3 Use time-base and y-gain on a cathode-ray oscilloscope (CRO) to determine frequency and amplitude

14.1.4 Derive and use v = f λ for the wave equation

14.1.5 Recall and use v = f λ

14.1.6 Understand energy transfer by a progressive wave

14.1.7 Recall and use intensity = power / area, intensity ∝ (amplitude)²

14.2 Transverse and Longitudinal Waves

14.2.1 Compare transverse and longitudinal waves

14.2.2 Analyze and interpret graphical representations of transverse and longitudinal waves

14.3 Doppler Effect for Sound Waves

14.3.1 Understand the change in frequency when a sound source moves relative to a stationary observer

14.3.2 Use the expression f₀ = fₛ(v / (v ± vₛ)) for the observed frequency

14.4 Electromagnetic Spectrum

14.4.1 State all electromagnetic waves are transverse waves traveling at the same speed in free space

14.4.2 Recall the range of wavelengths from radio waves to γ-rays

14.4.3 Recall that wavelengths 400–700 nm are visible to the human eye

14.5 Polarisation

14.5.1 Understand that polarisation is a phenomenon associated with transverse waves

14.5.2 Recall and use Malus's law (I = I₀ cos²θ) to calculate intensity of a plane-polarised wave after pas

15. The Mole

15.1 The Mole

15.1.1 Understand that amount of substance is an SI base quantity with the unit mol

15.1.2 Use molar quantities where one mole contains Avogadro's constant of particles

16. Equation of State

16.1 Ideal Gas Law

16.1.1 Understand that a gas obeying pV ∝ T is an ideal gas

16.1.2 Recall and use the equation pV = nRT for an ideal gas

16.1.3 Recall pV = NkT, where N = number of molecules and k is Boltzmann constant

16.1.4 Use the relationship k = R / NA

17. Kinetic Theory of Gases

17.1 Kinetic Theory

17.1.1 State the basic assumptions of the kinetic theory of gases

17.1.2 Explain pressure exerted by a gas due to molecular movement

17.1.3 Derive and use pV = (3/2)NmkT for pressure of a gas

17.1.4 Compare pV = (3/2)NmkT with pV = NkT to deduce average kinetic energy of a molecule

18. Particle Physics

18.1 Atoms, Nuclei and Radiation

18.1.1 Infer the existence and small size of the nucleus from α-particle scattering

18.1.2 Describe the nuclear atom model: protons, neutrons, and electrons

18.1.3 Distinguish between nucleon number and proton number

18.1.4 Understand isotopes as forms of the same element with different neutrons

18.1.5 Use the notation AZX for nuclides representation

18.1.6 Understand conservation of nucleon number and charge in nuclear processes

18.1.7 Describe the composition, mass, and charge of α-, β-, and γ-radiations

18.1.8 Understand antiparticles and positrons as antiparticles of electrons

18.1.9 State that (electron) antineutrinos are produced during β- decay and neutrinos during β+ decay

18.1.10 Represent α- and β-decay using radioactive decay equations

18.1.11 Use unified atomic mass unit (u) for mass

18.2 Fundamental Particles

18.2.1 Understand quarks as fundamental particles with six flavours

18.2.2 Understand hadrons as baryons (three quarks) or mesons (one quark and one antiquark)

18.2.3 Describe changes in quark composition during β- and β+ decay

18.2.4 Describe protons and neutrons as composed of quarks

18.2.5 Recall and use the charge of each quark and its respective antiquark

18.2.6 Recall electrons and neutrinos as fundamental particles (leptons)

19. Internal Energy

19.1 Internal Energy

19.1.1 Understand that internal energy is the sum of kinetic and potential energies of molecules in a syste

19.1.2 Relate rise in temperature to an increase in internal energy

20. Forces, Density and Pressure

20.1 Turning Effects of Forces

20.1.1 Weight acts at a single point: centre of gravity

20.1.2 Define and apply the moment of a force

20.1.3 Understand the concept of a couple

20.1.4 Define and apply torque of a couple

20.2 Equilibrium of Forces

20.2.1 State and apply the principle of moments

20.2.2 System in equilibrium has no resultant force or torque

20.2.3 Use a vector triangle to represent coplanar forces in equilibrium

20.3 Density and Pressure

20.3.1 Define and use density

20.3.2 Define and use pressure

20.3.3 Derive equation for hydrostatic pressure ∆p = ρg∆h

20.3.4 Use the equation ∆p = ρg∆h

20.3.5 Understand upthrust as the effect of hydrostatic pressure

20.3.6 Calculate upthrust using F = ρgV (Archimedes' principle)

21. Astronomy and Cosmology

21.1 Standard Candles

