The principle of moments is a fundamental concept in physics, particularly within the study of equilibrium of forces. It is essential for understanding how objects balance and rotate under the influence of various forces. This topic is highly relevant to the AS & A Level Physics curriculum (9702), providing students with the tools to analyze and solve real-world mechanical problems involving torque and balance.

Key Concepts

Understanding the Principle of Moments

The principle of moments, also known as the law of the lever, states that for a system to be in equilibrium, the sum of the clockwise moments about any pivot must equal the sum of the anticlockwise moments about that pivot. A moment, or torque, is the measure of the force causing an object to rotate about a pivot point.

Definitions and Fundamental Concepts

Moment of Force (Torque): The moment of a force about a pivot is calculated by the product of the force ($F$) and the perpendicular distance ($d$) from the pivot to the line of action of the force. It is given by the equation:

$$ \tau = F \cdot d $$

When multiple forces act on a body, each force creates its own moment. The direction of the moment (clockwise or anticlockwise) determines its positive or negative value, depending on the chosen convention.

Equilibrium: A body is said to be in equilibrium when it is either at rest or moving with constant velocity. For rotational equilibrium, the sum of all moments acting on the body must be zero:

$$ \sum \tau_{\text{clockwise}} = \sum \tau_{\text{anticlockwise}} $$

Applications of the Principle of Moments

The principle of moments is widely used in various applications, including:

Seesaws and Balance Scales: Ensuring that the moments on both sides of the pivot are equal for the system to balance.
Engineering Structures: Designing beams and bridges to withstand rotational forces.
Mechanical Systems: Analyzing the torque in engines and machinery to ensure proper function.

Calculating Moments

To calculate moments, identify the pivot point and measure the perpendicular distance from the pivot to the line of action of each force. The moment is then calculated using the formula:

$$ \tau = F \cdot d $$

For equilibrium, set the sum of clockwise moments equal to the sum of anticlockwise moments and solve for the unknown variable.

Examples and Problem Solving

Example 1: A 10 N force is applied 2 meters from a pivot in the clockwise direction. What force must be applied 3 meters from the pivot in the anticlockwise direction to maintain equilibrium?

Solution:

$$ \tau_{\text{clockwise}} = 10 \cdot 2 = 20 \, \text{Nm} $$ $$ \tau_{\text{anticlockwise}} = F \cdot 3 $$ Setting $\tau_{\text{clockwise}} = \tau_{\text{anticlockwise}}$: $$ 20 = F \cdot 3 \implies F = \frac{20}{3} \approx 6.67 \, \text{N} $$

Therefore, a force of approximately 6.67 N must be applied 3 meters from the pivot in the anticlockwise direction to maintain equilibrium.

Conditions for Equilibrium

For a body to remain in rotational equilibrium:

The sum of all vertical forces must be zero.
The sum of all horizontal forces must be zero.
The sum of all moments about any pivot must be zero.

These conditions ensure that the body is not experiencing any unbalanced forces or torques that would cause it to accelerate linearly or rotate.

Types of Equilibrium

There are two primary types of equilibrium:

Stable Equilibrium: If the system returns to its original position after a small displacement.
Unstable Equilibrium: If the system moves further away from its original position after a small displacement.

The principle of moments applies to both types, ensuring that the moments are balanced to maintain the state of equilibrium.

Centre of Mass and Its Role in Moments

The centre of mass is the point where the mass of a body is considered to be concentrated for the purpose of analysis. It plays a crucial role in calculating moments, as the distribution of mass affects the torque generated by gravitational forces.

Lever Systems and Mechanical Advantage

In lever systems, the principle of moments is used to achieve mechanical advantage, which allows a smaller input force to balance a larger load. The mechanical advantage ($MA$) is given by the ratio of the lengths of the lever arms:

$$ MA = \frac{d_{\text{load}}}{d_{\text{effort}}} $$

By adjusting the lengths, the lever can be optimized to require minimal effort to lift or balance heavy loads.

Moment Arms and Their Importance

The moment arm is the perpendicular distance from the pivot to the line of action of the force. Longer moment arms result in larger moments for the same amount of force, which is a key consideration in design and analysis of mechanical systems.

