A simple pendulum is a fascinating example of harmonic motion. It consists of a bob suspended by a string, swinging back and forth due to gravity. The pendulum's period depends on its length and local gravity, not on the bob's mass or swing amplitude.

Understanding pendulums helps us grasp key concepts in oscillatory motion. We can calculate a pendulum's period using a simple formula, and explore how changing its length affects its swing time. This knowledge has practical applications in timekeeping and beyond.

Simple Pendulum Mechanics

Pendulum as harmonic oscillator

• Simple pendulum consists of a bob (point mass) suspended by a massless string from a fixed point
• When displaced from equilibrium position, pendulum experiences a restoring force due to gravity
• For small angles of displacement (less than 15°), restoring force is approximately proportional to displacement
  • Linear relationship between force and displacement is characteristic of a harmonic oscillator
• Pendulum oscillates back and forth with a constant period, as long as angle remains small
  • Period is independent of bob mass and oscillation amplitude (maximum displacement from equilibrium)
• Motion of pendulum described by equation: $\frac{d^2\theta}{dt^2} + \frac{g}{L}\theta = 0$
  • $\theta$ is angular displacement
  • $L$ is pendulum length
  • $g$ is acceleration due to gravity (9.81 m/s^2 on Earth)

Calculation of pendulum period

• Period ($T$) of simple pendulum is time taken to complete one full oscillation
• Period depends on pendulum length ($L$) and acceleration due to gravity ($g$)
• Formula for period of simple pendulum: $T = 2\pi\sqrt{\frac{L}{g}}$
  • $\pi$ is mathematical constant pi (3.14159)
• To calculate period:
  1. Measure length ($L$) of pendulum from suspension point to center of bob
  2. Determine acceleration due to gravity ($g$) at pendulum location
  3. Substitute $L$ and $g$ values into formula and calculate period ($T$)
• Example: pendulum with $L = 1$ m on Earth ($g = 9.81$ m/s^2) has period $T = 2\pi\sqrt{\frac{1}{9.81}} \approx 2.01$ s
• Angular frequency ($\omega$) of the pendulum is related to the period: $\omega = \frac{2\pi}{T}$

Length vs period in pendulums

• Period of simple pendulum is directly proportional to square root of its length
  • Increasing pendulum length increases period
  • Decreasing pendulum length decreases period
• Relationship between period ($T$) and length ($L$) expressed as: $T \propto \sqrt{L}$
• If pendulum length is doubled, period increases by factor of $\sqrt{2}$ (1.41)
  • Example: 1 m pendulum with 2 s period, 2 m pendulum has 2.83 s period ($2 \times \sqrt{2}$)
• If pendulum length is halved, period decreases by factor of $\sqrt{2}$ (0.71)
  • Example: 1 m pendulum with 2 s period, 0.5 m pendulum has 1.41 s period ($2 \times \frac{1}{\sqrt{2}}$)
• Pendulum length can be adjusted to achieve desired period for applications like clocks

Energy and forces in a pendulum

• As the pendulum swings, energy is continuously converted between potential and kinetic forms
• At the highest points of swing, the pendulum has maximum potential energy and minimum kinetic energy
• At the lowest point (equilibrium position), the pendulum has maximum kinetic energy and minimum potential energy
• The tension in the string varies throughout the swing, being greatest at the bottom of the swing
• In real-world pendulums, damping effects cause a gradual decrease in amplitude over time due to air resistance and friction