AS Physics - 9702
1- Measurements & Physical Quantiites
1.1 & 1.2 Physical Quantities & SI Base Units
All physical quantities consist of a numerical magnitude and a unit. The International System of Units (SI) defines fundamental base quantities from which all other derived units are built.
SI Base Quantities
| Quantity | Base Unit | Symbol |
|---|---|---|
| Mass | Kilogram | $\text{kg}$ |
| Length | Meter | $\text{m}$ |
| Time | Second | $\text{s}$ |
| Electric Current | Ampere | $\text{A}$ |
| Temperature | Kelvin | $\text{K}$ |
| Amount of Substance | Mole | $\text{mol}$ |
| Luminous Intensity | Candela | $\text{cd}$ |
Interactive Prefix Explorer
Derived Units & Homogeneity
An equation is said to be homogeneous if the SI base units on both sides of the equation are equal.
Thermal energy formula: $E = \frac{cA(T_1 - T_2)t}{L}$[cite: 7]
Rearranging for $c$: $c = \frac{E \cdot L}{A \cdot (T_1 - T_2) \cdot t}$[cite: 7]
1.3 Scalars & Vectors
A scalar has magnitude only. A vector has both magnitude and direction[cite: 7].
Interactive Vector Addition Simulator
Simulate two perpendicular vectors acting on an object (e.g., Ship velocity vs. Water Current)[cite: 7].
Vector Resolution
Any vector $V$ at an angle $\theta$ to the horizontal can be resolved into two perpendicular components:
- Horizontal Component: $V_x = V \cos\theta$[cite: 7]
- Vertical Component: $V_y = V \sin\theta$
1.4 Errors & Uncertainties
Precision
How close repeated measurements are to each other. Affected by random errors[cite: 7].
Accuracy
How close a measured value is to the true value[cite: 7]. Affected by systematic errors.
Target Visualizer
Uncertainty Calculation Tool
For $E = \frac{kM}{T^2}$, percentage uncertainties add up as follows: $\% \text{Uncertainty in } E = \% \text{Uncertainty in } k + \% \text{Uncertainty in } M + 2 \times (\% \text{Uncertainty in } T)$.
Interactive Practice Question
Given: $\%k = \pm2.1\%$, $\%M = \pm0.6\%$, $\%T = \pm1.5\%$[cite: 7].
Calculated Value $E = 8.245 \times 10^9 \text{ Pa}$[cite: 7].
%
$\times 10^8 \text{ Pa}$
Lesson Summary
- Physical Quantities: Expressed as a magnitude and a unit[cite: 7]. There are 7 fundamental SI base units ($\text{kg}, \text{m}, \text{s}, \text{A}, \text{K}, \text{mol}, \text{cd}$).
- Homogeneity: Base units on both sides of a physical equation must be identical for it to be valid.
- Vectors vs. Scalars: Scalars have magnitude only; vectors have magnitude and direction. Vectors can be added head-to-tail or resolved into $V\cos\theta$ and $V\sin\theta$ components.
- Errors: Systematic errors shift all measurements consistently in one direction (affects accuracy). Random errors cause unpredictable fluctuations (affects precision).
- Uncertainties: When quantities multiply or divide, their percentage uncertainties are added together (powers are multiplied by the exponent).
2- kinematics
AS Physics (9702) Chapter 2: Kinematics
1. Core Quantities & Syllabus Objectives
Syllabus Objectives (2.1 Equations of Motion)
- Define and use distance, displacement, speed, velocity, and acceleration .
- Use graphical methods to represent distance, displacement, speed, velocity, and acceleration .
- Determine displacement from the area under a velocity-time graph .
- Determine velocity using the gradient of a displacement-time graph .
- Determine acceleration using the gradient of a velocity-time graph .
- Derive, from definitions, equations representing uniformly accelerated motion in a straight line .
- Solve problems using equations for uniform acceleration (including vertical fall without air resistance) .
- Describe an experiment to determine free fall acceleration $g$ using a falling object .
- Describe and explain 2D motion with uniform velocity in one direction and uniform acceleration in a perpendicular direction .
Speed ($v$)
The distance travelled by a body per unit time .
$$\text{Speed} = \frac{\text{Distance}}{\text{Time}} \quad \Rightarrow \quad v = \frac{s}{t}$$• Unit: $\text{m/s}$ or $\text{m s}^{-1}$
• Type: Scalar quantity (independent of direction) .
Velocity ($v$)
Distance travelled per unit time in a given direction (change in displacement per unit time) .
$$\text{Velocity} = \frac{\text{Change in Displacement}}{\text{Time Taken}}$$• Unit: $\text{m/s}$ or $\text{m s}^{-1}$
• Type: Vector quantity (requires both magnitude and direction) .
Acceleration ($a$) & Deceleration
When an object speeds up, it undergoes acceleration . When it slows down, it undergoes deceleration . Both are evaluated using :
$$\text{Acceleration} = \frac{\text{Change in Velocity}}{\text{Time}} = \frac{\text{Final Velocity } (v) - \text{Initial Velocity } (u)}{t}$$ $$a = \frac{v - u}{t}$$• Unit: $\text{m/s}^2$ or $\text{m s}^{-2}$
• If final velocity $v > u$, acceleration is positive .
• If final velocity $v < u$, final acceleration value is negative, indicating deceleration .
2. Graphical Representation of Motion
Displacement–Time ($s-t$) Graphs
Velocity–Time ($v-t$) Graphs
Worked Example: Lift Distance Calculation (Fig 1.2)
An elevator starts from rest at the ground floor and moves to the top floor according to the speed-time profile below :
3. SUVAT Equations & Derivations
The 4 Equations of Uniform Acceleration
Applicable exclusively when acceleration $a$ is constant straight-line motion :
- Eq 1: $$v = u + at$$
- Eq 2: $$s = ut + \frac{1}{2}at^2$$
- Eq 3: $$v^2 = u^2 + 2as$$
- Eq 4: $$s = \left(\frac{u + v}{2}\right)t$$
Syllabus Objective 6: Step-by-Step Derivations
1. Derivation of $v = u + at$
From acceleration definition: $a = \frac{v - u}{t} \Rightarrow at = v - u \Rightarrow \mathbf{v = u + at}$ .
2. Derivation of $s = \left(\frac{u + v}{2}\right)t$
Average velocity $= \frac{u + v}{2}$. Displacement = Average velocity $\times$ time $\Rightarrow \mathbf{s = \left(\frac{u + v}{2}\right)t}$ .
