Physics Professional

In calculus-based mechanics, position, velocity, and acceleration are vector functions of time. Their relationships are:

• v(t) = dr/dt (velocity is the time derivative of position)
• a(t) = dv/dt = d²r/dt²
• Δr = ∫v dt (displacement is the integral of velocity)

In two or three dimensions, vectors are added component-by-component. The magnitude of vector A = (A_x, A_y, A_z) is |A| = √(Ax² + Ay² + Az²).

Given v(t) = 3t² − 2t + 1 m/s, find displacement from t = 0 to t = 2 s:
  Δx = ∫₀² (3t² − 2t + 1) dt = [t³ − t² + t]₀² = (8 − 4 + 2) − 0 = 6 m

Newton’s second law in its most general form is F = dp/dt. For constant mass this reduces to F = ma = m d²x/dt², a second-order ordinary differential equation.

When force depends on position or velocity (e.g. spring, drag), we solve the ODE. For a linear restoring force F = −kx:

• ODE: m d²x/dt² + kx = 0
• Solution: x(t) = A cos(ωt + φ) with ω = √(k/m)

For a velocity-dependent drag F = −bv, the velocity decays exponentially: v(t) = v₀ e^(−bt/m).

Terminal velocity: set ma = mg − bv = 0
  v_terminal = mg/b

The work–energy theorem: Wnet = ΔKE = ½mvf² − ½mvi².

Work done by a variable force along a path: W = ∫ F · dr.

A conservative force has an associated potential energy: F = −dU/dx (in 1-D), and ΔU = −W. Total mechanical energy E = KE + U is conserved if only conservative forces act.

For a spring: Uspring = ½kx². For gravity near Earth’s surface: Ugrav = mgy.

Work done by a spring compressed from x = 0 to x = 0.2 m (k = 200 N/m):
  W = ∫₀^0.2 (−kx) dx = −½ × 200 × 0.04 = −4 J
  Energy stored in spring = +4 J

The impulse–momentum theorem: J = ∫ F dt = Δp.

In any collision, total momentum is conserved. Energy may or may not be:

• Elastic collision: both momentum and kinetic energy conserved. For 1-D head-on elastic collision of masses m₁ (velocity u₁) and m₂ (at rest): v₁ = (m₁ − m₂)/(m₁ + m₂) u₁, v₂ = 2m₁/(m₁ + m₂) u₁.
• Perfectly inelastic: objects stick together, maximum KE lost (consistent with momentum conservation).
• Coefficient of restitution e = relative speed of separation / relative speed of approach. Elastic: e = 1; perfectly inelastic: e = 0.

Equal masses (m each) elastic collision — first object stops:
  v₁ = (m − m)/(2m) u = 0,  v₂ = 2m/(2m) u = u  ✓

Rotational analogues of Newton’s second law:

• Torque: τ = r × F; magnitude τ = rF sinθ (N·m).
• Moment of inertia: I = Σmiri² = ∫ r² dm (kg·m²). Common values: solid disk I = ½MR²; solid sphere I = 2/5 MR².
• Rotational second law: τnet = Iα.
• Angular momentum: L = Iω; conserved when net external torque is zero.

Rotational KE: KErot = ½Iω². Total KE of a rolling object = translational + rotational.

A solid disk (M = 2 kg, R = 0.5 m) spins at ω = 10 rad/s.
  I = ½ × 2 × 0.25 = 0.25 kg·m²
  KE_rot = ½ × 0.25 × 100 = 12.5 J

Simple harmonic motion (SHM) arises whenever a restoring force is proportional to displacement: F = −kx.

• Angular frequency: ω = √(k/m) (rad/s)
• Period: T = 2π√(m/k) (s)
• Position: x(t) = A cos(ωt + φ)
• Max speed: vmax = Aω at x = 0
• Max acceleration: amax = Aω² at x = ±A

A simple pendulum of length L oscillates with T = 2π√(L/g) for small angles (θ < ~15°).

