Bio 5.2

14.1 Simple Harmonic Motion

• Periodic motion repeats at regular time intervals.
• Oscillatory (harmonic) motion is periodic motion around an equilibrium.
• Simple Harmonic Motion (SHM) occurs when restoring force is proportional to displacement.
• Displacement (x) is the object’s distance from equilibrium.
• Amplitude (A) is the maximum displacement from equilibrium.
• Period (T) is the time for one complete cycle.
• Frequency (f) is the number of cycles per unit time; $f = 1/T$.
• Angular frequency (ω) measures oscillation rate in radians per second; $ω = 2πf = 2π/T$.
• Displacement in SHM is described by $x(t) = A \cos(ωt + φ)$ or $x(t) = A \sin(ωt + φ)$.
  • $A$ represents amplitude.
  • $ω$ represents angular frequency.
  • $t$ represents the time.
  • $φ$ is the phase constant (initial phase angle).
• Velocity in SHM is defined as $v(t) = -Aω \sin(ωt + φ)$ or $v(t) = Aω \cos(ωt + φ)$.
• Acceleration in SHM is defined as $a(t) = -Aω^2 \cos(ωt + φ)$ or $a(t) = -Aω^2 \sin(ωt + φ)$.
• Maximum velocity in SHM: $v_{max} = Aω$.
• Maximum acceleration in SHM: $a_{max} = Aω^2$.
• Acceleration and displacement have the relationship $a(t) = -ω^2x(t)$.
• Example Problem
  • A block attached to a spring has an amplitude of 0.12 m and a period of 2.0 s.
  • Angular frequency calculates to ≈ 3.14 rad/s, maximum speed calculates to ≈ 0.38 m/s, and maximum acceleration calculates to ≈ 1.18 m/s².

14.2 Energy in Simple Harmonic Motion

• Potential energy (U) in SHM is associated with restoring force.
• For a spring-mass system: $U(x) = \frac{1}{2}kx^2$, where $k$ is the spring constant.
• Kinetic energy (K) in SHM is given by $K(t) = \frac{1}{2}mv^2 = \frac{1}{2}mA^2ω^2\sin^2(ωt + φ)$.
• Total mechanical energy (E) in SHM: $E = U + K = \frac{1}{2}kA^2 = \frac{1}{2}mA^2ω^2$.
• Total energy is constant and proportional to the square of the amplitude.
• Energy transforms between potential and kinetic, with total mechanical energy remaining constant without non-conservative forces.
• Example Problem
  • A 0.2 kg block attached to a spring (k = 100 N/m) has an amplitude of 0.05 m.
  • The total energy calculates to 0.125 J, potential energy at x = 0.02 m calculates to 0.02 J, and kinetic energy at x = 0.02 m calculates to 0.105 J.

14.3 The Simple Pendulum

• A simple pendulum is a point mass suspended by a string/rod of length L.
• For small angles (< 15°), pendulum motion approximates SHM.
• The restoring force is the component of gravitational force returning the pendulum to equilibrium.
• $F = -mg\sin\theta$ is the restoring force.
• Angular frequency (ω) is defined as $ω = \sqrt{\frac{g}{L}}$.
• Period (T) is defined as $T = 2\pi \sqrt{\frac{L}{g}}$.
• The period does not depend on mass (m).
• The period depends on length (L) and gravity (g).
• Example Problem
  • A simple pendulum with length 1.0 m has a period of ≈ 2.01 s and a frequency of ≈ 0.50 Hz.

14.4 Damped Oscillations

• Damped oscillations occur when non-conservative forces oppose motion, dissipating energy and reducing amplitude.
• Types of damping:
  • Underdamped: Oscillations with decreasing amplitude.
  • Critically damped: Quickest return to equilibrium without oscillation.
  • Overdamped: Slower return to equilibrium without oscillation.
• Damped oscillator motion: $m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0$, where b is the damping coefficient.
• Amplitude decays exponentially: $A(t) = A_0e^{-γt}$, where γ = b/(2m) is the damping constant.
• Example Problem
  • A damped oscillator with an initial amplitude of 0.2 m reduces to 0.1 m after 5 seconds, giving a damping constant of ≈ 0.139 s⁻¹.

14.5 Forced Oscillations and Resonance

• Forced oscillations occur when an external periodic force is applied, causing the system to oscillate at the driving frequency.
• Resonance occurs when the driving frequency nears the system's natural frequency, resulting in large amplitude oscillations.
• A resonance curve plots amplitude versus driving frequency, peaking at the resonance frequency.
• Resonance is used in musical instruments, radios, and MRI machines, but can be destructive, like bridge collapses from wind.
• Example: A mass-spring system at a natural frequency of 2 Hz undergoes increasing oscillation amplitude when driven near that frequency, reaching maximum amplitude at resonance.

Advanced Calculus

Real Numbers, Sequences, and Functions

1.1 The Real Number System

• Axiom 1 (Algebraic Axioms) defines the properties of addition (+) and multiplication (·) on the real numbers:

  • (a) Commutative laws: $x + y = y + x$, $x \cdot y = y \cdot x$.
  • (b) Associative laws: $(x + y) + z = x + (y + z)$, $(x \cdot y) \cdot z = x \cdot (y \cdot z)$.
  • (c) Distributive law: $x \cdot (y + z) = x \cdot y + x \cdot z$.
  • (d) Existence of additive identity: $\exists 0 \in \mathbb{R}$ such that $x + 0 = x, \forall x \in \mathbb{R}$.
  • (e) Existence of multiplicative identity: $\exists 1 \neq 0 \in \mathbb{R}$ such that $x \cdot 1 = x, \forall x \in \mathbb{R}$.
  • (f) Existence of additive inverse: $\forall x \in \mathbb{R}, \exists -x \in \mathbb{R}$ such that $x + (-x) = 0$.
  • (g) Existence of multiplicative inverse: $\forall x \neq 0 \in \mathbb{R}, \exists x^{-1} \in \mathbb{R}$ such that $x \cdot x^{-1} = 1$.

• Axiom 2 (Order Axiom) defines the existence of a subset $P$ of positive real numbers:

  • (a) Trichotomy law: For any $x \in \mathbb{R}$, exactly one of the following holds: $x \in P; x = 0; -x \in P$.
  • (b) Closure under addition and multiplication: If $x, y \in P$, then $x + y \in P$ and $x \cdot y \in P$.

• Axiom 3 (Completeness Axiom) states that every nonempty subset of $\mathbb{R}$ that is bounded above has a least upper bound in $\mathbb{R}$.

1.2 Mathematical Induction

• Mathematical induction proves a proposition $P(n)$ for all integers $n \geq n_0$ by:
  • (a) Showing $P(n_0)$ is true.
  • (b) Proving that if $P(k)$ is true for some $k \geq n_0$, then $P(k + 1)$ is also true.

1.3 Sequences of Real Numbers

• A sequence of real numbers is a function $f: \mathbb{N} \rightarrow \mathbb{R}$.
• Definition 1.1: A sequence $(x_n)$ converges to $x$ if, for every $\epsilon > 0$, there exists $N$ such that $n > N$ implies $|x_n - x| < \epsilon$.
• Theorem 1.1: The limit of a convergent sequence is unique.
• Theorem 1.2: Every convergent sequence is bounded.