3. Derivation of $s = ut + \frac{1}{2}at^2$
Substitute $v = u + at$ into $s = \left(\frac{u + v}{2}\right)t$:
$$s = \left(\frac{u + (u + at)}{2}\right)t = \left(\frac{2u + at}{2}\right)t = \left(u + \frac{1}{2}at\right)t = \mathbf{ut + \frac{1}{2}at^2}$$
4. Derivation of $v^2 = u^2 + 2as$
Rearrange $v = u + at \Rightarrow t = \frac{v - u}{a}$. Substitute into $s = \left(\frac{u + v}{2}\right)t$:
$$s = \left(\frac{v + u}{2}\right)\left(\frac{v - u}{a}\right) = \frac{v^2 - u^2}{2a} \Rightarrow 2as = v^2 - u^2 \Rightarrow \mathbf{v^2 = u^2 + 2as}$$
4. Motion under Free Fall & Terminal Velocity
Vacuum vs Air Fall Mechanics
• In air, a heavy coin falls faster than a light paper sheet because air drag has a larger relative effect on lighter objects .
• In a vacuum, both fall at the exact same rate regardless of mass .
• Acceleration of Free Fall ($g$): All objects falling freely under uniform gravity experience constant acceleration $g \approx 9.81\text{ m s}^{-2}$ (or $9.8\text{ m s}^{-2}$) .
Terminal Velocity Transition
1. Object is dropped: Weight $W$ acts downward, air resistance $F_D = 0 \Rightarrow a = g$ .
2. Speed increases: Fluid friction $F_D$ increases, reducing resultant force and acceleration .
3. Eventually upward air drag equals downward weight ($F_D = W$) .
4. Net force $\Sigma F = 0 \Rightarrow a = 0$. Object continues at constant Terminal Velocity .
Syllabus Objective 8: Free Fall ($g$) Experiment Setup
An electromagnet holds a steel sphere above a trapdoor connected to an electronic timer . When the switch cuts power to the magnet, the timer starts automatically . When the ball hits the trapdoor, it breaks the circuit, stopping the timer .
$$\text{Height dropped } = h, \quad \text{Initial velocity } u = 0, \quad \text{Time measured } = t$$ $$s = ut + \frac{1}{2}at^2 \quad \Rightarrow \quad h = 0 + \frac{1}{2}gt^2 \quad \Rightarrow \quad \mathbf{g = \frac{2h}{t^2}}$$5. Projectile Motion Mechanics & PDF Solution
Rules of 2D Projection[cite: 2, 3]
- Independence of Motion: Horizontal and vertical vector axes are analyzed completely separately[cite: 2, 3].
- Horizontal Component ($x$): No forces act horizontally ($a_x = 0$). Speed $v_H$ remains constant . $s_x = v_H \times t$[cite: 2, 3].
- Vertical Component ($y$): Subject to uniform acceleration under gravity ($a_y = g = 9.81\text{ m s}^{-2}$ downward)[cite: 2, 3].
- Time ($t$): Time is a scalar quantity and is identical for both horizontal and vertical components[cite: 3].
Steel Ball Horizontal Projection (PDF Fig 2.1)
Interactive PDF Exam Problem Solution[cite: 2, 3]
A ball is projected horizontally at $v_H = 4.9\text{ m s}^{-1}$ from a table and hits the ground $180\text{ cm}$ ($1.8\text{ m}$) away .
(a)(i) Time taken ($t$): $t = \frac{\text{distance}}{v_H} = \frac{1.8}{4.9} = \mathbf{0.37\text{ s}}$ .
(a)(ii) Vertical velocity component ($v_v$): $v_v = u_v + gt = 0 + (9.81 \times 0.37) = \mathbf{3.6\text{ m s}^{-1}}$ .
6. Complete Lesson Summary
Summary infographic of Chapter 2 principles, formulas, and projectile laws :
3 - Dynamics
3. Dynamics Overview
An understanding of forces from Cambridge IGCSE/O Level Physics or equivalent is assumed .
3.1 Momentum and Newton's Laws of Motion
- Understand that mass is the property of an object that resists change in motion .
- Recall \(F=ma\) and solve problems using it, understanding that acceleration and resultant force are always in the same direction .
- Define and use linear momentum as the product of mass and velocity .
- Define and use force as rate of change of momentum .
- State and apply each of Newton's laws of motion .
- Describe and use the concept of weight as the effect of a gravitational field on a mass (\(W=mg\)) .
3.2 Non-Uniform Motion
- Show a qualitative understanding of frictional forces and viscous/drag forces including air resistance .
- Describe and explain qualitatively the motion of objects in a uniform gravitational field with air resistance .
- Understand that objects moving against a resistive force may reach a terminal velocity .
3.3 Linear Momentum and its Conservation
- State the principle of conservation of momentum .
- Apply the principle of conservation of momentum to solve problems in 1D and 2D .
- Recall that for a perfectly elastic collision, relative speed of approach equals relative speed of separation .
- Understand that kinetic energy may change during interactions even if momentum is conserved .
3.1 Mass & Weight
| Feature | Mass | Weight |
|---|---|---|
| Definition | Measure of the amount of matter in an object . Resists change in motion (inertia) . | Gravitational force acting on an object that has mass . |
| Unit | Kilograms (\(\text{kg}\)) | Newtons (\(\text{N}\)) |
| Formula | — | \(W = mg\) |
| Quantity Type | Scalar | Vector |
| Constancy | Remains the same everywhere . | Changes according to the local value of gravity (\(g\)) . |
Note: Standard Earth gravitational field strength \(g = 9.8\text{ m/s}^2\) .
Gravitational Field
A region where a mass experiences a gravitational force. Gravitational field strength is defined as force per unit mass :
$$g = \frac{W}{m}$$
Interactive Concept: Spring Balance Measurement
Weight pulls down on the mass, stretching the spring proportional to \(W = mg\) .
Quick Check: Calculate Weight
Calculate the weight of a body with mass \(m = 5\text{ kg}\) on Earth (\(g = 9.8\text{ m/s}^2\)).
3.2 Linear Momentum & Newton's Laws
Linear Momentum (\(p\)): Defined as the product of mass and velocity .
$$p = mv$$
Units: \(\text{kg}\cdot\text{m/s}\) or \(\text{N}\cdot\text{s}\) .