A mass of 0.4 kg on a spring (k = 160 N/m):
  ω = √(160/0.4) = √400 = 20 rad/s
  T = 2π/20 ≈ 0.314 s

Newton’s law of universal gravitation: F = Gm₁m₂/r², where G = 6.674 × 10⁻¹¹ N·m²/kg².

Kepler’s three laws (derived from Newton’s law):

• First: Planets orbit the Sun in ellipses with the Sun at one focus.
• Second: A line joining a planet to the Sun sweeps equal areas in equal times (conservation of angular momentum).
• Third: T² ∝ a³, where a is the semi-major axis.

Circular orbit: gravitational force provides centripetal force → vorbit = √(GM/r). Escape velocity: vesc = √(2GM/r).

Orbital speed at radius r = 6.4 × 10⁶ m around Earth (M = 6×10²⁴ kg):
  v = √(6.674×10⁻¹¹ × 6×10²⁴ / 6.4×10⁶) ≈ 7 900 m/s

Coulomb’s law: F = kq₁q₂/r², where k = 8.99 × 10⁹ N·m²/C².

The electric field due to a point charge: E = kq/r² (N/C), directed radially outward for positive q.

Gauss’s law: ∮ E · dA = Qenc/ε₀. For a uniformly charged sphere of charge Q and radius R:

• Outside (r > R): E = kQ/r² (same as point charge)
• Inside (r < R): E = kQr/R³ (grows linearly)

Electric potential: V = kq/r; relationship to field: E = −dV/dx.

Force between two charges q = 2 μC separated by 0.3 m:
  F = 8.99×10⁹ × (2×10⁻⁶)² / 0.09 ≈ 0.40 N (repulsive)

The Lorentz force on a charge moving in combined electric and magnetic fields: F = q(E + v × B).

Faraday’s law of induction: EMF = −dΦB/dt, where Φ_B = ∫ B · dA is magnetic flux.

Ampère’s law (with Maxwell’s displacement-current correction): ∮ B · dl = μ₀(I + ε₀ dΦE/dt).

Maxwell’s four equations (integral form) unify all classical electromagnetism and predict electromagnetic waves travelling at c = 1/√(μ₀ε₀) ≈ 3 × 10⁸ m/s.

An EMF is induced in a coil of 50 turns when flux changes by 0.02 Wb in 0.1 s:
  EMF = −N × ΔΦ/Δt = −50 × 0.02/0.1 = −10 V  (magnitude 10 V)

The laws of thermodynamics:

• Zeroth law: If A is in thermal equilibrium with B, and B with C, then A is in equilibrium with C (defines temperature).
• First law: ΔU = Q − W (change in internal energy = heat added − work done by system).
• Second law: The entropy of an isolated system never decreases: ΔS ≥ 0. Equivalently, no process can be 100 % efficient.
• Third law: Entropy approaches zero as temperature approaches absolute zero.

Entropy change for a reversible process: dS = dQ/T. The Carnot efficiency gives the upper bound: η = 1 − Tcold/Thot.

Carnot engine between 500 K (hot) and 300 K (cold):
  η = 1 − 300/500 = 0.40 = 40 %

Einstein’s special relativity (1905) is built on two postulates: (1) the laws of physics are the same in all inertial frames; (2) the speed of light c is the same in all inertial frames.

Key results (Lorentz factor γ = 1/√(1 − v²/c²)):

• Time dilation: Δt = γ Δt₀ (moving clocks run slow)
• Length contraction: L = L₀/γ
• Mass–energy equivalence: E = γmc² → E = mc² at rest
• Relativistic momentum: p = γmv

The photoelectric effect (Einstein, 1905): photons of energy E = hf (h = 6.626 × 10⁻³⁴ J·s) eject electrons when hf > the work function φ. Maximum kinetic energy of ejected electron: KEmax = hf − φ.

Bohr model: electron energy levels in hydrogen:
  E_n = −13.6 / n² eV
  Photon emitted when electron drops from n=3 to n=2:
  ΔE = 13.6(1/4 − 1/9) = 13.6 × 5/36 ≈ 1.89 eV  (red light, λ ≈ 656 nm)