Balancing Torques in Complex Systems

In systems with multiple forces, it's essential to consider the contribution of each force's torque to ensure overall balance. This involves calculating individual moments and ensuring their vector sum equals zero.

Practical Considerations in Applying the Principle of Moments

When applying the principle of moments in real-world scenarios, factors such as friction, material strength, and environmental conditions must be considered to ensure accurate and safe outcomes.

Graphical Methods for Solving Moments Problems

Graphical methods, such as free-body diagrams, are invaluable tools for visualizing and solving moments problems. These diagrams help in identifying forces, their points of application, and the corresponding distances from the pivot.

Impact of Angle on Moment Calculation

The angle at which a force is applied affects the effective component of the force that contributes to the torque. The effective force is the product of the applied force and the sine of the angle between the force and the lever arm:

$$ \tau = F \cdot d \cdot \sin(\theta) $$

Understanding this relationship is crucial for accurately calculating moments in systems where forces are not perpendicular to the lever arm.

Units and Dimensions of Moments

The SI unit of torque is the Newton-meter (N.m). It is a derived unit that combines the units of force (Newtons) and distance (meters). Ensuring consistent units is vital for accurate calculations and comparisons.

Summarizing Key Concepts

To effectively apply the principle of moments:

Identify the pivot point.
Determine all forces acting on the system and their respective distances from the pivot.
Calculate the moments generated by each force.
Set the sum of clockwise moments equal to the sum of anticlockwise moments for equilibrium.
Solve for unknown variables as needed.

Advanced Concepts

Mathematical Derivation of the Principle of Moments

The principle of moments can be derived from Newton's First Law of Motion, which states that a body remains at rest or in uniform motion unless acted upon by a net external force. For rotational equilibrium, this implies that the net external torque must be zero. Mathematically, this is expressed as:

$$ \sum \tau = 0 $$

Where $\tau$ represents the torque. Breaking this down for a system with multiple forces:

$$ \sum \tau_{\text{clockwise}} - \sum \tau_{\text{anticlockwise}} = 0 $$

Rearranging:

$$ \sum \tau_{\text{clockwise}} = \sum \tau_{\text{anticlockwise}} $$>

This fundamental principle ensures that there is no net rotation, maintaining the system in equilibrium.

Advanced Problem-Solving Techniques

When dealing with complex systems involving multiple forces and pivots, advanced problem-solving techniques are required. These include:

Simultaneous Equations: Solving multiple torque equations simultaneously to find unknown forces or distances.
Vector Analysis: Using vectors to account for directionality of forces and torques.
Calculus-Based Approaches: Applying integration to determine moments in systems with continuous force distributions.

Example 2: A beam is supported at two points, A and B, with A being 3 meters from the left end and B 5 meters from the right end. A force of 150 N is applied downward at the left end, and the beam is in equilibrium. Calculate the force exerted by support B.

Solution:

Length of beam = 3 + 5 = 8 meters
Taking moments about point A:
- Clockwise moment due to support B: $F_B \cdot 8$
- Clockwise moment due to the 150 N force: $150 \cdot 0 = 0$ (since it's applied at the pivot)
- Anticlockwise moment is zero (no other forces)

Setting clockwise moments equal to anticlockwise moments:

$$ F_B \cdot 8 = 150 \cdot 3 $$> $$ F_B = \frac{450}{8} = 56.25 \, \text{N} $$>

Therefore, support B exerts an upward force of 56.25 N.

Interdisciplinary Connections

The principle of moments extends beyond physics into various engineering disciplines. In mechanical engineering, it is crucial for designing rotating machinery and structures. In civil engineering, it aids in the analysis of bridges and buildings to ensure stability. Additionally, in biomechanics, it helps in understanding human movements and the forces exerted by muscles.

Dynamics of Rotational Systems

While the principle of moments primarily deals with static equilibrium, its extension into dynamics involves analyzing how torques affect rotational motion. Newton's Second Law for rotation states:

$$ \sum \tau = I \cdot \alpha $$

Where $I$ is the moment of inertia and $\alpha$ is the angular acceleration. This relationship governs how forces can cause objects to start rotating, change their rotational speed, or alter their rotational axis.