Change in Momentum (\(\Delta p\)):
$$\Delta p = mv - mu$$
Impulse
Impulse is defined as the force acting on an object for a duration of time, producing a change in momentum :
$$\text{Impulse} = F \Delta t = \Delta p = mv - mu$$
Newton's Laws of Motion
- First Law: A body will remain at rest or move with constant velocity unless acted on by a resultant force .
- Second Law: A resultant force acting on a body causes a rate of change of momentum proportional to and in the direction of the force (\(F = ma\)) .
$$F = \frac{\Delta p}{\Delta t} = \frac{mv - mu}{t}$$
- Third Law: If body A exerts a force on body B, body B exerts an equal magnitude force on body A in the opposite direction .
- Pairs act on different objects .
- Pairs must be of the same type (e.g., both gravitational or both contact) .
Momentum in Two Dimensions
When bodies collide or separate at angles, momentum must be conserved independently along perpendicular axes (x and y) .
Collision Vector Diagram
Worked Exam Problem (N-19 P23 Q4)
Ball X (\(m_X = 0.300\text{ kg}\)) at velocity \(v_X\) at \(60.0^\circ\) to line AB, collides with Ball Y (\(m_Y = 0.200\text{ kg}\)) moving at \(6.00\text{ m/s}\) at \(60.0^\circ\) to AB . They stick together and travel along line AB .
(i) Perpendicular component of Y's initial momentum:
$$p_{\perp Y} = m_Y v_Y \sin(60.0^\circ) = 0.200 \times 6.00 \times \sin(60.0^\circ) = 1.04\text{ kg}\cdot\text{m/s}$$
(ii) Find initial speed \(v_X\):
Since combined final motion is purely along line AB, initial total perpendicular momentum = \(0\) :
$$0.300 v_X \sin(60.0^\circ) - 1.04 = 0 \implies v_X = \frac{1.04}{0.300 \sin(60.0^\circ)} = 4.00\text{ m/s}$$
(iii) Combined speed \(v\) along line AB after collision:
$$0.300 v_X \cos(60.0^\circ) + 0.200(6.00)\cos(60.0^\circ) = (0.300 + 0.200)v$$
$$0.300(4.00)(0.5) + 0.200(6.00)(0.5) = 0.500 v \implies v = 2.4\text{ m/s}$$
3.3 Types of Collisions
| Collision Type | Total Momentum | Total Energy | Total Kinetic Energy |
|---|---|---|---|
| Elastic | Conserved | Conserved | Conserved |
| Inelastic | Conserved | Conserved | Not conserved |
Perfectly Elastic Collision Property:
$$\text{Relative speed of approach} = \text{Relative speed of separation}$$
$$u_1 - u_2 = v_2 - v_1$$
Interactive Practice: Steel Ball & Wooden Block
A wooden block (\(85\text{ g}\)) moves left at \(2.0\text{ m/s}\). A steel ball (\(4.0\text{ g}\)) is fired right at \(45\text{ m/s}\) and embeds inside .
What is the combined final speed \(v\)?
Chapter Summary: Dynamics
- Mass vs. Weight: Mass is an intrinsic scalar property resisting acceleration ; weight is a gravitational vector force (\(W = mg\)) .
- Newton's 1st Law: Constant velocity unless acted on by a net force .
- Newton's 2nd Law & Momentum: Net force equals the rate of change of momentum (\(F = \frac{\Delta p}{\Delta t} = ma\)) .
- Newton's 3rd Law: Action-reaction pairs are equal, opposite, of the same force type, and act on different objects .
- Impulse: Force multiplied by time interval (\(F\Delta t\)), which equals change in momentum (\(\Delta p\)) .
- 2D Momentum: Resolve vectors into perpendicular components and conserve momentum independently along x and y axes .
- Collision Types: Momentum is conserved in all closed systems . Kinetic energy is conserved only in elastic collisions . Relative speed of approach equals relative speed of separation for perfectly elastic interactions .
4 - Forces
4. Forces, Density and Pressure Overview
4.1 Turning Effects of Forces
- Understand that the weight of an object may be taken as acting at a single point known as its center of gravity.
- Define and apply the moment of a force.
- Understand that a couple is a pair of forces that acts to produce rotation only.
- Define and apply the torque of a couple.
4.2 Equilibrium of Forces
- State and apply the principle of moments.
- Understand that, when there is no resultant force and no resultant torque, a system is in equilibrium.
- Use a vector triangle to represent coplanar forces in equilibrium.
4.3 Density and Pressure
- Define and use density.
- Define and use pressure.
- Derive, from the definitions of pressure and density, the equation for hydrostatic pressure \(\Delta p = \rho g \Delta h\).
- Use the equation \(\Delta p = \rho g \Delta h\).
- Understand that the upthrust acting on an object in a fluid is due to a difference in hydrostatic pressure.
- Calculate the upthrust acting on an object in a fluid using the equation \(F = \rho g V\) (Archimedes' principle).
4.1.1 Combining Forces & Components
Force is defined as the rate of change of momentum:
$$F = ma \quad \text{or} \quad F = \frac{mv - mu}{t}$$
1. Forces in a Straight Line
Parallel forces add directly when in the same direction, and subtract when opposing.
2. Resolving Vector Components
Any vector \(F\) acting at an angle \(\theta\) can be split into two perpendicular components:
- Horizontal Component: \(F_x = F \cos\theta\)
- Vertical Component: \(F_y = F \sin\theta\)
Object on an Inclined Plane
Component of weight down the slope \(= W \cos(90^\circ - \theta) = W \sin\theta\).
Component of Normal Reaction \(N\) down the slope \(= N \cos(90^\circ) = 0\).
4.2 Center of Gravity & Stability
Center of Gravity: The single point through which all of an object's weight can be considered to act.
- For symmetrical shapes, the center of gravity lies at its geometrical center.
- For non-symmetrical shapes, it can be located experimentally using a plumb line.
Stability States
The stability of a body is increased by:
- Lowering its center of gravity.
- Increasing the area of its base.
- Stable equilibrium: Displaced slightly, the object returns to its original position.
- Unstable equilibrium: Displaced slightly, the object moves further away from its original position.
- Neutral equilibrium: Displaced, the object remains in its new position.
