Fundamental Knowledge: Key Formulas & Concepts
Part 1: Mechanics
| Concept | Formula | Description |
|---|---|---|
| Newton's Second Law | $$F_{net} = ma$$ | Net force equals mass times acceleration. |
| Kinematics (const. accel.) | $$v = v_0 + at$$ | Final velocity. |
| Kinematics (const. accel.) | $$x = v_0t + \frac{1}{2}at^2$$ | Displacement. |
| Kinematics (const. accel.) | $$v^2 = v_0^2 + 2ax$$ | Velocity-Displacement relation. |
| Kinetic Energy | $$KE = \frac{1}{2}mv^2$$ | Energy of motion. |
| Gravitational Potential Energy | $$PE_g = mgh$$ | Energy of position in a gravitational field. |
| Work Done by a Constant Force | $$W = Fd\cos\theta$$ | Force multiplied by displacement. |
| Linear Momentum | $$p = mv$$ | Mass in motion. |
| Impulse | $$\Delta p = F \Delta t$$ | Change in momentum. |
| Centripetal Acceleration | $$a_c = \frac{v^2}{r}$$ | Acceleration towards the center in circular motion. |
| Hooke's Law (SHM) | $$F = -kx$$ | Restoring force of a spring. |
| Universal Gravitation | $$F_g = G\frac{m_1m_2}{r^2}$$ | Force between two masses. |
Concept: Work & Force
Part 2: Thermodynamics
| Concept | Formula | Description |
|---|---|---|
| First Law of Thermodynamics | $$\Delta U = Q - W$$ | Change in internal energy. |
| Ideal Gas Law | $$PV = nRT$$ | Relates P, V, T for ideal gases. |
| Specific Heat | $$Q = mc\Delta T$$ | Heat for temperature change. |
| Latent Heat | $$Q = mL$$ | Heat for phase change. |
| Thermal Efficiency | $$e = \frac{W}{Q_H}$$ | Ratio of work done to heat absorbed. |
Concept: P-V Diagram (Work)
Part 3: Electricity & Magnetism
| Concept | Formula | Description |
|---|---|---|
| Ohm's Law | $$V = IR$$ | Relates voltage, current, and resistance. |
| Coulomb's Law | $$F_e = k\frac{|q_1q_2|}{r^2}$$ | Force between two charges. |
| Electric Field | $$E = \frac{F}{q}$$ | Force per unit charge. |
| Capacitance | $$Q = CV$$ | Charge stored on a capacitor. |
| Energy in Capacitor | $$U_C = \frac{1}{2}CV^2$$ | Potential energy stored. |
| Series Resistance | $$R_T = R_1 + R_2 + \dots$$ | Total resistance is the sum. |
| Parallel Resistance | $$\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \dots$$ | Reciprocal sum. |
| Magnetic Force (on charge) | $$F_B = qvB\sin\theta$$ | Force on a moving charge in a B-field. |
| Faraday's Law of Induction | $$\mathcal{E} = -N\frac{\Delta\Phi_B}{\Delta t}$$ | Induced EMF from changing magnetic flux. |
| Motional EMF | $$\mathcal{E} = vBL$$ | EMF induced in a conductor moving in B-field. |
Concept: Series vs Parallel
Part 4: Waves & Optics
| Concept | Formula | Description |
|---|---|---|
| Wave Speed | $$v = f\lambda$$ | Relates speed, frequency, and wavelength. |
| Snell's Law | $$n_1\sin\theta_1 = n_2\sin\theta_2$$ | Law of refraction. |
| Refractive Index | $$n = c/v$$ | Ratio of light speed in vacuum to medium. |
| Diffraction Grating | $$d\sin\theta = m\lambda$$ | Condition for constructive interference. |
| Young's Double Slit (Fringe Spacing) | $$\Delta x = \frac{L\lambda}{d}$$ | Distance between bright fringes. |
| Photon Energy | $$E = hf$$ | Energy of a single photon. |
Concept: Wave Properties
Part 5: Algebra & Functions
| Concept | Formula | Description |
|---|---|---|
| Quadratic Formula | $$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$ | Solves $ax^2+bx+c=0$. |
| Standard Form (Quadratic) | $$y = a(x-p)^2+q$$ | Vertex form, vertex at $(p,q)$. |
| Logarithm Definition | $$y = b^x \iff x = \log_b(y)$$ | Logarithms are exponents. |
Part 6: Geometry & Trigonometry
| Concept | Formula | Description |
|---|---|---|
| Pythagorean Theorem | $$a^2 + b^2 = c^2$$ | For right-angled triangles. |
| Distance Formula | $$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$ | Distance between two points. |
| Circle Equation | $$(x-h)^2 + (y-k)^2 = r^2$$ | Center $(h,k)$, radius $r$. |
| Trigonometric Identity | $$\sin^2\theta + \cos^2\theta = 1$$ | Fundamental identity. |
| Sine Rule | $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ | Relates sides and angles. |
| Cosine Rule | $$c^2 = a^2 + b^2 - 2ab\cos C$$ | Relates three sides and an angle. |
Concept: Right Triangle (SOH CAH TOA)
Part 7: Probability & Statistics
| Concept | Formula | Description |
|---|---|---|
| Permutation | $${}_n P_r = \frac{n!}{(n-r)!}$$ | Ordered arrangements. |
| Combination | $${}_n C_r = \frac{n!}{r!(n-r)!}$$ | Unordered selections. |
| Conditional Probability | $$P(A|B) = \frac{P(A \cap B)}{P(B)}$$ | Probability of A given B. |
Part 8: Fundamental Constants
| Constant | Value (Approximate) |
|---|---|
| Speed of Light in Vacuum (c) | $$3.00 \times 10^8 \text{ m/s}$$ |
| Gravitational Acceleration on Earth (g) | $$9.8 \text{ m/s}^2$$ |
| Planck's Constant (h) | $$6.63 \times 10^{-34} \text{ J} \cdot \text{s}$$ |
| Universal Gravitational Constant (G) | $$6.67 \times 10^{-11} \text{ N m}^2/\text{kg}^2$$ |
Part 9: SI Fundamental & Derived Units
SI Base Units
