Physical Pendulum Calculator: Period, Length, Mass, Moment of Inertia

What is a Torsional Pendulum?

A torsional pendulum is a mechanical system consisting of an object suspended by a wire or rod that provides a restoring torque when twisted. Unlike a simple pendulum that swings back and forth, a torsional pendulum oscillates by twisting back and forth. This system is crucial in various scientific and engineering applications, including the measurement of material properties and the study of rotational motion.

Formula

The period of a torsional pendulum is given by the following equation:

\[ T = 2\pi \sqrt{\\frac{I}{\kappa}} \]

Where:

• \(T\) is the period of oscillation (time for one complete twist and return), measured in seconds (s)
• \(I\) is the moment of inertia of the suspended object, measured in kg·m²
• \(\kappa\) (kappa) is the torsion constant of the wire or rod, measured in N·m/rad
• \(\pi\) is the mathematical constant pi, approximately 3.14159

Calculation Steps

Let's calculate the period of a torsional pendulum with a moment of inertia of 0.1 kg·m² and a torsion constant of 0.05 N·m/rad:

1. Identify the known values:
   • Moment of inertia (I) = 0.1 kg·m²
   • Torsion constant (\(\kappa\)) = 0.05 N·m/rad
2. Apply the period formula: \[ T = 2\pi \sqrt{\frac{I}{\kappa}} \]
3. Substitute the known values: \[ T = 2\pi \sqrt{\frac{0.1 \text{ kg·m²}}{0.05 \text{ N·m/rad}}} \]
4. Perform the calculation: \[ T = 2\pi \sqrt{2} \approx 8.886 \text{ s} \]

Example and Visual Representation

Let's visualize a torsional pendulum with our calculated period:

This visual representation shows:

• The support structure at the top
• The torsion wire or rod suspending the disk
• The disk (representing the object with moment of inertia I)
• The angle of twist θ
• The period of oscillation (approximately 8.886 seconds)