Overview
This lecture covers periodic motion, focusing on the mass-spring system, Hooke's Law, energy analysis, and equations of motion for simple harmonic oscillators. Key concepts include forces, kinematics, energy, and resonance in spring-mass systems, with sample problems and formulas.
Periodic and Simple Harmonic Motion
- Periodic motion repeats itself in regular intervals (oscillation).
- Examples: mass-spring system and simple pendulum.
- Restoring force always acts opposite to displacement, returning the system to equilibrium.
Hooke's Law and Forces in Springs
- Hooke's Law: Restoring force ( F_r = -kx ), where ( x ) is displacement from equilibrium, and ( k ) is the spring constant (N/m).
- Stiffer springs have higher ( k ), requiring more force for the same displacement.
- Negative sign shows restoring force direction opposes displacement.
Spring Constant, Force, and Work Calculations
- Force to stretch/compress: ( F = kx ).
- Work done (or elastic potential energy): ( W = U = \frac{1}{2}kx^2 ).
- For variable force, work equals area under the force vs. displacement graph.
Kinematics and Dynamics of Oscillators
- At max displacement, velocity is zero; acceleration is maximum.
- At equilibrium, velocity is maximum; acceleration is zero.
- If friction is present, oscillations are damped (amplitude decreases).
Energy in Spring-Mass Systems
- Mechanical energy ( E_{mech} = KE + PE_{spring} ).
- ( KE = \frac{1}{2}mv^2 ); ( PE_{spring} = \frac{1}{2}kx^2 ).
- Total energy constant if no friction; depends on amplitude: ( E_{mech} = \frac{1}{2}ka^2 ).
Velocity and Acceleration in SHM
- Maximum velocity: ( v_{max} = a\sqrt{\frac{k}{m}} ).
- Maximum acceleration: ( a_{max} = \frac{k}{m}a ).
- General velocity at ( x ): ( v = \pm v_{max}\sqrt{1 - \left(\frac{x^2}{a^2}\right)} ).
Period and Frequency of Oscillation
- Period: ( T = 2\pi\sqrt{\frac{m}{k}} ); Frequency: ( f = \frac{1}{T} ).
- Increasing mass increases period; increasing ( k ) decreases period.
- Total distance in ( n ) periods: ( 4a \times n ).
Position, Velocity, and Acceleration Functions
- Position: ( x(t) = a\cos(2\pi ft) ) or ( a\sin(2\pi ft) ) depending on initial conditions.
- Velocity: ( v(t) = -v_{max}\sin(2\pi ft) ).
- Acceleration: ( a(t) = -a_{max}\cos(2\pi ft) ).
Damped Harmonic Motion and Resonance
- Damping due to friction causes amplitude to decrease over time.
- Types of damping: underdamped (oscillates before stopping), overdamped (returns slowly), critically damped (returns fastest without oscillation).
- Resonant frequency: driving a system at its natural frequency increases amplitude dramatically.
Key Terms & Definitions
- Periodic Motion — Motion that repeats at regular intervals.
- Simple Harmonic Motion (SHM) — Type of periodic motion where restoring force is proportional to displacement.
- Hooke's Law — ( F = -kx ), relation between force, displacement, and spring constant.
- Spring Constant (k) — Measure of spring stiffness, unit N/m.
- Amplitude (a) — Maximum displacement from equilibrium.
- Period (T) — Time to complete one cycle.
- Frequency (f) — Number of cycles per second (Hz).
- Mechanical Energy — Total energy (kinetic + potential) in the oscillating system.
- Damping — Reduction of amplitude due to friction/resistance.
- Resonance — Large amplitude oscillation when driven at natural frequency.
Action Items / Next Steps
- Practice applying Hooke's Law and energy formulas to spring problems.
- Memorize SHM equations for position, velocity, acceleration.
- Complete assigned homework problems on mass-spring oscillators.
- Review key definitions and solve additional examples on period, frequency, and resonance.