Newton's Law of Gravitation is a fundamental principle in physics that describes the attractive force between two masses. This law is pivotal in understanding various phenomena, from the motion of celestial bodies to everyday gravitational interactions. For students enrolled in the AS & A Level Physics course (9702), mastering this concept is essential for both academic success and practical applications in the physical world.

Key Concepts

Understanding the Gravitational Force

Gravitational force is one of the four fundamental forces of nature, responsible for the attraction between objects with mass. Unlike other forces, gravity has an infinite range but becomes weaker with increasing distance. Newton's formulation provides a quantitative description of this force, enabling precise calculations in various contexts.

Newton's Law of Universal Gravitation

Isaac Newton introduced the Law of Universal Gravitation in 1687, positing that every pair of masses attracts each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centers. Mathematically, this is expressed as: $$F = \frac{G m_1 m_2}{r^2}$$ where:

F is the gravitational force between the two masses.
G is the gravitational constant, approximately $6.674 \times 10^{-11} \, \text{N}\cdot\text{m}²/\text{kg}²$.
m₁ and m₂ are the masses of the two objects.
r is the distance between the centers of the two masses.

This universal law implies that gravity acts equally on all masses, regardless of their composition, providing a consistent framework for predicting gravitational interactions.

Gravitational Constant (G)

The gravitational constant, denoted by G, is a key parameter in Newton's law. Its precise measurement has historically been challenging due to the weakness of gravity compared to other fundamental forces. The accepted value of G is: $$G = 6.674 \times 10^{-11} \, \text{N}\cdot\text{m}²/\text{kg}²$$ This constant ensures that the gravitational force calculated using Newton's equation aligns with experimental observations.

Derivation of Gravitational Force

To derive the gravitational force between two point masses, consider the following: 1. **Proportionality to Masses**: The force is directly proportional to the product of the two masses. This means that if either mass increases, the gravitational force increases proportionally. 2. **Inverse Square Law**: The force diminishes with the square of the distance between the masses. Doubling the distance reduces the force by a factor of four. Combining these two principles, we arrive at the formula: $$F = \frac{G m_1 m_2}{r^2}$$ This equation succinctly captures the essence of gravitational interactions between point masses.

Applications of Newton's Law of Gravitation

Newton's law is extensively applied in various fields:

Astronomy: Calculating the orbits of planets, moons, and artificial satellites.
Engineering: Designing spacecraft trajectories and understanding the gravitational forces acting on structures.
Geophysics: Studying Earth's gravitational field to explore internal structures.
Everyday Phenomena: Explaining why objects fall towards the Earth.

By providing a universal framework, Newton's law allows for the prediction and analysis of gravitational effects across different scales and environments.

Gravitational Potential Energy

The gravitational force also relates to gravitational potential energy, the energy an object possesses due to its position in a gravitational field. The potential energy (**U**) between two masses is given by: $$U = -\frac{G m_1 m_2}{r}$$ The negative sign indicates that work is required to separate the two masses against the gravitational pull, reflecting the attractive nature of gravity.

Gravitational Field Strength

The concept of a gravitational field simplifies the analysis of gravitational forces. The gravitational field strength (**g**) at a point is the force experienced by a unit mass placed at that point: $$g = \frac{F}{m} = \frac{G m}{r^2}$$ This equation shows that the gravitational field strength depends on the mass creating the field and the distance from its center, decreasing with the square of the distance.

Limitations of Newton's Law of Gravitation

While Newton's law accurately describes gravitational interactions at most scales, it has limitations:

Relativity: At extremely high masses or velocities, Einstein's General Theory of Relativity provides a more accurate description.
Quantum Scale: Newtonian gravity does not account for quantum effects, which become significant at microscopic scales.
Non-spherical Masses: The law assumes point masses or spherically symmetric bodies; irregular mass distributions require more complex analysis.

Understanding these limitations is crucial for applying gravitational principles correctly in advanced physics contexts.

Calculating Gravitational Forces: Examples

**Example 1: Earth and Moon** Calculate the gravitational force between the Earth and the Moon. Given:

Mass of Earth, $m_1 = 5.972 \times 10^{24} \, \text{kg}$
Mass of Moon, $m_2 = 7.348 \times 10^{22} \, \text{kg}$
Distance between centers, $r = 3.844 \times 10^{8} \, \text{m}$

Using Newton's formula: $$F = \frac{(6.674 \times 10^{-11}) (5.972 \times 10^{24}) (7.348 \times 10^{22})}{(3.844 \times 10^{8})^2}$$ Calculating the above expression gives: $$F \approx 1.982 \times 10^{20} \, \text{N}$$ This force keeps the Moon in orbit around the Earth. **Example 2: Two Identical Masses** Determine the gravitational force between two identical masses of $10 \, \text{kg}$ each, separated by a distance of $2 \, \text{m}$. Given:

Masses, $m_1 = m_2 = 10 \, \text{kg}$
Distance, $r = 2 \, \text{m}$

Applying the formula: $$F = \frac{6.674 \times 10^{-11} \times 10 \times 10}{2^2} = \frac{6.674 \times 10^{-11} \times 100}{4} = 1.6685 \times 10^{-9} \, \text{N}$$ This minuscule force exemplifies why gravitational interactions between everyday objects are imperceptible.

