| Case | Mass (kg) | d (m) | Mode | Key input | g (m/s^2) | Estimated T (s) |
|---|---|---|---|---|---|---|
| Lab rig | 2.50 | 0.35 | Direct Ip | Ip = 0.028 kg*m^2 | 9.80665 | ~1.80 |
| COM known | 1.20 | 0.25 | Icm + axis | Icm = 0.006 kg*m^2 | 9.80665 | ~1.48 |
| Design check | 0.80 | 0.18 | Gyration | k = 0.09 m | 3.711 | ~1.82 |
The period of a physical pendulum for small oscillations is: T = 2*pi * sqrt( Ip / (m * g * d) )
- T is the period (seconds).
- Ip is the moment of inertia about the pivot (kg*m^2).
- m is the mass (kg).
- g is gravitational acceleration (m/s^2).
- d is distance from pivot to center of mass (m).
When you only know inertia about the center of mass, the parallel-axis rule applies: Ip = Icm + m*d^2. If you know the radius of gyration, use Icm = m*k^2.
- Select an inertia mode that matches your data.
- Enter mass and pivot-to-center distance with units.
- Choose a gravity preset or type a custom value.
- Provide the required inertia input for your selected mode.
- Press Calculate to see period, frequency, and w.
- Use the download buttons to export your result.
Physical Pendulum Period Guide
Physical pendulum overview
A physical pendulum is any rigid body that swings about a fixed pivot. Its mass is distributed through the body, so the swing rate depends on geometry as well as weight. For small angles, the motion is close to sinusoidal and the period stays nearly constant.
What this calculator returns
Enter your inputs and the tool reports period T in seconds, frequency f in hertz, and angular frequency w in rad/s. These three values are linked: f = 1/T and w = 2*pi*f. Use them for timing, vibration checks, and control tuning.
Period equation used
The small-angle period is T = 2*pi*sqrt(Ip /(m*g*d)). Ip is the moment of inertia about the pivot axis, m is total mass, g is gravitational acceleration, and d is the pivot-to-center distance. Larger Ip increases T, while larger g or d reduces T.
Flexible unit handling
Mass can be typed in kilograms, grams, or pounds. Distance accepts meters, centimeters, millimeters, inches, and feet. Inertia supports kg*m^2 and g*cm^2, then converts everything to SI units before computing. This lets you mix shop measurements with lab data safely.
Three inertia entry modes
Use Direct Ip if you already know inertia about the pivot. Use Icm + parallel axis if you know inertia about the center of mass; the calculator applies Ip = Icm + m*d^2. Use Radius of gyration if you know k, with Icm = m*k^2.
Gravity presets with real values
Select Earth (9.80665 m/s^2), Moon (1.62), Mars (3.711), Jupiter (24.79), or the Sun (274). A custom option supports test rigs, drop towers, or simulations where g differs from standard environments.
Example numbers you can verify
Try m = 2.50 kg, d = 0.35 m, and Ip = 0.028 kg*m^2 on Earth. The result is about T = 1.80 s, f = 0.56 Hz, and w = 3.49 rad/s. Save the output as CSV for spreadsheets or PDF for reports and lab notebooks.
FAQs
1) When is the small-angle model accurate?
It is most accurate for small swings, commonly under 10–15 degrees. At larger angles the period increases slightly. For high precision, measure the period experimentally or use a large-angle correction model.
2) What if I only know the shape and dimensions?
Compute Icm from standard inertia formulas or CAD. Then choose the Icm + parallel axis mode, enter d and m, and the calculator will form Ip automatically using Ip = Icm + m*d^2.
3) Why does increasing d sometimes shorten the period?
Restoring torque grows with m*g*d, so a larger d can reduce T when Ip is fixed. But moving the pivot usually also increases Ip, which can counteract the change.
4) What is the radius of gyration k?
k is defined by Icm = m*k^2. It summarizes how far mass is distributed from the center-of-mass axis. A smaller k means mass is concentrated nearer the axis and oscillations are faster.
5) Can I combine multiple parts or attachments?
Yes. Find each part’s mass, center offset, and inertia, then sum inertias about the same pivot using the parallel-axis idea. Enter the final Ip directly, or enter combined Icm and d.
6) Why show f and w as well as T?
Frequency f helps with sensor sampling, filtering, and resonance checks. Angular frequency w is common in vibration and control equations. Both are derived from T, so they update instantly when inputs change.