Introduction
A seconds pendulum is defined as a simple pendulum whose time period is exactly 2 seconds. In this question, we calculate its length, angular frequency, and frequency. Additionally, we differentiate between a simple and compound pendulum.
Given
- Time period (T) = 2 s
- Acceleration due to gravity (g) = 9.8 m/s²
Step 1: Calculate Length of the Pendulum
The time period of a simple pendulum is given by:
T = 2π√(L/g)
Rearranging to find L:
L = (T² × g) / (4π²)
L = (2² × 9.8) / (4 × π²) = (4 × 9.8) / (39.478)
L ≈ 39.2 / 39.478 ≈ 0.993 m
Step 2: Calculate Angular Frequency (ω)
ω = 2π / T = 2π / 2 = π rad/s ≈ 3.14 rad/s
Step 3: Calculate Frequency (f)
f = 1 / T = 1 / 2 = 0.5 Hz
Step 4: Difference Between Simple and Compound Pendulum
Simple Pendulum
- Consists of a point mass suspended by a weightless, inextensible string.
- Theoretical concept—idealized system.
- Oscillates in a single vertical plane.
- Time period: T = 2π√(L/g)
Compound Pendulum
- A real, rigid body that swings about a horizontal axis.
- Has mass distributed along its length.
- Time period depends on moment of inertia: T = 2π√(I / mgh)
- Used in real-world applications like pendulum clocks.
Final Answers
- Length: 0.993 m
- Angular frequency (ω): 3.14 rad/s
- Frequency (f): 0.5 Hz
- Difference: A simple pendulum is an idealized model with a point mass, while a compound pendulum is a real body with extended mass.
Conclusion
The calculations demonstrate key physical quantities for a seconds pendulum, and the conceptual comparison highlights the practical differences between simple and compound pendulums in oscillatory motion studies.