4.3 Turning Effects of Forces
Moment of a Force: The turning effect produced when a force acts at a distance from a fixed pivot.
$$\text{Moment} = \text{Force} \times \text{perpendicular distance to pivot} = F \times d$$
Unit: \(\text{N}\cdot\text{m}\)
Non-Perpendicular Force Moment Calculation
$$\text{Moment} = F_2 \times d \sin\theta$$
Principle of Moments
For a body in equilibrium, the sum of clockwise moments about any point equals the sum of anticlockwise moments about that same point:
$$\sum \text{Clockwise Moments} = \sum \text{Anticlockwise Moments}$$
Torque of a Couple
A couple consists of two equal in magnitude, parallel forces acting in opposite directions separated by a distance \(d\). It produces rotation only (no linear acceleration).
Steering Wheel Couple
$$\text{Torque} = (15 \times 0.20) + (15 \times 0.20) = 6.0\text{ N}\cdot\text{m}$$
Interactive Practice: Calculate Torque of a Couple
Two equal and opposite parallel forces of \(25\text{ N}\) are separated by a distance of \(0.4\text{ m}\). What is the total torque produced?
4.5 Density & Hydrostatic Pressure
Density (\(\rho\)): Mass per unit volume.
$$\rho = \frac{m}{V} \quad [\text{kg}\cdot\text{m}^{-3}]$$
Pressure (\(p\)): Normal force applied per unit cross-sectional area.
$$p = \frac{\text{Force}}{\text{Area}} = \frac{F}{A} \quad [\text{N}\cdot\text{m}^{-2} \text{ or Pa}]$$
Derivation of Hydrostatic Pressure Equation \(\Delta p = \rho g \Delta h\)
- \(\text{Pressure} = \frac{\text{Force}}{\text{Area}}\)
- \(\text{Force} = \text{Weight} = mg\)
- \(\text{Mass} = \rho \times V\)
- \(\text{Volume } V = A \times \Delta h\)
- \(\text{Force} = \rho (A \Delta h) g\)
- $$\Delta p = \frac{\rho A \Delta h g}{A} = \rho g \Delta h$$
Interactive Calculator: Hydrostatic Pressure
Calculate the pressure exerted at a depth \(\Delta h\) in water (\(\rho = 1000\text{ kg/m}^3\), \(g = 9.81\text{ m/s}^2\)):
Hydrostatic Pressure: 49050 Pa
4.7 Upthrust & Archimedes' Principle
Upthrust (Buoyancy Force): An upward resultant force acting on an object submerged in a fluid due to the difference in hydrostatic pressure at the top and bottom surfaces of the object.
Origin of Upthrust Force
\(F_{\text{bottom}} > F_{\text{top}}\), producing a net upward force called Upthrust.
- Upthrust in liquids is significantly larger than in gases because liquids are much denser.
- An object floats if upthrust equals its total weight.
Archimedes' Principle
An object submerged in a fluid at rest experiences an upward upthrust equal to the weight of the fluid displaced by the object.
$$\text{Upthrust } F = \rho g V$$
Where \(\rho\) is the fluid density, \(g\) is acceleration due to gravity, and \(V\) is the volume of fluid displaced.
Lesson summary: Forces, Density & Pressure
- Forces & Components: Forces are vector quantities that can be combined or resolved into perpendicular components (\(F_x = F \cos\theta\), \(F_y = F \sin\theta\)).
- Center of Gravity: The single point where total weight is considered to act. Lowering center of gravity and widening the base increases stability.
- Moments & Couples: Moment is force times perpendicular distance (\(F \times d\)). A couple produces rotation only, with \(\text{Torque} = \text{Force} \times \text{perpendicular distance between forces}\).
- Equilibrium: Achieved when both resultant force is zero and total resultant moment is zero (\(\sum \text{Clockwise} = \sum \text{Anticlockwise}\)).
- Density & Pressure: \(\rho = \frac{m}{V}\) and \(p = \frac{F}{A}\). Hydrostatic fluid pressure increases with depth: \(\Delta p = \rho g \Delta h\).
- Upthrust & Archimedes' Principle: Upthrust arises from pressure differences on submerged surfaces. Magnitude of upthrust is calculated by \(F = \rho g V\), matching the weight of displaced fluid.
5 - Work, Energy & Power
Chapter 5: Work, Energy and Power Overview
An understanding of the forms of energy and energy transfers from Cambridge IGCSE/O Level Physics or equivalent is assumed .
5.1 Energy Conservation
- Understand the concept of work, and recall and use \(\text{Work Done} = \text{Force} \times \text{displacement in the direction of the force}\) .
- Recall and apply the principle of conservation of energy .
- Recall and understand that the efficiency of a system is the ratio of useful energy output from the system to the total energy input .
- Use the concept of efficiency to solve problems .
- Define power as work done per unit time .
- Solve problems using \(P = \frac{W}{t}\) .
- Derive \(P = Fv\) and use it to solve problems .
5.2 Gravitational Potential Energy and Kinetic Energy
- Derive, using \(W = Fs\), the formula \(\Delta E_p = mg\Delta h\) for gravitational potential energy changes in a uniform gravitational field .
- Recall and use the formula \(\Delta E_p = mg\Delta h\) for gravitational potential energy changes in a uniform gravitational field .
- Derive, using the equations of motion, the formula for kinetic energy \(E_k = \frac{1}{2}mv^2\) .
- Recall and use \(E_k = \frac{1}{2}mv^2\) .
5.1 Work Done
Work is done on a body when a force moves (displaces) the body in the direction of the force . Energy is then transferred from one body to another .
$$\text{Work done} = \text{force} \times \text{parallel distance moved by object} $$
$$W = F \times d \quad [\text{N}\cdot\text{m} \text{ or Joules, J}] $$
Scenarios for Work Done
Scenario 1: Smooth Horizontal Surface
$$W = F \times s $$
Scenario 2: Rough Surface (Work Against Friction)
$$\text{Work done against friction} = \text{Resistive force} \times \text{distance} $$
Scenario 3: Force at an Angle to Displacement
$$\text{Work done} = F \cos\theta \times \text{displacement} $$
Worked Example: Barrel on an Inclined Plane
A barrel of weight \(2.5 \times 10^3\text{ N}\) is moved up a frictionless slope inclined at \(40^\circ\) to the horizontal . A force \(F\) is applied parallel to the slope to move it at constant speed .