21.1.1 Understand luminosity as the total power radiated by a star

21.1.2 Recall and use the inverse square law F = L / (4πd²) for radiant flux intensity

21.1.3 Understand that an object with known luminosity is a standard candle

21.1.4 Use standard candles to determine distances to galaxies

21.2 Stellar Radii

21.2.1 Recall and use Wien’s displacement law λmax ∝ 1 / T to estimate the surface temperature of a star

21.2.2 Use Stefan–Boltzmann law L = 4πσr²T⁴

21.2.3 Use Wien’s law and Stefan–Boltzmann law to estimate the radius of a star

21.3 Hubble’s Law and Big Bang Theory

21.3.1 Understand redshift and the increase in wavelength of light from distant objects

21.3.2 Use ∆λ / λ = ∆f / f = v / c for redshift

21.3.3 Explain why redshift suggests the Universe is expanding

21.3.4 Recall and use Hubble’s law v = H₀d to explain the Big Bang theory

22. First Law of Thermodynamics

22.1 First Law

22.1.1 Recall and use W = p∆V for work done when the volume of a gas changes at constant pressure

22.1.2 Understand the difference between work done by a gas and work done on a gas

22.1.3 Recall and use the first law of thermodynamics: ∆U = q + W

23. Alternating Currents

23.1 Characteristics of Alternating Currents

23.1.1 Understand and use terms period, frequency, and peak value for alternating current or voltage

23.1.2 Use x = x₀ sin(ωt) for sinusoidal alternating current or voltage

23.1.3 Recall that the mean power in a resistive load is half the maximum power for sinusoidal current

23.1.4 Distinguish between root-mean-square (r.m.s.) and peak values and recall I₀r.m.s = I₀ / √2 and V₀r.m

23.2 Rectification and Smoothing

23.2.1 Distinguish graphically between half-wave and full-wave rectification

23.2.2 Explain the use of a single diode for half-wave rectification

23.2.3 Explain the use of four diodes (bridge rectifier) for full-wave rectification

23.2.4 Analyze the effect of a capacitor in smoothing, including the effect of capacitance and load resista

24. Superposition

24.1 Stationary Waves

24.1.1 Explain and use the principle of superposition

24.1.2 Understand experiments demonstrating stationary waves using microwaves, stretched strings, and air c

24.1.3 Explain the formation of stationary waves using graphical methods and identify nodes and antinodes

24.1.4 Determine wavelength from the positions of nodes or antinodes

24.2 Diffraction

24.2.1 Explain diffraction and show understanding of related experiments

24.2.2 Understand the effect of gap width relative to wavelength in diffraction experiments

24.3 Interference

24.3.1 Understand the terms interference and coherence

24.3.2 Demonstrate two-source interference using water waves, sound, light, and microwaves

24.3.3 Understand the conditions required for two-source interference fringes

24.3.4 Recall and use λ = ax / D for double-slit interference with light

24.4 Diffraction Grating

24.4.1 Recall and use d sin θ = nλ for diffraction grating calculations

24.4.2 Describe the use of a diffraction grating to determine the wavelength of light

25. Simple Harmonic Oscillations

25.1 Simple Harmonic Motion

25.1.1 Understand and use terms displacement, amplitude, period, frequency, angular frequency, and phase di

25.1.2 Understand that simple harmonic motion occurs when acceleration is proportional to displacement from

25.1.3 Use a = -ω²x and recall x = x₀ sin(ωt) as the solution

25.1.4 Use v = v₀ cos(ωt) and v = ±ω√(x₀² - x²) for velocity

26. Energy in Simple Harmonic Motion

26.1 Energy in SHM

26.1.1 Describe the interchange between kinetic and potential energy during SHM

26.1.2 Recall and use E = ½mω²x₀² for the total energy of a system undergoing SHM

27. Quantum Physics

27.1 Energy and Momentum of a Photon

27.1.1 Understand that electromagnetic radiation has a particulate nature

27.1.2 Recall and use E = hf for the energy of a photon

27.1.3 Use the electronvolt (eV) as a unit of energy

27.1.4 Understand that a photon has momentum and use p = E / c for photon momentum

27.2 Photoelectric Effect

27.2.1 Understand that photoelectrons are emitted from a metal surface when illuminated by electromagnetic

27.2.2 Understand and use the terms threshold frequency and threshold wavelength

27.2.3 Explain photoelectric emission in terms of photon energy and work function

27.2.4 Recall and use hf = Φ + ½mv² for the maximum kinetic energy of photoelectrons

