Restoring acceleration proportional to displacement.

$a = -\omega^2 x$

Differential Equation: d2x/dt2 = -ω2x:

$\frac{d^2x}{dt^2} = -\omega^2 x$

Simple harmonic oscillator equation.

$\frac{d^2x}{dt^2} = -\omega^2 x$

Simple harmonic oscillator equation.

Position as function of time.

Where:

• A = amplitude (maximum displacement)
• $\omega$ = angular frequency
• $\phi$ = phase constant

Velocity as function of time.

$v(t) = \frac{dx}{dt} = -\omega A\sin(\omega t + \phi)$

Acceleration as function of time.

$a(t) = \frac{dv}{dt} = -\omega^2 A\cos(\omega t + \phi) = -\omega^2 x(t)$

Spring provides restoring force: $F = -kx$ (Hooke's Law)

Equation of motion: $m\frac{d^2x}{dt^2} = -kx$

$\frac{d^2x}{dt^2} = -\frac{k}{m}x$

Compare to SHM equation: $\omega^2 = \frac{k}{m}$

$F = -kx$

Where k = spring constant (N/m).

Period: T = 2π√(m/k):

$T = 2\pi\sqrt{\frac{m}{k}}$

Frequency: $f = \frac{1}{T} = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$

Angular frequency: $\omega = 2\pi f = \sqrt{\frac{k}{m}}$

$T = 2\pi\sqrt{\frac{L}{g}}$

Valid for small angles (θ < 15 degrees ), where $\sin\theta \approx \theta$.

Properties:

• Independent of mass!
• Independent of amplitude (small angles)
• Period increases with length

$E_{total} = K + U = \text{constant} = \frac{1}{2}kA^2$

At any position: $\frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \frac{1}{2}kA^2$

Energy exchange:

• x = 0: K maximum, U = 0
• x = ±A: K = 0, U maximum
• Continuous exchange between K and U

For small angles, gravity provides restoring torque.

$\tau = -mgL\sin\theta \approx -mgL\theta$

Equation of motion: $mL^2\frac{d^2\theta}{dt^2} = -mgL\theta$

$\frac{d^2\theta}{dt^2} = -\frac{g}{L}\theta$

Compare to SHM: $\omega^2 = \frac{g}{L}$

$T = 2\pi\sqrt{\frac{m}{k}}$

Frequency: $f = \frac{1}{T} = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$

Angular frequency: $\omega = 2\pi f = \sqrt{\frac{k}{m}}$

Damping small enough for oscillations.

Differential equation: $m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0$

Solution: Oscillation with exponentially decaying amplitude.

Minimum damping without oscillation.

Condition: $b^2 = 4mk$

Returns to equilibrium fastest without overshooting.

Large damping, no oscillations.

Condition: $b^2 > 4mk$

Returns to equilibrium slowly, no overshoot.

F = G m1 m2 / r2:

Newton's law of universal gravitation.

$F = G\frac{m_1 m_2}{r^2}$

Where:

• G = 6.674 × 10-11 N - m2/kg2
• r = separation distance

$U = -\frac{Gm_1 m_2}{r}$

Reference: U = 0 at r -> ∞

Escape velocity: $v_{esc} = \sqrt{\frac{2GM}{R}}$

First Law: Planetary (and satellite) orbits are ellips with central mass at one focus.

Second Law (Equal Area): Radius vector sweeps equal areas in equal times (consequence of angular momentum conservation).

Third Law: $T^2 \propto a^3$ (period squared ∝ semi-major axis cubed)

$\frac{T^2}{a^3} = \frac{4\pi^2}{GM} = \text{constant}$

Special case of elliptical orbit with e = 0.

Circular orbit condition: $F_g = F_c$

$\frac{GMm}{r^2} = \frac{mv^2}{r}$

Orbital speed: $v = \sqrt{\frac{GM}{r}}$

Orbital period: $T = \frac{2\pi r}{v} = 2\pi\sqrt{\frac{r^3}{GM}}$

Total mechanical energy: $E = K + U = -\frac{GMm}{2r}$ (negative for bound orbits)