⏱️ Simple Pendulum Calculator
Calculate pendulum period
How to Use This Calculator
Enter Length
Input the length (L) of the pendulum in meters (m). This is the distance from the pivot point to the center of mass of the bob. For example, a 1-meter pendulum has L = 1 m. The length must be greater than zero.
Enter Gravity (Optional)
Input the gravitational acceleration (g) in m/s². Default is 9.81 m/s² (Earth's surface). Use different values for other planets or locations: Moon (1.62 m/s²), Mars (3.71 m/s²), Jupiter (24.79 m/s²). Gravity must be greater than zero.
Calculate and Review
Click the "Calculate" button to compute the period (time for one complete swing), frequency (swings per second), and angular frequency. The period is independent of mass and amplitude (for small angles), depending only on length and gravity.
Formula
T = 2π√(L/g)
f = 1/T, ω = 2πf
Where:
• T = Period (s)
• L = Length of pendulum (m)
• g = Gravitational acceleration (m/s²)
• f = Frequency (Hz)
• ω = Angular frequency (rad/s)
• π ≈ 3.14159
Example 1: Standard Pendulum
A simple pendulum has length 1 m. Calculate its period on Earth (g = 9.81 m/s²).
Given:
• Length (L) = 1 m
• Gravity (g) = 9.81 m/s²
Solution:
T = 2π√(L/g)
T = 2π√(1 / 9.81)
T = 2π√(0.102) ≈ 2π × 0.319
T ≈ 2.00 s
f = 1/T = 1/2.00 = 0.5 Hz
Example 2: Pendulum on Moon
The same 1 m pendulum is taken to the Moon where g = 1.62 m/s². What is its period?
Given:
• Length (L) = 1 m
• Gravity (g) = 1.62 m/s² (Moon)
Solution:
T = 2π√(1 / 1.62)
T = 2π√(0.617) ≈ 2π × 0.786
T ≈ 4.94 s
The period is longer on the Moon due to weaker gravity, making the pendulum swing slower.
Frequently Asked Questions
What is a simple pendulum?
A simple pendulum is an idealized model consisting of a point mass (bob) suspended from a massless, inextensible string or rod. When displaced from its equilibrium position and released, it oscillates back and forth under gravity. The simple pendulum model assumes small angle oscillations and ignores air resistance and friction.
Why doesn't the period depend on mass?
The period formula T = 2π√(L/g) shows that period depends only on length and gravity, not mass. This is because both the restoring force (gravity) and inertia (mass) are proportional to mass, so mass cancels out. Heavier and lighter pendulums of the same length have the same period, which is a remarkable property of pendulums.
What is the small angle approximation?
The formula T = 2π√(L/g) is exact only for small angles (typically less than 15°). For larger angles, the period increases slightly. The approximation uses sin(θ) ≈ θ for small angles. For angles up to about 20°, the error is less than 1%. For larger amplitudes, more complex formulas are needed, but this calculator works well for most practical applications.
How does length affect the period?
Period is proportional to the square root of length: T ∝ √L. Doubling the length increases the period by √2 ≈ 1.41 times. Quadrupling the length doubles the period. Longer pendulums swing slower because they have longer paths to travel. This is why grandfather clocks use long pendulums for accurate timekeeping.
How does gravity affect the period?
Period is inversely proportional to the square root of gravity: T ∝ 1/√g. Stronger gravity means shorter period (faster swinging). On Jupiter (high gravity), a pendulum swings faster than on Earth. On the Moon (low gravity), it swings slower. This is why pendulum clocks run differently at different elevations or locations.
Where are simple pendulums used in real-world applications?
Simple pendulums are used in: pendulum clocks and timekeeping devices (accurate time measurement), seismometers (detecting earthquakes), metronomes (musical timing), physics demonstrations (energy conservation), amusement park rides, and educational purposes (teaching oscillatory motion and energy conversion).
About Simple Pendulum Calculator
The simple pendulum calculator computes the period and frequency of a simple pendulum using T = 2π√(L/g) where L is length and g is gravity. A simple pendulum is an idealized model of a mass suspended from a pivot, oscillating under gravity.
This calculator is essential for students studying oscillatory motion, engineers designing timing devices, and anyone analyzing pendulum behavior. The simple pendulum demonstrates important physics principles: the period is independent of mass, depends only on length and gravity, and provides a classic example of harmonic motion and energy conservation.