Derivation of Orbital Velocity

To understand how Newton's law applies to orbital motion, consider deriving the orbital velocity (**v**) required for a mass to orbit without falling into the central mass. Starting with the gravitational force providing the necessary centripetal force: $$\frac{G m_1 m_2}{r^2} = \frac{m_2 v^2}{r}$$ Simplifying: $$v = \sqrt{\frac{G m_1}{r}}$$ This equation calculates the velocity needed for a stable orbit, highlighting the dependence on the central mass and orbital radius.

Newton's Law in the Context of Kepler's Laws

Newton's Law of Gravitation provides the theoretical foundation for Kepler's empirical laws of planetary motion:

First Law (Elliptical Orbits): Planets move in elliptical orbits with the Sun at one focus.
Second Law (Equal Areas): A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
Third Law (Harmonic Law): The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

By applying Newton's law, these laws can be derived and understood within a unified gravitational framework.

Gravitational Interactions in Multi-body Systems

In systems with more than two masses, gravitational interactions become complex due to the additive nature of forces. The total gravitational force on a mass is the vector sum of forces exerted by all other masses. This complexity leads to phenomena such as orbital resonances and chaotic motion, which require advanced methods to analyze and predict accurately.

Gravitational Waves and Modern Physics

While Newton's law does not account for the dynamic aspects of gravity, the concept of gravitational waves emerges from Einstein's General Relativity. These ripples in spacetime propagate at the speed of light and have been observed by detectors like LIGO, providing insights into cosmic events such as black hole mergers. Understanding the transition from Newtonian gravity to relativistic gravity is essential for contemporary physics studies.

Advanced Concepts

Mathematical Derivation of Gravitational Force from Newtonian Principles

Deriving Newton's law from first principles involves calculus and vector analysis. Consider two point masses, $m_1$ and $m_2$, separated by a distance $r$. The gravitational force vector $\vec{F}$ exerted on $m_2$ by $m_1$ is given by: $$\vec{F} = -G \frac{m_1 m_2}{r^2} \hat{r}$$ where $\hat{r}$ is the unit vector pointing from $m_2$ to $m_1$. The negative sign indicates the force is attractive. This vector form allows the application of Newton's second law to determine acceleration and resulting motion. Deriving orbital characteristics from this force involves setting the gravitational force equal to the centripetal force required for circular motion: $$\frac{G m_1 m_2}{r^2} = \frac{m_2 v^2}{r}$$ Simplifying provides the orbital velocity: $$v = \sqrt{\frac{G m_1}{r}}$$ This derivation showcases the interplay between gravitational theory and kinematic principles in physics.

Inverse Square Law and Its Implications

The inverse square nature of gravitational force has profound implications:

Flux Conservation: The gravitational flux passing through a spherical surface remains constant, leading to the inverse square law.
Dilution of Force: As distance increases, the force decreases rapidly, which explains why gravity is weak at large scales in quantum contexts.
Newtonian Potential: The potential energy associated with gravity also follows an inverse relationship, influencing the stability of orbits.

Understanding the inverse square law is crucial for applications in fields ranging from astrophysics to engineering.

Gravitational Lensing and Light Deflection

Though primarily a relativistic effect, Newtonian gravity can approximate light deflection under certain conditions. Gravitational lensing occurs when massive objects bend the path of light passing nearby, acting like a lens. This phenomenon allows astronomers to observe distant objects and infer the presence of dark matter. Although General Relativity provides a more accurate description, Newtonian gravity offers foundational insights into the behavior of light in gravitational fields.

Escape Velocity and Its Calculation

Escape velocity is the minimum speed required for an object to break free from a celestial body's gravitational influence without further propulsion. It is derived by equating kinetic energy to gravitational potential energy: $$\frac{1}{2} m v^2 = \frac{G m M}{r}$$ Solving for $v$ gives: $$v = \sqrt{\frac{2 G M}{r}}$$ This concept is essential in space exploration, determining the energy required for spacecraft to leave planetary surfaces.

Gravitational Binding Energy

Gravitational binding energy is the energy required to disperse a system of masses against gravitational attraction. For a spherical body of uniform density, it is calculated as: $$U = -\frac{3 G M^2}{5 R}$$ where $M$ is the mass and $R$ is the radius of the body. This energy concept is critical in understanding the stability of celestial objects like stars and planets.

Gravitational Equilibrium and Hydrostatic Balance

In celestial bodies, gravitational force is balanced by internal pressure, maintaining structural integrity. This equilibrium, known as hydrostatic balance, ensures that stars and planets do not collapse under their own gravity or disperse due to internal pressures. Mathematical modeling of hydrostatic balance involves differential equations that incorporate gravitational forces and pressure gradients, providing insights into stellar structures and evolution.

Gravitational Potential and Equipotential Surfaces

Gravitational potential energy per unit mass defines the gravitational potential ($\Phi$): $$\Phi = -\frac{G M}{r}$$ Equipotential surfaces are surfaces where gravitational potential is constant. For spherically symmetric masses, these surfaces are concentric spheres. Understanding equipotential surfaces aids in analyzing orbital paths and satellite dynamics.