Question: What is the work done in moving the barrel a distance of \(6.0\text{ m}\) up the slope?
$$\text{Work done} = F \times s = W \sin(40^\circ) \times s $$
$$\text{Work done} = (2.5 \times 10^3 \times \sin 40^\circ) \times 6.0 = 9.6 \times 10^3\text{ J} $$
5.2 Energy Derivations
Law of Conservation of Energy: Energy cannot be created or destroyed, it can only change from one form to another . Work done equals energy transferred .
1. Kinetic Energy (\(E_k\))
Energy present in moving objects :
$$E_k = \frac{1}{2}mv^2 $$
Derivation of Kinetic Energy Formula
- $$\text{Work Done} = \text{Force} \times \text{distance} = F \times s $$
- From Newton's second law: $$F = ma $$
- Using equations of motion: $$v^2 = u^2 + 2as \quad \text{where } u = 0 $$
- Rearranging for acceleration: $$a = \frac{v^2}{2s} $$
- Substituting $F$ and $a$ into Work Done equation: $$\text{Work Done} = (m \times \frac{v^2}{2s}) \times s = \frac{1}{2}mv^2 $$
2. Gravitational Potential Energy (\(\Delta E_p\))
Energy stored due to the position of an object in a gravitational field :
$$\Delta E_p = mg\Delta h $$
Derivation of Gravitational Potential Energy
- $$\text{Work Done} = \text{Force} \times \text{displacement} $$
- Force required to lift body at constant speed: $$F = \text{Weight} = mg $$
- Displacement in direction of force is height change: $$s = \Delta h $$
- Substituting into work formula: $$\Delta E_p = mg\Delta h $$
Interactive Practice: Kinetic Energy Calculator
Calculate the kinetic energy of an object:
Kinetic Energy: 125 J
5.3 Power and Efficiency
Power
Power is the rate at which energy is transferred or the rate at which work is being done .
$$\text{Power} = \frac{\text{Energy}}{\text{time}} = \frac{\text{Work Done}}{t} $$
Unit: Watts (\(\text{W}\) or \(\text{J/s}\))
Derivation of Moving Power Equation \(P = Fv\)
For a constant force causing movement at speed \(v\) :
$$\text{Power} = \frac{\text{Work Done}}{t} = \frac{F \times s}{t} $$
Since velocity \(v = \frac{s}{t}\) :
$$P = Fv $$
Efficiency
The efficiency of a system is the ratio of useful energy output from the system to the total energy input .
- High Efficiency: Most of the energy transferred is useful .
- Low Efficiency: Most of the energy transferred is wasted .
$$\text{Efficiency} = \frac{\text{Useful Energy Output}}{\text{Total Energy Input}} \times 100\% $$
$$\text{Efficiency} = \frac{\text{Power Output}}{\text{Power Input}} \times 100\% $$
Interactive Practice: Calculate System Efficiency
An electric motor receives \(500\text{ W}\) of electrical power input and provides \(350\text{ W}\) of useful mechanical power output. What is its efficiency?
Lesson summary: Work, Energy & Power
- Work Done: Product of force and displacement in the direction of the force (\(W = F \cdot d\)) . On inclines or angles, parallel components must be resolved (\(W = F s \cos\theta\) or \(W = mg \Delta h\)) .
- Conservation of Energy: Energy cannot be created or destroyed, only transferred between forms .
- Kinetic Energy (\(E_k\)): Energy of motion derived from equations of motion and Newton's second law: \(E_k = \frac{1}{2}mv^2\) .
- Gravitational Potential Energy (\(\Delta E_p\)): Energy stored due to height position in a uniform gravitational field: \(\Delta E_p = mg\Delta h\) .
- Power (\(P\)): Rate of work done or energy transfer (\(P = \frac{W}{t}\)) . For moving systems under constant force, \(P = Fv\) .
- Efficiency: Ratio of useful output energy/power to total input energy/power, expressed as a fraction or percentage .
6 - Deformation
Chapter 6: Deformation of Solids Overview
6.1 Stress and Strain
- Understand that deformation is caused by tensile or compressive forces (forces and deformations assumed to be in one dimension only) .
- Understand and use the terms load, extension, compression, and limit of proportionality .
- Recall and use Hooke's law .
- Recall and use the formula for the spring constant \(k = \frac{F}{x}\) .
- Define and use the terms stress, strain, and the Young modulus .
- Describe an experiment to determine the Young modulus of a metal in the form of a wire .
6.2 Elastic and Plastic Behaviour
- Understand and use the terms elastic deformation, plastic deformation, and elastic limit .
- Understand that the area under the force-extension graph represents the work done .
- Determine the elastic potential energy of a material deformed within its limit of proportionality from the area under the force-extension graph .
- Recall and use \(E_p = \frac{1}{2}Fx = \frac{1}{2}kx^2\) for a material deformed within its limit of proportionality .
6.1 Hooke's Law & Deformation Basics
Deformation: The change in shape or length of an object when a force is applied to it .
- Tensile Force: A force that stretches an object .
- Compressive Force: A force that compresses an object .
$$\text{Extension } (x) = \text{Final Length} - \text{Initial Length} \quad [\text{meters, m}] $$
Hooke's Law
Force and extension are directly proportional to each other up to the limit of proportionality :
$$F \propto x \implies F = kx $$
Where \(k\) is the spring constant (\(\text{N}\cdot\text{m}^{-1}\) or \(\text{N/m}\)) .
Force vs. Extension Behavior
Types of Deformation & Materials
- Elastic Deformation: The deformation is reversible and disappears completely after the applied force is removed .
- Plastic Deformation: The deformation is irreversible and permanent, remaining even after the force is removed .
- Brittle Materials: Have little to no plastic region (e.g., glass, concrete). They break with minimal plastic deformation .
- Ductile Materials: Have a large plastic region (e.g., copper). They can be drawn into wires or stretched before breaking .
Combinations of Springs
$$\text{Parallel: } k_{eq} = k_1 + k_2 \qquad \text{Series: } \frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2} $$
Interactive Practice: Equivalent Spring Constant
Two identical springs with spring constant \(k = 100\text{ N/m}\) are connected in parallel. What is their combined spring constant \(k_{eq}\)?
