Overview
This lecture covers the fundamentals of simple harmonic motion (SHM), focusing on the mass-spring system. Key concepts include periodic motion, Hooke's Law, restoring force, calculations of frequency and period, and energy relationships in oscillating systems.
Introduction to Periodic Motion
- Periodic motion repeats itself at regular intervals, such as the motion of a mass attached to a spring.
- Simple harmonic motion (SHM) is a specific type of periodic motion where the restoring force is proportional to displacement.
Mass-Spring System Example
- A mass attached to a spring oscillates back and forth when displaced from equilibrium.
- The equilibrium position is where the net force on the mass is zero.
Hooke's Law and Restoring Force
- Hooke's Law: restoring force, F = -kx, where k is the spring constant and x is displacement from equilibrium.
- The negative sign indicates the force always acts toward the equilibrium position.
Negative Symbol for Restoring Force
- The negative sign in F = -kx means that the direction of the force opposes displacement.
Frequency and Period of Oscillator
- Frequency (f) is the number of cycles per second; period (T) is the time to complete one cycle (T = 1/f).
- For a mass-spring system: ( f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} ), and ( T = 2\pi \sqrt{\frac{m}{k}} ).
Maximum Displacement and Velocity
- Amplitude (A) is the maximum displacement from equilibrium.
- Maximum velocity occurs as the mass passes through equilibrium.
Restoring Force and Acceleration
- The restoring force causes acceleration always directed toward equilibrium.
- Acceleration in SHM: ( a = -\frac{k}{m}x ).
Vertical Spring Example
- When a mass hangs vertically, gravity and the spring both contribute to the forces on the mass.
- The new equilibrium position accounts for both gravitational and spring forces.
Key Terms & Definitions
- Simple Harmonic Motion (SHM) — Motion where the restoring force is proportional and opposite to displacement.
- Hooke's Law — The relationship ( F = -kx ) for springs.
- Amplitude (A) — Maximum displacement from equilibrium.
- Frequency (f) — Number of oscillations per second.
- Period (T) — Time taken for one complete oscillation.
- Restoring Force — Force that brings the mass back to equilibrium.
Action Items / Next Steps
- Practice applying Hooke’s Law and calculating frequency and period for various mass-spring systems.
- Review SHM formulas and solve related practice problems.
- Study vertical spring systems and review derivations for equilibrium and oscillation.