Newtonian Gravitational Dynamics vs. Einsteinian Relativity

While Newtonian gravity provides accurate predictions for many scenarios, Einstein's General Relativity offers a more comprehensive framework, especially in strong gravitational fields and high-velocity contexts. Key differences include:

Space-Time Curvature: General Relativity describes gravity as the curvature of spacetime caused by mass and energy.
Precession of Orbits: Relativistic corrections explain anomalies like the precession of Mercury's orbit, which Newtonian gravity cannot fully account for.
Gravitational Time Dilation: Time runs slower in stronger gravitational fields, a phenomenon absent in Newtonian theory.

Studying these differences enhances the understanding of gravitational phenomena across different scales and conditions.

Numerical Methods in Gravitational Simulations

Simulating gravitational interactions, especially in multi-body systems, requires advanced numerical techniques. Methods like the Runge-Kutta algorithm and symplectic integrators are employed to solve differential equations governing motion. These simulations are essential in astrophysics for modeling galaxy formations, star clusters, and planetary systems, providing insights that are often inaccessible through analytical solutions alone.

Gravitational Anomalies and Dark Matter

Observations of gravitational anomalies, such as the rotation curves of galaxies, suggest the presence of unseen mass, termed dark matter. Newtonian gravity alone cannot explain these discrepancies, leading to hypotheses about dark matter's properties and distribution. Studying these anomalies involves combining Newtonian principles with observational data to infer the existence and influence of dark matter in the universe.

Comparison Table

Aspect	Newton's Law of Gravitation	Einstein's General Relativity
Fundamental Concept	Gravity as a force between masses.	Gravity as the curvature of spacetime.
Applicability	Accurate for most macroscopic and low-velocity scenarios.	Essential for high-velocity, massive, and cosmological scales.
Mathematical Framework	Inverse square law: $F = Gm_1m_2 / r^2$.	Einstein's field equations describing spacetime geometry.
Predictive Power	Predicts planetary motions, satellite orbits accurately.	Explains gravitational lensing, black holes, gravitational waves.
Limitations	Does not account for spacetime curvature or relativistic effects.	More complex, requires advanced mathematics; reduces to Newtonian gravity in weak fields.

Summary and Key Takeaways

Newton's Law of Gravitation quantitatively describes the gravitational force between two masses.
The law is fundamental in understanding celestial mechanics and various physical phenomena.
Gravitational interactions follow an inverse square law, weakening with distance.
Advanced concepts extend the basic law to complex systems and bridge to modern physics.
Comparison with General Relativity highlights the evolution and limitations of gravitational theories.

Examiner Tip

Tips

To remember the inverse square law, use the mnemonic "Force Falls Fast with Distance." This emphasizes that force decreases rapidly as distance increases. When calculating gravitational forces, always double-check your exponent on the radius. Practice with real-world examples, like Earth-Moon systems, to solidify your understanding. Additionally, visualize gravitational fields using diagrams to better grasp how mass and distance influence gravitational strength.

Did You Know

Despite its subtlety, gravity played a crucial role in the discovery of the planet Neptune. Astronomers noticed irregularities in Uranus's orbit, which led them to predict and subsequently discover Neptune using Newton's gravitational equations. Additionally, gravity isn't just a force; according to Einstein, it's a warping of spacetime, a concept that revolutionized our understanding of the universe.

Common Mistakes

One frequent error is confusing mass with weight. Remember, mass is the amount of matter in an object, measured in kilograms, while weight is the gravitational force acting on that mass. Another mistake students make is neglecting to square the distance in the denominator when calculating gravitational force, leading to incorrect force values. Lastly, assuming gravitational force can be felt over infinite distances without considering the inverse square relationship often results in misconceptions about its reach and strength.

FAQ

What is Newton's Law of Gravitation?

Newton's Law of Gravitation states that every two masses attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centers, mathematically expressed as $F = \frac{G m_1 m_2}{r^2}$.

How is the gravitational constant (G) determined?

The gravitational constant, G, is determined experimentally through precise measurements of gravitational forces between known masses. One of the first experiments to measure G was conducted by Henry Cavendish using a torsion balance apparatus.

Why does gravitational force decrease with distance?

Gravitational force decreases with the square of the distance between two masses because it follows the inverse square law. As the distance increases, the force spreads out over a larger area, reducing its strength proportionally.

Can Newton's Law of Gravitation explain all gravitational phenomena?

While Newton's Law accurately describes gravitational interactions in most everyday scenarios, it cannot account for relativistic effects near massive objects or at high velocities. Einstein's General Relativity provides a more comprehensive framework for such conditions.

What is the significance of the inverse square relationship in gravity?

The inverse square relationship means that gravitational force decreases rapidly as distance increases. This relationship is crucial for determining orbital paths, the strength of celestial interactions, and understanding the distribution of gravitational fields in space.

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

Get PDF

PDF

Explore

Your Flashcards are Ready!

Newton's Law of Gravitation: F = Gm₁m₂ / r²

Introduction