Exam 1 Phys

Simple Harmonic Motion (SHM)

Occur when the restoring force is proportional and opposite to the displacement: F = -kx.

Angular frequency (ω0)

Defined as ω_o = √(k/m), where k is the spring constant and m is the mass.

Equations of Motion for SHM

The equation is x¨ + ω0²x = 0, with the general solution being x(t) = A cos(ω_ot) + B sin(ω_o)t).

Kinetic Energy in SHM

Given by K = (1/2) m ẋ², where m is mass and ẋ is velocity.

Potential Energy in SHM

Given by U = (1/2) k x², where k is the spring constant and x is the displacement.

Total Energy in undamped SHM

Etotal = K + U = (1/2) k A² = constant.

Damped SHM

Involves damping forces, leading to the equation: x¨ + 2ζω₀ ẋ + ω₀² x = 0.

Damping Ratio (ζ)

Defined as ζ = b/(2mω₀), where b is the damping coefficient and m is mass.

Under-Damped System

When ζ < 1, the system oscillates with exponentially decaying amplitude.

Forced SHM Equation of Motion

Describes the motion with an additional driving force: m x¨ + b ẋ + k x = F₀ cos(ωt).

Steady-State Amplitude (X(ω))

For under-damped systems, given by X(ω) = F_-/m / √(((ω₀² - ω²)² + (2ζω₀ω)²)).

Resonance Frequency (ωr)

Occurs near ω ≈ ω₀√(1 - 2ζ²) for small damping (ζ ≪ 1).

Quality Factor (Q)

Measures the sharpness of resonance, given by Q = ω₀/(2ζ) or Q = (1/(2ζ)).

Normal Modes

Independent oscillation patterns in coupled oscillators where the entire system moves with a single frequency.

Energy Exchange in SHM

In undamped SHM, total energy is constant; in damped, it decreases exponentially; in driven, it balances input and output.

Equations for Coupled Oscillators

Described by simultaneous second-order ODEs, and normal modes can be found by solving system equations.

Damped Frequency (ωd)

In an under-damped system, ωd = ω₀√(1 - ζ²).

Phasor Representation

Using e^(iωt) to treat cos(ωt) as the real part, useful for adding oscillations.

Exponential Decay in Damped SHM

Energy decreases exponentially over time due to the damping force.

Graphical Analysis in SHM

Plotting x, v, a vs. time reveals the phase relationships (e.g., v leads x by π/2).

Dimensionless Damping Ratio (ζ)

Often used in problems to simplify analysis and measure damping relative to the natural frequency.

Practical Approach to SHM Problems

Identify type of motion, use correct differential equation, and check boundary/initial conditions.