What is Pendulum Length?
Pendulum length is the distance between the pivot point and the center of mass of the pendulum bob. This length is a crucial factor that determines the oscillation characteristics of a pendulum, including how long it takes to complete one full swing (the period).
Understanding pendulum length is fundamental to timekeeping, understanding harmonic motion, and even some amusement park attractions.
How to Calculate Pendulum Length
The formula for calculating pendulum length is:
[\text{Pendulum Length} = \frac{g}{4 \cdot \pi^2 \cdot f^2}]
Where:
- Pendulum Length (L) is in meters
- g is the acceleration due to gravity in m/s² (typically 9.81 m/s² on Earth)
- f is the frequency in hertz (Hz)
Calculation Example
Let's calculate the pendulum length for a frequency of 1.5 Hz on Earth.
Given:
- Acceleration due to gravity (g) = 9.81 m/s²
- Frequency (f) = 1.5 Hz
Step 1: Apply the formula:
[\text{Pendulum Length} = \frac{9.81}{4 \cdot \pi^2 \cdot (1.5)^2}]
Step 2: Calculate the denominator:
[4 \cdot \pi^2 \cdot 2.25 = 4 \cdot 9.87 \cdot 2.25 = 88.83]
Step 3: Divide:
[\text{Pendulum Length} = \frac{9.81}{88.83} \approx 0.11 \text{ m}]
The pendulum length is approximately 0.11 meters (11 cm).
Common Pendulum Lengths
| Frequency (Hz) | Period (s) | Length (m) |
|---|---|---|
| 0.5 | 2.0 | 0.994 |
| 1.0 | 1.0 | 0.248 |
| 1.5 | 0.67 | 0.110 |
| 2.0 | 0.5 | 0.062 |