Energy Considerations in Rotational Equilibrium

In rotational systems, energy considerations involve both potential and kinetic energy. Ensuring equilibrium often involves minimizing potential energy or analyzing energy transfer through work done by torques. Understanding these energy dynamics is essential for designing efficient mechanical systems.

Probe into Rigid Body Dynamics

Rigid body dynamics studies the motion of solid objects without deformation. The principle of moments is integral to this field, particularly in analyzing how different forces and torques interact to produce rotational motion. This includes studying the balance of multiple torques and understanding complex motion behaviors.

Impact of Variable Forces on Moments

When forces vary in magnitude or direction, calculating moments becomes more complex. Techniques such as integrating variable forces or breaking them down into components are necessary to accurately determine the resulting torque and ensure equilibrium.

Statics vs. Dynamics in Moment Analysis

Statics involves analyzing systems in equilibrium, focusing on balancing moments and forces. Dynamics, on the other hand, deals with systems in motion, examining how moments cause changes in rotational motion. Understanding the distinction is crucial for applying the principle of moments appropriately in different scenarios.

Free-Body Diagrams for Complex Systems

Free-body diagrams are essential tools for visualizing all forces and moments acting on a system. For complex systems, detailed free-body diagrams help in identifying all contributing factors and simplifying the analysis required to apply the principle of moments effectively.

Applications in Aerospace Engineering

In aerospace engineering, the principle of moments is vital for the stability and control of aircraft. It ensures that the distribution of mass and aerodynamic forces keeps the aircraft balanced during flight, preventing unwanted rotations and maintaining desired flight paths.

Lever Arm Adjustments for Maximum Efficiency

Optimizing lever arms in mechanical systems can enhance efficiency by maximizing the torque produced with minimal input force. This involves precise calculations and adjustments based on the principles of moments to achieve desired mechanical advantages.

Real-World Problem: Balancing a Ladder

Consider a ladder leaning against a wall. To ensure it doesn't slip, the moments caused by the ladder's weight and the friction at the base must balance. Calculating these moments involves assessing the distances from the pivot point (base of the ladder) to the points where forces act, ensuring equilibrium to prevent slipping.

Moment of Inertia and Its Role in Rotational Motion

The moment of inertia ($I$) quantifies an object's resistance to changes in its rotational motion. It depends on the mass distribution relative to the axis of rotation. A higher moment of inertia means more torque is required to achieve the same angular acceleration, playing a crucial role in designing rotational systems.

Advanced Applications in Robotics

Robotics relies heavily on the principle of moments to design limb movements and ensure stability. Calculating the required torques for joints and actuators involves applying moments to achieve precise and controlled movements in robotic systems.

Non-Uniform Mass Distribution

In systems where mass is not uniformly distributed, calculating moments requires integrating mass distribution or summing moments from discrete mass points. This complexity is common in real-world applications, necessitating advanced mathematical techniques for accurate analysis.

Environmental Factors Affecting Moments

Environmental conditions such as wind, temperature, and humidity can influence the forces acting on a system, thereby affecting the moments. Engineers must consider these factors to ensure structures remain in equilibrium under varying conditions.

Computational Methods for Analyzing Moments

Modern engineering often employs computational tools and software to model and analyze moments in complex systems. These tools can handle intricate calculations and provide visualizations, enhancing the accuracy and efficiency of moment analysis.

Designing Equilibrium in Architectural Structures

Architectural designs must consider the principle of moments to ensure structural integrity. Balancing moments within buildings and other structures prevents collapse and ensures they can withstand various forces acting upon them.

Real-World Applications: Nautical Engineering

In nautical engineering, the principle of moments is applied to balance ships and submarines. Proper distribution of weight and buoyancy ensures stability in water, preventing capsizing and maintaining navigational control.

Material Properties and Their Influence on Moments

The material properties, such as elasticity and tensile strength, influence how forces and moments affect a structure. Selecting appropriate materials is crucial for ensuring that moments do not cause unwanted deformations or failures.

Historical Development of the Principle of Moments

The principle of moments has its roots in ancient engineering, notably in the work of Archimedes and Galileo. Over centuries, it has evolved into a cornerstone of modern physics and engineering, underpinning countless technological advancements.

