With the physical pendulum calculator, you can compute the frequency and period of a physical pendulum. The text below describes what exactly a physical pendulum is and how the moment of inertia affects its oscillations. You will also learn the equation describing the period of a physical pendulum and what the radius of its swings is.
On a side note, if you are looking to refresh basic concepts, we've got you covered with our frequency calculator and pendulum period (simple pendulum) calculator.
What is a physical pendulum?
A physical pendulum is any rigid object performing small oscillations around its equilibrium position. An example of a physical pendulum is a swing in a kids' playground or the swinging weight of a pendulum clock. Unlike an idealized simple pendulum — a point mass on a massless string — a physical pendulum has its mass distributed over its whole body, so its swing depends on its moment of inertia.
Oscillations are considered small when the maximal angle doesn't exceed about 15°. If it does, the physics of the pendulum becomes more complicated and the simple formula below no longer holds.
During this movement, kinetic energy is transformed into potential energy, and vice versa, but the sum of these energies is conserved (ignoring losses such as friction and air resistance).
The period of a physical pendulum
The period T of a physical pendulum is:
T = 2π √( I / (g · m · R) )
In this equation:
- I [kg·m²] — moment of inertia of the object about the pivot (see the moment of inertia calculator);
- g [m/s²] — acceleration due to gravity;
- m [kg] — mass of the object; and
- R [m] — distance from the center of mass to the pivot point.
The moment of inertia in the formula must be computed with respect to the pivot, not the center of mass. On the Earth's surface, the acceleration due to gravity is g = 9.81 m/s².
Radius of oscillation
The combination
L = I / (m · R)
that appears in the equation for the period of a physical pendulum is called the radius of oscillation L. It has a dimension of length. With it, the period takes the same form as the simple pendulum, T = 2π √(L / g). Two different pendulums with the same radius of oscillation have the same period.
Frequency and angular frequency
Once you know the period, the frequency (how many full swings happen per second) is simply its reciprocal, and the angular frequency follows directly:
- f = 1 / T — frequency in hertz (Hz);
- ω = 2π / T = 2π·f — angular frequency in radians per second (rad/s).
How to use the physical pendulum calculator
- Choose your unit system — metric (SI) or American (imperial).
- Enter the moment of inertia of the body about the pivot point.
- Enter the mass of the object.
- Enter the distance R from the center of mass to the pivot.
- Pick the gravitational acceleration — Earth, Moon, Mars, Jupiter, or a custom value.
- Click Calculate to instantly get the period, frequency, angular frequency, and radius of oscillation.
Worked example
Suppose a rigid body has a moment of inertia about its pivot of I = 0.5 kg·m², a mass of m = 2 kg, and the center of mass sits R = 0.3 m from the pivot, on Earth (g = 9.81 m/s²). The period is:
T = 2π √( 0.5 / (9.81 × 2 × 0.3) ) ≈ 1.832 s
The frequency is f = 1/T ≈ 0.546 Hz, the angular frequency is ω = 2π·f ≈ 3.43 rad/s, and the radius of oscillation is L = I / (m·R) = 0.5 / (2 × 0.3) ≈ 0.833 m.
FAQs
What is the difference between a simple and a physical pendulum?
A simple pendulum is an idealization: a point mass on a massless, inextensible string. A physical pendulum is any real rigid body that swings about a pivot, with its mass spread across the whole object. Its period depends on the body's moment of inertia about the pivot rather than just a single length.
Does the period of a physical pendulum depend on its mass?
Indirectly. Mass appears in the denominator, but the moment of inertia in the numerator usually grows with mass too, so the two effects partly cancel. What truly sets the period is the radius of oscillation L = I / (m·R) together with gravity g.
Why must the angle stay small?
The formula T = 2π √(I / (g·m·R)) is derived using the small-angle approximation sin(θ) ≈ θ. Beyond roughly 15°, this approximation breaks down, the restoring torque is no longer proportional to the angle, and the real period becomes slightly longer and amplitude-dependent.
What is the radius of oscillation used for?
It is the length of an equivalent simple pendulum that would have the same period. It lets you compare different physical pendulums directly: two bodies with the same L oscillate with the same period, regardless of their shape.