6.1 Stress, Strain & The Young Modulus
Tensile Stress (\(\sigma\)): Applied force per unit cross-sectional area of the material .
$$\sigma = \frac{\text{Force}}{\text{Area}} = \frac{F}{A} \quad [\text{N}\cdot\text{m}^{-2} \text{ or Pascal, Pa}] $$
Tensile Strain (\(\varepsilon\)): Fractional extension of the material .
$$\varepsilon = \frac{\text{Extension}}{\text{Original Length}} = \frac{x}{L} \quad [\text{Dimensionless / Unitless}] $$
Young Modulus (\(E\)): Ratio of stress to strain within the limit of proportionality .
$$E = \frac{\text{Stress}}{\text{Strain}} = \frac{\sigma}{\varepsilon} = \frac{F/A}{x/L} = \frac{FL}{Ax} \quad [\text{N}\cdot\text{m}^{-2} \text{ or Pa}] $$
Determining the Young Modulus Experimentally
1. Measure original length \(L\) with a meter rule and diameter with a micrometer screw gauge to calculate area \(A = \frac{\pi d^2}{4}\).
2. Apply varying load masses (\(F = mg\)) and record extension \(x\) using a marker and vernier scale.
3. Plot a graph of Force \(F\) against Extension \(x\). The gradient equals \(\frac{FL}{Ax} \cdot \frac{A}{L} = \frac{EA}{L}\).
Interactive Calculator: Young Modulus Calculation
Young Modulus: 1.00e+11 Pa
6.2 Elastic Potential Energy & Work Done
The work done in stretching or compressing a material is equal to the force multiplied by the distance moved .
The area under a force-extension graph within the limit of proportionality represents the work done, which is stored as elastic potential energy (\(E_p\)) in the material .
Force-Extension Graph Area
$$E_p = \text{Area under Force-Extension Graph} = \frac{1}{2} F x $$
Substituting Hooke's Law (\(F = kx\)) :
$$E_p = \frac{1}{2} (kx) x = \frac{1}{2} k x^2 $$
Interactive Practice: Elastic Potential Energy
A spring with a spring constant \(k = 200\text{ N/m}\) is stretched by an extension \(x = 0.05\text{ m}\). Calculate the stored elastic potential energy .
Lesson summary: Deformation of Solids
- Deformation & Forces: Tensile forces stretch an object, while compressive forces compress it . Extension is calculated as \(x = \text{final length} - \text{initial length}\) .
- Hooke's Law: Force is directly proportional to extension (\(F = kx\)) up to the limit of proportionality . The spring constant is \(k = \frac{F}{x}\) .
- Combinations: Parallel springs add directly (\(k_{eq} = k_1 + k_2\)) , while series springs add reciprocally (\(\frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2}\)) .
- Deformation Types: Elastic deformation is reversible upon removing load . Plastic deformation is permanent . Brittle materials fracture with little plastic strain , while ductile materials show significant plastic deformation .
- Stress, Strain & Young Modulus: Tensile Stress \(\sigma = \frac{F}{A}\) , Tensile Strain \(\varepsilon = \frac{x}{L}\) , Young Modulus \(E = \frac{\sigma}{\varepsilon} = \frac{FL}{Ax}\) .
- Elastic Potential Energy: Work done is represented by the area under the force-extension graph , given by \(E_p = \frac{1}{2}Fx = \frac{1}{2}kx^2\) within the limit of proportionality .
7 - Waves
7.1 Progressive Waves
A progressive wave (or travelling wave) is a disturbance that carries energy from one place to another without transferring matter. Points on the wave vibrate back and forth about fixed positions.
- Displacement (): Distance and direction of a particle from its equilibrium position.
- Amplitude (): Maximum displacement from the equilibrium position.
- Wavelength (): Distance between two consecutive crests or troughs.
- Frequency (): Number of waves passing a point per unit time (). Measured in Hertz ().
- Period (): Time taken for one complete oscillation.
- Speed (): Speed at which energy is transmitted through the medium.
Wave Equation Derivation
Using the fundamental definitions of speed, frequency, and wavelength:
Wave Intensity
Intensity () is defined as the power transmitted per unit area perpendicular to the direction of wave travel:
Intensity is directly proportional to the square of amplitude () and frequency ():
7.2 Transverse and Longitudinal Waves
| Feature | Transverse Waves | Longitudinal Waves |
|---|---|---|
| Vibration Direction | Perpendicular to direction of wave energy travel | Parallel to direction of wave energy travel |
| Structure | Crests and Troughs | Compressions (high concentration) & Rarefactions (low concentration) |
| Examples | EM Waves (Light, UV, IR), rope waves | Sound waves, Ultrasound, Infrasound |
| Polarisation | Can be polarised | Cannot be polarised |
CRO Measurements & Phase Difference
A Cathode-Ray Oscilloscope displays voltage against time to analyze waveforms.
- Time-base: Controls the x-axis scale ( or ).
- Y-Gain: Controls the y-axis scale ().
Phase Difference Formula
Worked Example: CRO & Phase Calculation
Problem: A wave on a CRO has a time-base setting of and Y-gain of . A full wave cycle occupies 4 grid divisions. Find , , and Peak Amplitude.
Solution:
7.3 Doppler Effect for Sound Waves
The apparent change in frequency observed when a wave source moves relative to a stationary observer.
- Moving Towards Observer: Use minus sign () $\rightarrow$ Higher observed frequency ().
- Moving Away from Observer: Use plus sign () $\rightarrow$ Lower observed frequency ().
7.4 Electromagnetic Spectrum
All electromagnetic waves are transverse waves traveling at speed in free space.
| Region | Typical Wavelength Range () |
|---|---|
| Radio Waves | |
| Microwaves | |
| Infrared | |
| Visible Light | |
| Ultraviolet | |
| X-rays | |
| -rays |
7.5 Polarisation & Malus's Law
Polarisation restricts wave oscillations to a single plane perpendicular to propagation. Only transverse waves can be polarised.
Malus's Law Formula
Worked Solution: Polarisation Question (From PDF)
Scenario: Vertically polarised light of intensity passes through Filter 1 at to vertical, then Filter 2 at to vertical.
(i) Calculate ratio :
(ii) Calculate ratio :
Angle between Filter 1 () and Filter 2 () is .
Interactive Knowledge Check
A plane polarized beam of intensity passes through a polarizer angled at . What is the output intensity?
Chapter summary
- Progressive Waves: Transfer energy through vibrating particles without transferring physical matter. Key relationship: .
- Wave Types: Transverse (perpendicular vibrations) vs Longitudinal (parallel vibrations).
- Intensity: Power per unit area; directly proportional to the square of amplitude ().
- Doppler Shift: Observed frequency changes when source moves relative to observer ().
- EM Spectrum: All EM waves travel at in vacuum; visible light ranges between and .