27.2.5 Explain why the maximum kinetic energy of photoelectrons is independent of intensity, while current

27.3 Wave-Particle Duality

27.3.1 Understand that the photoelectric effect shows the particulate nature of light, while diffraction sh

27.3.2 Describe and interpret electron diffraction as evidence for the wave nature of particles

27.3.3 Understand the de Broglie wavelength as the wavelength associated with a moving particle

27.3.4 Recall and use λ = h / p for de Broglie wavelength

27.4 Energy Levels in Atoms and Line Spectra

27.4.1 Understand discrete electron energy levels in atoms (e.g., atomic hydrogen)

27.4.2 Understand the appearance and formation of emission and absorption line spectra

27.4.3 Recall and use hf = E₁ – E₂ for the energy difference between electron energy levels

28. Damped and Forced Oscillations, Resonance

28.1 Damped Oscillations

28.1.1 Understand that damping occurs due to resistive forces

28.1.2 Understand the types of damping: light, critical, and heavy damping

28.2 Resonance

28.2.1 Understand that resonance occurs when an oscillating system is forced to oscillate at its natural fr

29. Work, Energy and Power

29.1 Energy Conservation

29.1.1 Understand the concept of work, and recall W = F × s

29.1.2 Apply the principle of conservation of energy

29.1.3 Understand efficiency as useful energy output / total energy input

29.1.4 Use efficiency concept to solve problems

29.1.5 Define power as work done per unit time

29.1.6 Solve problems using P = W / t

29.1.7 Derive P = Fv and solve problems

29.2 Gravitational Potential Energy and Kinetic Energy

29.2.1 Derive ∆EP = mg∆h for gravitational potential energy changes

29.2.2 Recall and use ∆EP = mg∆h for gravitational potential energy changes

29.2.3 Derive EK = ½mv² for kinetic energy

29.2.4 Recall and use EK = ½mv² for kinetic energy

30. Electric Fields

30.1 Electric Fields and Field Lines

30.1.1 Understand that an electric field is a field of force and define it as force per unit positive charg

30.1.2 Represent an electric field using field lines

30.2 Uniform Electric Fields

30.2.1 Recall and use E = ∆V / ∆d to calculate electric field strength between charged parallel plates

30.2.2 Describe the effect of a uniform electric field on the motion of charged particles

30.3 Electric Force Between Point Charges

30.3.1 Understand that, for a point outside a spherical conductor, the charge may be considered a point mas

30.3.2 Recall and use Coulomb’s law F = Q₁Q₂ / (4πε₀r²) for the force between two point charges

30.4 Electric Field of a Point Charge

30.4.1 Recall and use E = Q / (4πε₀r²) for the electric field strength due to a point charge

30.5 Electric Potential

30.5.1 Define electric potential as work done per unit positive charge in bringing a test charge from infin

30.5.2 Recall and use V = Q / (4πε₀r) for electric potential due to a point charge

30.5.3 Understand how electric potential leads to electric potential energy of two point charges and use EP

31. Physical Quantities and Units

31.1 Physical Quantities

31.1.1 Physical quantities consist of magnitude and unit

31.1.2 Estimate physical quantities from the syllabus

31.2 SI Units

31.2.1 SI base units: mass, length, time, current, temperature

31.2.2 Express derived units from SI base units

31.2.3 Check homogeneity of physical equations

31.2.4 Use prefixes (pico, nano, micro, kilo, etc.)

31.3 Errors and Uncertainties

31.3.1 Understand systematic and random errors

31.3.2 Assess uncertainty in derived quantities

31.4 uniform Electric Fields

31.4.1 Distinguish between precision and accuracy

31.5 Scalars and Vectors

31.5.1 Difference between scalar and vector quantities

31.5.2 Add and subtract coplanar vectors

31.5.3 Represent a vector as components

32. Motion in a Circle

32.1 Centripetal Acceleration

32.1.1 Understand centripetal acceleration and its role in circular motion

32.1.2 Recall and use a = rω² and a = v² / r

32.1.3 Recall and use F = mrω² and F = mv² / r

32.1.4 Understand force causing centripetal acceleration is always perpendicular to motion

32.2 Kinematics of Uniform Circular Motion

32.2.1 Recall and use ω = 2π / T and v = rω

32.2.2 Understand and use angular speed

32.2.3 Define and express angular displacement in radians

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Understand the Types of Damping: Light, Critical, and Heavy Damping

Introduction