Exploring Advanced Theoretical Models

Advanced theoretical models incorporate factors like rotational inertia, damping, and external forces to provide a more comprehensive understanding of systems in equilibrium. These models are essential for high-precision applications and cutting-edge research.

Case Study: Balancing a Crane

Analyzing the moments in a crane involves ensuring that the load is balanced to prevent tipping. By calculating the moments generated by the load and the counterweights, engineers can design cranes that operate safely and efficiently.

Integration with Control Systems

Modern control systems use the principles of moments to automate and regulate rotational movements in machinery and robotics. Feedback mechanisms adjust torques in real-time to maintain desired positions and movements.

Future Trends in Moment Analysis

Advancements in materials science, computational modeling, and automation are pushing the boundaries of moment analysis. These trends promise more efficient, accurate, and versatile applications across various engineering fields.

Comparison Table

Aspect	Basic Principle of Moments	Advanced Applications
Definition	The balance of clockwise and anticlockwise moments for equilibrium.	Involves complex systems with multiple forces and pivots.
Calculation	Simple multiplication of force and distance ($\tau = F \cdot d$).	Includes variable forces, moments of inertia, and integration techniques.
Applications	Seesaws, balance scales, basic lever systems.	Engineering structures, aerospace stability, robotics.
Problem Complexity	Linear problems with single pivot points.	Non-linear problems with multiple interacting forces.
Theoretical Depth	Fundamental concept based on equilibrium.	Incorporates dynamics, material science, and advanced mathematics.

Summary and Key Takeaways

The principle of moments is essential for analyzing rotational equilibrium.
Torque is calculated as the product of force and its perpendicular distance from the pivot.
Equilibrium occurs when clockwise and anticlockwise moments balance.
Advanced applications extend the principle to complex engineering and real-world systems.
Understanding moments is crucial for designing stable and efficient structures and machinery.

Examiner Tip

Tips

1. Visualize the Problem: Draw free-body diagrams to clearly identify forces and their distances from the pivot.
2. Consistent Sign Convention: Always define a consistent direction for positive and negative moments (e.g., clockwise as positive).
3. Check Units: Ensure all measurements are in compatible units (e.g., Newtons for force, meters for distance) to avoid calculation errors.
Mnemonic: "Clockwise Counts Positive, Anticlockwise Negative" to remember sign conventions during calculations.

Did You Know

1. The principle of moments was first formalized by the ancient Greek mathematician Archimedes, who used it to design the famous Archimedean screw for moving water.
2.. In ballet, dancers utilize the principle of moments to maintain balance and perform graceful movements by adjusting their center of mass.
3. Space engineers apply the principle of moments to ensure the stability of satellites and space stations, preventing unwanted rotations in microgravity environments.

Common Mistakes

Mistake 1: Ignoring the direction of moments. Students often forget to account for clockwise and anticlockwise directions, leading to incorrect equilibrium equations.
Incorrect: Summing all moments without considering their rotational direction.
Correct: Separating moments into clockwise and anticlockwise before setting them equal.

Mistake 2: Using incorrect distances. Using the total length instead of the perpendicular distance from the pivot to the force's line of action can result in wrong torque calculations.
Incorrect: $\tau = F \cdot \text{Total Length}$
Correct: $\tau = F \cdot d$

FAQ

What is the principle of moments?

The principle of moments states that for an object to be in rotational equilibrium, the sum of clockwise moments must equal the sum of anticlockwise moments about any pivot point.

How do you calculate torque?

Torque is calculated by multiplying the force applied ($F$) by the perpendicular distance ($d$) from the pivot point to the force's line of action: $\tau = F \cdot d$.

Why is the direction of the moment important?

The direction determines whether the moment is clockwise or anticlockwise, affecting how moments are summed to achieve equilibrium.

Can moments balance without equilibrium?

No, for moments to balance, the system must be in rotational equilibrium, meaning there is no net torque causing rotation.

How does the principle of moments apply to everyday objects?

It applies to objects like see-saws, doors, and wrenches, where balancing forces and their distances from a pivot are essential for functionality and safety.

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!

State and Apply the Principle of Moments

Introduction