- Polarisation: Applies strictly to transverse waves. Output intensity governed by Malus's Law: .
8 - Superposition
8.1 Principle of Superposition
When two or more waves meet at a point, the resultant displacement is the vector sum of the displacements of the individual waves.
- Constructive Interference: Waves arrive in phase (peaks line up with peaks). The resultant amplitude doubles to .
- Destructive Interference: Waves arrive in anti-phase (peaks line up with troughs). The resultant amplitude is zero.
8.2 Stationary Waves
Stationary (standing) waves are produced by the superposition of two waves of the same frequency and amplitude travelling in opposite directions.
- Nodes (N): Points along the wave where there is zero displacement/vibration. Distance between adjacent nodes is .
- Antinodes (A): Points where vibrations reach maximum amplitude. Distance between adjacent antinodes is .
- Nodes and antinodes do not travel along the wave. Between adjacent nodes, all particles vibrate in phase.
Stationary Waves in Air Columns
A node is always formed at a closed end, and an antinode is formed at an open end.
Both Ends Open / String Fixed at Both Ends
| Harmonic Sequence | Length Relation () | Frequency () |
|---|---|---|
| 1st Harmonic (Fundamental) | ||
| 2nd Harmonic | ||
| 3rd Harmonic |
One End Closed & One End Open
| Harmonic Sequence | Length Relation () | Frequency () |
|---|---|---|
| 1st Harmonic | ||
| 3rd Harmonic | ||
| 5th Harmonic |
8.3 Diffraction & Interference
Diffraction: The spreading of a wave as it passes through a gap or around an obstacle.
Coherence: Two wave sources are coherent if they maintain a constant phase difference and have the same frequency.
Conditions for Interference Fringes
| Interference Type | Path Difference | Phase Difference |
|---|---|---|
| Constructive | () | or |
| Destructive | or |
Given sources and with wavelength . Path distances to point are and .
Path difference is , resulting in maximum displacement (constructive interference) at point .
Young's Double-Slit Experiment
Monochromatic light diffracted through double slits acts as two coherent sources, producing alternating bright and dark interference fringes on a distant screen.
- : Slit separation (distance between centres of the two slits).
- : Fringe separation (distance between adjacent bright or dark fringes).
- : Distance from double-slit to screen.
Key Proportionalities & Intensity Adjustments
- Fringe separation and .
- If overall light intensity is increased: Bright fringes become brighter, dark fringes remain unchanged, fringe spacing stays constant.
- If intensity of only one slit is reduced: Bright fringes become dimmer, dark fringes become partially bright, fringe spacing stays constant.
8.4 The Diffraction Grating
A diffraction grating consists of a large number of parallel, identical, closely spaced slits. It produces much sharper and brighter maxima than a double-slit setup.
- : Grating spacing (distance between adjacent slit centers).
- : Number of lines per unit length (e.g., lines per meter).
- : Angle of diffraction for the -th order maximum.
- : Order number ().
Chapter 8 summary
- Principle of Superposition: Resultant displacement is the vector sum of individual displacements.
- Stationary Waves: Formed by opposite-travelling waves of equal frequency and amplitude. Nodes ( amplitude) and Antinodes (max amplitude) are separated by .
- Air Columns: Closed ends form nodes, open ends form antinodes.
- Two-Source Interference: Requires monochromatic and coherent sources. Constructive path difference is ; destructive path difference is .
- Double-Slit Formula: determines wavelength from fringe patterns.
- Diffraction Grating: Uses with to measure light wavelength with high precision.
9 - Electric current
Chapter 9: Electric Current & Resistivity
Electric Current & Quantisation of Charge
Electric current is the flow of charge carriers, measured in Amperes ($\text{A}$) . In conducting wires, current is the flow of electrons moving from the negative terminal to the positive terminal . However, conventional current is defined as flowing from positive to negative .
Quantisation of Charge
Charge is quantised; it exists in discrete packets . The elementary charge $e$ is:
$$ e = 1.60 \times 10^{-19} \text{ C} \text{ }$$
The relationship between charge $Q$, current $I$, and time $t$ is given by:
$$ Q = I \times t \text{ }$$
Interactive Simulation: Charge Flow in a Wire
Red dots represent electrons moving right-to-left, while Yellow arrow shows conventional current direction .
Current in a Conductor & Drift Velocity
Current moving through a conductor can be calculated using the microscopic parameters of the material :
$$ I = n A v q \text{ }$$
- $I$: Current ($\text{A}$)
- $n$: Charge carrier number density ($\text{m}^{-3}$)
- $A$: Cross-sectional area ($\text{m}^2$)
- $v$: Drift speed ($\text{m s}^{-1}$)
- $q$: Charge per carrier ($\text{C}$)
Derivation of $I = nAvq$
1. Number of charge carriers in length $l$: $\text{Number} = n \times A \times l$
2. Total charge: $\Delta Q = n \times A \times l \times q$
3. Since current $I = \frac{\Delta Q}{t} = \frac{n A l q}{t}$ and drift speed $v = \frac{l}{t}$, we get:
$$ I = n A v q \text{ }$$
Interactive Simulation: Tapered Wire Drift Speed Ratio
Compare two connected wires (Wire P and Wire Q) carrying the same current :
Ratio of drift speed $\frac{v_P}{v_Q} = \left(\frac{d_Q}{d_P}\right)^2 = $ 4
Ohm's Law & I-V Characteristics
Ohm's Law: For a conductor at constant temperature, current is directly proportional to the potential difference across it ($V = IR$) .
Component Selector
Resistivity & Temperature Dependence
Resistivity ($\rho$) is a material property defining how strongly it opposes electric current . Measured in $\Omega \cdot \text{m}$ :
$$ R = \frac{\rho L}{A} \text{ }$$
Interactive Resistance & Heating Simulation
Calculated Resistance ($R$): 0.085 $\Omega$
Note: Higher temperatures increase atomic vibrations, causing more electron collisions and increasing resistance .
Practice Questions
Q1: What do the symbols represent in the electric current equation $I = nAvq$?
Q2: Two copper wires P and Q are joined in series. Wire P has diameter $d$ and wire Q has diameter $2d$. What is the ratio of average drift speed in P to that in Q ($v_P / v_Q$)?
Lesson Summary
- Electric Current ($I$): Rate of charge flow ($I = Q/t$), measured in Amperes ($\text{A}$) .
