⏱️ Physical Pendulum Calculator
Calculate physical pendulum period
How to Use This Calculator
Enter Moment of Inertia
Input the moment of inertia (I) in kg·m² about the pivot point. This measures the object's resistance to rotational motion. For regular shapes, use the mass moment of inertia calculator first, then apply the parallel axis theorem if needed.
Enter Mass
Input the mass (m) in kilograms (kg). This is the total mass of the pendulum object. All values must be positive.
Enter Distance to Center of Mass
Input the distance (d) in meters from the pivot point to the center of mass. This is the distance along which the gravitational force acts. For a uniform rod pivoted at one end, d = L/2 where L is the length.
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. All values must be positive.
Calculate and Review
Click the "Calculate" button to compute the period, frequency, and angular frequency. The period is the time for one complete oscillation, frequency is oscillations per second, and angular frequency is 2π times the frequency.
Formula
T = 2π√(I / (mgd))
f = 1/T, ω = 2πf
Where:
• T = Period (s)
• I = Moment of inertia about pivot (kg·m²)
• m = Mass (kg)
• g = Gravitational acceleration (m/s²)
• d = Distance from pivot to center of mass (m)
• f = Frequency (Hz)
• ω = Angular frequency (rad/s)
Example 1: Uniform Rod
A uniform rod of mass 1 kg and length 1 m is pivoted at one end. Moment of inertia about the pivot is I = mL²/3 = 0.333 kg·m². Calculate the period.
Given:
• Moment of inertia (I) = 0.333 kg·m²
• Mass (m) = 1 kg
• Distance to center (d) = 0.5 m (L/2)
• Gravity (g) = 9.81 m/s²
Solution:
T = 2π√(I / (mgd))
T = 2π√(0.333 / (1 × 9.81 × 0.5))
T = 2π√(0.333 / 4.905)
T = 2π√(0.0679) ≈ 1.64 s
T ≈ 1.64 s
Example 2: Physical Pendulum
A physical pendulum has I = 0.5 kg·m², m = 2 kg, d = 0.3 m. What is its frequency?
Given:
• Moment of inertia (I) = 0.5 kg·m²
• Mass (m) = 2 kg
• Distance (d) = 0.3 m
Solution:
T = 2π√(0.5 / (2 × 9.81 × 0.3))
T = 2π√(0.5 / 5.886) ≈ 1.83 s
f = 1/T = 1/1.83
f ≈ 0.546 Hz
Frequently Asked Questions
What is a physical pendulum?
A physical pendulum (also called a compound pendulum) is any rigid object that swings about a pivot point. Unlike a simple pendulum (point mass on a string), a physical pendulum accounts for the object's size, shape, and mass distribution. The moment of inertia and center of mass location determine its behavior.
What's the difference between a simple and physical pendulum?
A simple pendulum assumes all mass is concentrated at a point, giving T = 2π√(L/g). A physical pendulum accounts for the actual mass distribution and uses T = 2π√(I/(mgd)). Physical pendulums are more accurate for real objects with extended mass. The simple pendulum is a special case where I = mL² and d = L.
How do I find the moment of inertia for my object?
Use the mass moment of inertia calculator for common shapes. If the pivot isn't at the center of mass, apply the parallel axis theorem: I_pivot = I_cm + md², where I_cm is moment about center of mass and d is distance from center of mass to pivot.
What affects the period of a physical pendulum?
The period depends on: moment of inertia (larger I means longer period), mass (cancels out in the formula), distance to center of mass (smaller d means longer period), and gravity (stronger gravity means shorter period). Unlike simple pendulums, the period depends on mass distribution, not just length.
Can this calculator be used for any rotating object?
Yes, if the object swings under gravity about a pivot. The formula applies to any rigid body oscillating in a gravitational field, including: swinging doors, metronomes, balance wheels in clocks, and any object that rotates back and forth under gravity.
Where are physical pendulums used in real-world applications?
Physical pendulums are used in: pendulum clocks and timekeeping devices, seismometers (detecting earthquakes), metronomes (musical timing), swinging bridges and structures, analyzing sports equipment (golf clubs, bats), and understanding the dynamics of any rotating object under gravity.
About Physical Pendulum Calculator
The physical pendulum calculator computes the period and frequency of a physical pendulum using T = 2π√(I/(mgd)) where I is moment of inertia, m is mass, g is gravity, and d is distance to center of mass. A physical pendulum accounts for the actual mass distribution of the swinging object.
This calculator is essential for engineers designing oscillating systems, students studying rotational dynamics, and anyone analyzing the motion of rigid bodies under gravity. Understanding physical pendulums helps explain the behavior of real-world swinging objects from clocks to bridges.