Overview
This lecture introduces the fundamentals of simple harmonic motion (SHM), exploring its definition, key properties, and the behavior of springs and pendulums in SHM, along with related graphs and energy concepts.
Definition and Core Concepts of SHM
- Simple harmonic motion is periodic motion around an equilibrium point where there is no net force.
- A restoring force proportional and opposite to the displacement acts on the object in SHM.
- The equilibrium position is the point where net force is zero.
Examples of SHM: Spring and Pendulum
- In a spring, equilibrium is at the spring’s natural length; net force is zero there.
- In a pendulum, equilibrium is when gravity and tension forces balance.
- Displacement from equilibrium increases the restoring force.
Hooke’s Law and Linear Restoring Force
- Hooke’s Law: Fs = -kx, where Fs is the spring force, k is the spring constant, and x is displacement.
- The negative sign indicates force is opposite to displacement.
- The restoring force is linearly proportional to displacement (fits y=mx+b).
SHM in Pendulums
- Restoring force for a pendulum: F = -mg sinθ, where θ is the angle from equilibrium.
- For small angles, sinθ ≈ θ, making the restoring force linearly related to displacement.
Amplitude, Period, and Frequency
- Amplitude is the maximum displacement from equilibrium, not the total path length.
- Period (T): time to complete one full cycle (both extremes and return), measured in seconds.
- Frequency (f): cycles per second (1/T), measured in Hertz (Hz).
Graphs in SHM
- Displacement-time graphs for SHM are sine waves.
- Amplitude is measured from center to one extreme on the graph.
- Period is the time for a complete sine wave (both up and down, returning to start).
Velocity and Acceleration in SHM
- Slope of displacement-time graph gives velocity; slope of velocity-time graph gives acceleration.
- Velocity is maximum at equilibrium, zero at maximum displacement (extrema).
- Acceleration is maximum at extrema, zero at equilibrium.
Force and Energy in SHM
- Force-time graph matches acceleration-time graph (F=ma).
- Kinetic energy is highest at equilibrium; potential energy is highest at extrema.
- Total energy (kinetic + potential) remains constant (conservation of energy).
- Springs use elastic potential energy; pendulums use gravitational potential energy.
Key Terms & Definitions
- Simple Harmonic Motion (SHM) — Periodic motion with a linear restoring force toward equilibrium.
- Equilibrium Position — The point where net force on the object is zero.
- Restoring Force — The force that acts to bring the object back to equilibrium, proportional to displacement.
- Amplitude — The maximum displacement from equilibrium.
- Period (T) — Time for one complete cycle of motion.
- Frequency (f) — Number of cycles per second; f = 1/T.
- Hooke’s Law — Fs = -kx; spring force is proportional and opposite to displacement.
Action Items / Next Steps
- Review and memorize definitions and key equations for SHM.
- Practice identifying amplitude, period, and frequency from SHM graphs.
- Prepare for upcoming assignments or quizzes on SHM properties and calculations.