- Quantisation of Charge: Charge exists in integer multiples of elementary charge $e = 1.60 \times 10^{-19} \text{ C}$ .
- Drift Velocity Equation: $I = nAvq$, linking macroscopic current with microscopic carrier density ($n$) and drift speed ($v$) .
- Potential Difference ($V$): Energy transferred per unit charge ($V = W/Q$) .
- Power: Dissipative electrical power is given by $P = IV = I^2 R = \frac{V^2}{R}$ .
- I-V Characteristics:
- Resistors: Ohmic (straight line through origin) .
- Filament Lamps: Non-ohmic; resistance increases as current heats filament .
- Diodes: Conducts only in forward bias after threshold voltage .
- Resistivity ($\rho$): Material-dependent property given by $R = \frac{\rho L}{A}$ . Increases with temperature in metallic conductors due to increased lattice vibrations .
10 - DC Circuits
Chapter 10: DC Circuits
9.1 Circuit Symbols & Functions
Circuit symbols allow complex systems to be represented cleanly in schematic diagrams.
Cell
Battery of Cells
Fixed Resistor
Variable Resistor
Thermistor
LDR
Diode
Ammeter
Voltmeter
- Switch: Turns the circuit on (closed) or off (open).
- Fixed Resistor: Limits the flow of electric current with a constant resistance.
- Variable Resistor: Adjustable slider allows dynamic change of resistance.
- Thermistor: Non-linear sensor where resistance decreases as temperature increases.
- Light-Dependent Resistor (LDR): Light-sensitive device where resistance decreases with higher light intensity.
- Diode: Allows current to flow in one direction only (converts AC to DC).
- Light-Emitting Diode (LED): Emits light when current flows in forward bias.
- Ammeter: Measures current flowing through components; connected in series.
- Voltmeter: Measures potential difference across components; connected in parallel.
Electromotive Force (E.M.F) & Potential Difference (P.D)
When charges circulate through a circuit power source, chemical energy converts into electrical energy.
Electromotive Force (E.M.F)
The e.m.f ($E$) represents energy transferred from chemical/other forms to electrical energy per coulomb of charge ($C$):
$$\text{e.m.f} = \frac{\text{Energy transformed from other forms to electrical}}{\text{Charge}}\text{}$$
Measured in Volts ($\text{V}$), e.m.f corresponds to the terminal voltage across an open circuit when no current flows.
Potential Difference (P.D)
The potential difference ($V$) measures electrical energy transferred into other energy forms (like thermal energy) per unit charge:
$$\text{P.D} = \frac{\text{Energy transformed from electrical to other forms}}{\text{Charge}}\text{}$$
• E.M.F: Energy transferred into electrical system from supply source.
• P.D: Energy transferred out of electrical system into circuit load components.
Resistors in Series & Parallel
1. Resistors in Series
The total resistance of components connected in series equals the direct sum of individual resistances:
$$R_{\text{total}} = R_1 + R_2 + R_3\text{}$$
Proof:
$$V = V_1 + V_2 + V_3\text{}$$
$$I R_{\text{total}} = I R_1 + I R_2 + I R_3\text{}$$
$$R_{\text{total}} = R_1 + R_2 + R_3\text{}$$
2. Resistors in Parallel
The reciprocal of total equivalent resistance equals the sum of individual reciprocals:
$$\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}\text{}$$
Proof:
$$I = I_1 + I_2 + I_3\text{}$$
$$\frac{V}{R_{\text{total}}} = \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3}\text{}$$
$$\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}\text{}$$
Key Note: Total combined resistance in parallel is always less than any individual parallel branch resistance.
Interactive Resistor Combination Simulator
Series Equivalent ($R_{\text{series}}$): 50.00 $\Omega$
Parallel Equivalent ($R_{\text{parallel}}$): 12.00 $\Omega$
Potential Dividers
Potential dividers are circuit arrangements designed to provide an output voltage ($V_{\text{out}}$) as a fraction of supply input voltage ($V_{\text{in}}$).
$$V_{\text{out}} = \left( \frac{R_2}{R_1 + R_2} \right) \cdot V_{\text{in}}\text{}$$
Interactive Potential Divider Simulation
Calculated Output Voltage ($V_{\text{out}}$): 8.00 V
Internal Resistance
Internal resistance ($r$) within a power source dissipates energy, reducing the ideal supply voltage down to an effective terminal voltage ($V$).
$$E = I(R + r) = IR + Ir\text{}$$
$$V = E - Ir\text{}$$
- $E$: Electromotive Force ($\text{V}$)
- $V = IR$: Terminal Potential Difference ($\text{V}$)
- $Ir$: "Lost Volts" dissipated within the cell ($\text{V}$)
Interactive Terminal Voltage & Lost Volts Simulator
Circuit Current ($I$): 1.00 A
Terminal Voltage ($V = IR$): 5.00 V
Lost Volts ($Ir$): 1.00 V
Kirchhoff's Laws
Kirchhoff's First Law (Conservation of Charge)
The total electrical current flowing into any junction must equal the total current flowing out:
$$\sum I_{\text{in}} = \sum I_{\text{out}} \implies I_1 = I_2 + I_3\text{}$$
Kirchhoff's Second Law (Conservation of Energy)
In any closed circuit loop, the algebraic sum of e.m.f.s equals the sum of potential drops across components:
$$\sum E = \sum V = \sum I R\text{}$$
Interactive Kirchhoff Loop Solver
Calculated Net Loop Current ($I$): 0.10 A
Lesson Summary
- Circuit Components: Circuit diagrams rely on standardized symbols (e.g., resistors, switches, LDRs, diodes, ammeters, voltmeters).
- Energy Transfers:
- e.m.f: Non-electrical energy transformed into electrical energy per unit charge.
- p.d: Electrical energy transformed into other output energy forms per unit charge.
- Resistor Networks:
- Series: $R_{\text{total}} = R_1 + R_2 + R_3$\text{}
- Parallel: $\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$\text{}
- Potential Division: Output voltage scaled according to component resistance ratios: $V_{\text{out}} = \left(\frac{R_2}{R_1 + R_2}\right) V_{\text{in}}$.
- Internal Losses: Real power sources experience inner voltage drops governed by $V = E - Ir$.
- Kirchhoff's Laws:
- First Law: $\sum I_{\text{in}} = \sum I_{\text{out}}$ (Conservation of Charge).
- Second Law: $\sum E = \sum IR$ (Conservation of Energy).