Simple harmonic motion is the back-and-forth movement of objects around a central point. It's everywhere, from swinging pendulums to vibrating guitar strings. This motion follows Hooke's law, where the restoring force is proportional to displacement.

Understanding simple harmonic motion helps explain many natural phenomena. Key concepts include periodic motion, oscillation, and phase. We'll explore how energy transforms between potential and kinetic forms during this motion, and how to solve problems involving spring-mass systems and pendulums.

Simple Harmonic Motion

Application of Hooke's law

• Hooke's law states that the restoring force $F$ is directly proportional to the displacement $x$ from equilibrium and acts in the opposite direction $F = -kx$
• The spring constant $k$ measures the stiffness of the spring and is expressed in units of N/m (newtons per meter)
• In a spring-mass system undergoing simple harmonic motion, the restoring force causes the mass to oscillate back and forth around its equilibrium position
• As the spring is stretched or compressed, it stores potential energy $PE = \frac{1}{2}kx^2$
• The kinetic energy $KE = \frac{1}{2}mv^2$ is maximum at the equilibrium position when the velocity is highest and zero at the extremes when the velocity is zero
• The total energy $E = PE + KE$ remains constant throughout the motion, with energy continuously converting between potential and kinetic forms (conservation of energy)

Characteristics of periodic motion

• Periodic motion involves an object or system that repeatedly returns to its starting position after a fixed time interval
• The amplitude $A$ represents the maximum displacement from the equilibrium position
• The period $T$ is the time taken for one complete cycle of the motion and is related to the frequency $f$ by $T = \frac{1}{f}$
• Frequency $f$ measures the number of complete cycles per second and is expressed in hertz (Hz)
• The angular frequency $\omega$ describes the rate of change of the angle in radians per second and is related to the period by $\omega = \frac{2\pi}{T}$
• Examples of periodic motion include a swinging pendulum, a vibrating guitar string, and the oscillations of a spring-mass system

Oscillation and Phase

• Oscillation refers to the repetitive variation of a quantity or condition about a central value (equilibrium position)
• Phase describes the position of an oscillating object within its cycle of motion
• Harmonic motion is a type of oscillation where the restoring force is directly proportional to the displacement from equilibrium

Simple harmonic oscillator problems

• For a spring-mass system, the period $T$ depends on the mass $m$ and spring constant $k$ according to $T = 2\pi \sqrt{\frac{m}{k}}$
  1. Identify the given values for mass and spring constant
  2. Substitute these values into the equation for period
  3. Calculate the period using the simplified equation
• For a simple pendulum, the period $T$ depends on the length $L$ and acceleration due to gravity $g$ according to $T = 2\pi \sqrt{\frac{L}{g}}$
  1. Identify the given value for the pendulum length
  2. Substitute this value and the acceleration due to gravity (9.81 m/s²) into the equation for period
  3. Calculate the period using the simplified equation
• When comparing spring-mass and pendulum systems, note that both exhibit simple harmonic motion but have different restoring forces (spring force vs. gravity)
• The period of a spring-mass system is affected by changes in mass or spring constant, while the period of a pendulum is affected by changes in length or gravitational acceleration

Advanced Concepts

• Resonance occurs when an oscillating system is driven at its natural frequency, resulting in increased amplitude
• Damping is the gradual reduction in the amplitude of oscillations due to energy dissipation
• In real-world systems, damping and resonance can significantly affect the behavior of oscillating objects