| Quantity | Unit Name | Symbol |
|---|---|---|
| Length | Meter | m |
| Mass | Kilogram | kg |
| Time | Second | s |
| Electric Current | Ampere | A |
| Temperature | Kelvin | K |
| Amount of Substance | Mole | mol |
| Luminous Intensity | Candela | cd |
Common Derived Units
| Quantity | Unit Name | Symbol | In Base Units |
|---|---|---|---|
| Force | Newton | N | $$\text{kg} \cdot \text{m/s}^2$$ |
| Energy, Work | Joule | J | $$\text{N} \cdot \text{m}$$ |
| Power | Watt | W | $$\text{J/s}$$ |
| Pressure | Pascal | Pa | $$\text{N/m}^2$$ |
| Electric Charge | Coulomb | C | $$\text{A} \cdot \text{s}$$ |
| Voltage | Volt | V | $$\text{W/A}$$ |
| Resistance | Ohm | Ω | $$\text{V/A}$$ |
Theme 1: Mechanics - Newton's Laws
#1
According to Newton's Third Law, for every action, there is an equal and opposite reaction. Why don't these action-reaction force pairs cancel each other out?
Theme 2: Mechanics - Work and Energy
#2
What is the conceptual difference between kinetic energy and potential energy? Explain the principle of conservation of mechanical energy.
Theme 3: Thermodynamics
#3
What is the fundamental difference between heat and temperature?
Theme 4: Electricity and Magnetism
#4
Explain Ohm's Law. From a microscopic perspective, what causes electrical resistance in a metal conductor?
Theme 5: Waves
#5
What is the principle of superposition, and how does it explain the phenomena of constructive and destructive interference?
Theme 6: Functions and Graphs
#6
What is the conceptual difference between the domain and the range of a function?
Theme 7: Geometry and Trigonometry
#7
Explain the relationship between the sine, cosine, and tangent of an angle in a right-angled triangle, and state the Pythagorean theorem.
Theme 8: Probability and Statistics
#8
Explain the difference between a permutation and a combination.
Theme 9: Mechanics - Circular Motion
#9
An object is moving in a circle at a constant speed. Is the object accelerating? Explain your answer.
Theme 10: Electricity - Circuits
#10
What is the difference between a series circuit and a parallel circuit in terms of the paths for current and the total resistance?
Theme 11: Mechanics - Mass vs. Weight
#11
What is the fundamental conceptual difference between mass and weight?
Theme 12: Mechanics - Simple Harmonic Motion
#12
What is the defining characteristic of the force that causes simple harmonic motion (SHM)?
Theme 13: Thermodynamics
#13
What is the difference between specific heat capacity and heat capacity?
Theme 14: Thermodynamics
#14
The Second Law of Thermodynamics states that the entropy of an isolated system always increases. What is entropy, and what does this law imply about the direction of spontaneous processes?
Theme 15: Electricity
#15
What is the conceptual difference between electric potential (voltage) and electric potential energy?
Theme 16: Magnetism
#16
How does a permanent magnet, like a bar magnet, create a magnetic field from a microscopic perspective?
Theme 17: Electromagnetism
#17
Explain the basic principle of how an electric generator works, based on Faraday's Law of Induction.
Theme 18: Waves
#18
What is the difference between a transverse wave and a longitudinal wave? Provide an example of each.
Theme 19: Waves - Sound
#19
Why does sound travel faster in solids and liquids than it does in gases?
Theme 20: Waves - Light
#20
Why does the sky appear blue on a clear day?
Theme 21: Algebra
#21
Why is division by zero considered 'undefined' in mathematics?
Theme 22: Algebra - Logarithms
#22
What is the primary purpose of using logarithms? Explain their conceptual relationship to exponents.
Theme 23: Functions
#23
What is the key conceptual difference in the way an exponential function (like $y = 2^x$) and a power function (like $y = x^2$) grow?
Theme 24: Geometry
#24
Why must the sum of the interior angles of any triangle always be $180^\circ$ (in Euclidean geometry)? Provide a brief proof.
Theme 25: Trigonometry
#25
Why are the sine and cosine functions considered 'periodic'? What is the fundamental period of both functions?
Theme 26: Thermodynamics - Ideal Gases
#26
State the Ideal Gas Law equation and describe what each term represents. From a microscopic perspective, how does increasing the temperature of a gas in a sealed, rigid container lead to an increase in pressure?
Theme 27: Mechanics - Momentum
#27
What is the principle of conservation of linear momentum? Under what condition does it apply?
Theme 28: Thermodynamics
#28
Describe the three mechanisms of heat transfer: conduction, convection, and radiation.
Theme 29: Electromagnetism
#29
A charged particle moves through a uniform magnetic field. What is the direction of the magnetic force on the particle relative to its velocity and the magnetic field?
Theme 30: Waves - Refraction
#30
When a light wave enters a denser medium (like from air to water), what happens to its speed, wavelength, and frequency?