South 고등학교 2학년 Mechanics and Energy
Chapters: 3
1. Space-time and motion
Resultant force in 2D · Motion with Newton's laws · Circular motion · Kepler's laws and gravity · Energy of planets and satellites · Equivalence principle and curved space-time
- Scalars and Vectors for Motion in a Plane (Class 11) – A scalar has only size; a vector has size and direction. Vectors are equal if their size and direction match. Multiplying by a number changes the length (a negative number flips it). Vectors add tail-to-head (triangle or parallelogram law); A − B = A + (−B). Any vector in a plane is A = Ax î + Ay ĵ with Ax = A cos θ, Ay = A sin θ. A·B = AB cos θ is a scalar; A×B has size AB sin θ and is perpendicular to both.
- Inertia, First Law, Momentum, Second Law and Impulse – A body keeps its state of rest or uniform motion unless a net external force acts on it (first law). Inertia is this laziness to change, and mass measures it. Momentum p = mv. The rate of change of momentum equals the net force: F = dp/dt, which gives F = ma when mass is constant (second law). Impulse J = F × Δt = Δp; a longer stopping time means a smaller force.
- Projectile Motion and Uniform Circular Motion (Class 11) – In a plane, r = r₀ + v₀t + ½at² and v = v₀ + at, applied separately along x and y. A projectile has constant horizontal velocity u cos θ and a vertical velocity that changes by g each second, so its path is a parabola: y = x tan θ − gx²/(2u²cos²θ). T = 2u sin θ/g, H = u² sin²θ/2g, R = u² sin 2θ/g (maximum at 45°). In uniform circular motion speed is constant but the velocity turns, giving a centripetal acceleration a = v²/r = ω²r towards the centre.
- Kepler's Laws of Planetary Motion – Kepler gave three rules for how planets move. 1) Each planet moves on an ellipse with the Sun at one focus. 2) The line from the Sun to the planet sweeps equal areas in equal times, so the planet moves faster when it is near the Sun. 3) The square of the time for one round (T²) is proportional to the cube of the semi-major axis (a³). The second law is really conservation of angular momentum. The third law follows from Newton's law of gravitation.
- Escape Speed, Orbital Velocity and Energy of an Orbiting Satellite – Throw something fast enough sideways and it keeps falling around the Earth without landing: that speed is the orbital velocity, v₀ = √(GM/r), about 7.9 km/s just above the surface. Throw it faster, at the escape speed vₑ = √(2GM/R) = √(2gR) ≈ 11.2 km/s, and it leaves Earth for ever. vₑ = √2 × v₀. A satellite in a circular orbit has KE = GMm/2r, PE = −GMm/r and total energy E = −GMm/2r. The total is negative, so the satellite is bound. Higher orbits are slower and take longer: T = 2π√(r³/GM).
- General Relativity: Gravity as Curved Space-Time – General relativity says that mass and energy bend space-time, and things move along the bent paths. That is what we call gravity. It predicts that light bends near a mass (gravitational lensing), that clocks run slower in strong gravity, and that a very dense mass can form a black hole.
2. Heat and energy
Heat management in buildings · Phase change and heat · First law of thermodynamics · Heat engine cycles · Irreversibility and entropy
- Thermal Expansion, Specific Heat, Calorimetry and Latent Heat – Temperature tells how hot a body is; heat is energy that flows because of a temperature difference. Most things expand when heated: ΔL = αLΔT, ΔA = βAΔT, ΔV = γVΔT, with β = 2α and γ = 3α. Water is an exception between 0 °C and 4 °C, where it shrinks on heating, so it is densest at 4 °C. The heat needed to warm a body is Q = mcΔT, where c is the specific heat; gases have two, Cp > Cv, with Cp − Cv = R per mole. In calorimetry, heat lost by hot bodies equals heat gained by cold ones. During melting or boiling the temperature stays constant and heat Q = mL goes into changing the state.
- First Law of Thermodynamics – Internal energy U is the total energy of the molecules inside a system. It changes in only two ways: by heat Q (energy that flows because of a temperature difference) and by work W (energy moved by a force, like a moving piston). First law: ΔQ = ΔU + ΔW. Heat given to a gas partly raises its internal energy and partly lets it do work. Work by a gas at constant pressure is PΔV. For an ideal gas Cp − Cv = R.
- Second Law of Thermodynamics, Heat Engines and Refrigerators – The first law says energy is conserved; the second law says which way heat and energy can go. Heat flows by itself only from hot to cold (Clausius). No engine can turn all the heat it takes into work; some must be thrown into a colder body (Kelvin–Planck). A heat engine takes Q₁ from a hot source, does work W and rejects Q₂: efficiency η = W/Q₁ = 1 − Q₂/Q₁. A refrigerator uses work W to move Q₂ from cold to hot: COP α = Q₂/W. The best possible engine, the Carnot engine, has η = 1 − T₂/T₁.
3. Elastic waves and sound
Simple harmonic motion · Elastic waves: transmission and reflection · Doppler effect · Acoustics of spaces · How instruments make sound
- Simple Harmonic Motion (SHM) – A motion that repeats after a fixed time is periodic. If the object goes to and fro about a middle point and the force pulling it back is proportional to its distance from the middle (F = −kx), the motion is simple harmonic. Its position is x = A sin(ωt + φ), with ω = 2π/T = 2πf. Speed is largest in the middle, acceleration is largest at the ends, and total energy ½kA² stays constant.
- Progressive Waves: Types, Speed and Equation – A wave carries energy from place to place without carrying the matter along. In a transverse wave the particles move at right angles to the wave; in a longitudinal wave they move along it. Speed v = fλ. On a string v = √(T/μ); for sound in a gas v = √(γP/ρ). A wave moving along +x is y = A sin(kx − ωt), with k = 2π/λ and ω = 2πf.
- The Doppler Effect – The Doppler effect is the change in the frequency we hear (or detect) when the source of a wave and the observer move toward or away from each other. Moving closer squeezes the crests together: shorter wavelength, higher frequency, higher pitch. Moving apart stretches them: longer wavelength, lower frequency. For sound, f′ = f × (v ± vo) ÷ (v ∓ vs), using the upper signs when they approach. It is used in speed guns, weather radar, ultrasound scans of blood flow, bat echolocation and in astronomy (red shift and blue shift of light).
- Acoustics: How Rooms, Ears and Audio Tools Handle Sound – Acoustics is the science of sound: how it is made, how it travels and reflects in rooms, how we hear it, and how we record it. In a room we hear the direct sound plus many reflections; the time the sound takes to die away is the reverberation time, T = 0.161 V / A (Sabine). Soft materials absorb sound and shorten T. Loudness is measured in decibels, pitch depends on frequency and timbre on the mix of overtones. Microphones turn sound into electrical signals, which computers sample into numbers; MIDI sends note instructions, not sound.
- Superposition, Reflection and Standing Waves – When waves overlap, their displacements add (superposition). A wave reflected from a fixed end comes back upside down; from a free end it comes back upright. A wave and its reflection make a standing wave with nodes (no motion) and antinodes (most motion). A string fixed at both ends allows fₙ = n·v/2L (all harmonics). An open pipe allows fₙ = n·v/2L; a pipe closed at one end allows only odd harmonics, fₙ = n·v/4L (n = 1, 3, 5…).