Pendulum Calculator

The Physics of Pendulum Oscillations

A pendulum is a simple mechanical system consisting of a mass suspended from a pivot point so that it can swing freely. When displaced from its resting position, gravity acts as a restoring force, pulling it back towards the equilibrium point, causing it to oscillate back and forth. Pendulums have historical significance in timekeeping (clocks), seismology, and the measurement of local gravity.

Simple Pendulum Calculations

A simple pendulum is idealized as a point mass (bob) suspended from a massless, rigid cord of length $L$. Assuming small swing angles (typically below 15 degrees), the system can be modeled using simple harmonic motion equations:

1. Pendulum Period (T)

The period is the time required for the bob to complete one full back-and-forth oscillation:

$$T = 2\pi \sqrt{\frac{L}{g}}$$

Where $L$ is the length of the cord and $g$ is the local acceleration due to gravity. Note that the mass of the bob does not affect the period.

2. Oscillation Frequency (f)

The frequency is the number of complete cycles per second, measured in Hertz (Hz):

$$f = \frac{1}{T} = \frac{1}{2\pi} \sqrt{\frac{g}{L}}$$

3. Finding Pendulum Length (L)

To design a pendulum with a specific target period, the required cord length is computed by rearranging the period formula:

$$L = g \cdot \left(\frac{T}{2\pi}\right)^2$$

Physical (Compound) Pendulum Calculations

Real-world pendulums are not point masses suspended by massless strings. A physical pendulum consists of a rigid body of total mass $m$ pivoting about a fixed point. Its period is determined by its moment of inertia $I$ and the distance $d$ from the pivot to the center of mass:

$$T = 2\pi \sqrt{\frac{I}{m \cdot g \cdot d}}$$

Where:

• $I$ is the moment of inertia about the pivot axis ($kg \cdot m^2$)
• $m$ is the total mass of the object ($kg$)
• $d$ is the distance from the pivot to the center of mass ($m$)
• $g$ is the local gravity ($m/s^2$)