Overview
This lesson covers the simple pendulum, including definitions of period and frequency, their formulas, and how pendulum behavior depends on length and gravitational acceleration.
Complete Swing Definition
- A complete swing is motion from point A to C and back to A
- One complete cycle is used to define period and frequency
- Understanding this is essential for calculations
Period and Frequency
- Period (T): time for one complete swing; measured in seconds
- Calculate period: T = time / number of cycles
- Frequency (f): reciprocal of period; measured in Hz (s⁻¹)
- Calculate frequency: f = number of cycles / time
- Period: time per cycle; frequency: cycles per second
- Period and frequency are inversely related
Period Formula and Dependencies
- Period formula: T = 2π√(L/g)
- L is pendulum length; g is gravitational acceleration
- Earth's g = 9.8 m/s²
- Period is independent of mass (mass of bob does not affect period)
- Increasing L increases T (direct relationship)
- Increasing g decreases T (inverse relationship)
- Longer strings take more time to complete a swing
Frequency Formula
- Frequency formula: f = (1/2π)√(g/L)
- Increasing L decreases frequency
- Increasing g increases frequency
- Frequency and period are inversely related
Key Formulas and Relationships
| Variable | Formula | Units | Relationship |
|---|
| Period (T) | T = 2π√(L/g) | seconds (s) | T ∝ √L, T ∝ 1/√g |
| Frequency (f) | f = 1/T or f = (1/2π)√(g/L) | hertz (Hz) or s⁻¹ | f ∝ √g, f ∝ 1/√L |
| Length (L) | L = gT²/(4π²) | meters (m) | Derived from period formula |
| Gravity (g) | g = 4π²L/T² | m/s² | Derived from period formula |
Example 1: Period and Frequency on Earth and Moon
- Pendulum length: 70 cm = 0.7 m
- On Earth (g = 9.8 m/s²): T = 1.679 s, f = 0.5956 Hz
- On Moon (g = 1.6 m/s²): T = 4.16 s, f = 0.24 Hz
- Lower gravity increases period and decreases frequency
Example 2: Calculating Period from Cycles
- Pendulum makes 42 cycles in 63 seconds
- Period: T = 63 s / 42 cycles = 1.5 s
- Frequency: f = 1/1.5 = 0.67 Hz
- Length on Earth: L = (9.8 × 1.5²)/(4π²) = 0.5585 m
Example 3: Determining Unknown Planet's Gravity
- Pendulum makes 28 swings in 45 seconds
- Period: T = 45/28 = 1.607 s per cycle
- Pendulum length: 80 cm = 0.80 m
- Gravitational acceleration: g = 4π²L/T² = 12.2 m/s²
- This equals 1.24 g's (1.24 times Earth's gravity)
Example 4: Grandfather Clock Pendulum
- One second between tick and tock represents half a cycle
- Complete period is 2 seconds (tick to tock and back)
- Length: L = (9.8 × 2²)/(4π²) = 0.993 m
- Common mistake: confusing half-cycle time with full period
Example 5: Period on Different Planet
- Initial period on Earth (T₁): 1.7 s with g₁ = 9.8 m/s²
- New planet has g₂ = 15 m/s²
- Formula: T₂ = T₁√(g₁/g₂)
- New period: T₂ = 1.7√(9.8/15) = 1.37 s
- Higher gravity reduces period
Key Concepts
- Period is independent of mass; changing bob mass does not change period
- Mass of string is assumed negligible relative to bob
- Simple pendulum can determine unknown planet's gravitational acceleration
- Need only length and period (or time for multiple cycles) to calculate g
- Period and frequency always move inversely: if one increases, other decreases