Torsional Pendulum Calculator

Understanding Torsional Pendulum Calculator

A torsional pendulum oscillates by twisting rather than swinging. Examples include Cavendish experiment apparatus and balance wheels in watches.

How to use this calculator

1. Enter the object's moment of inertia, \(I\), in \(\mathrm{kg\cdot m^2}\).
2. Enter the torsion constant, \(\kappa\), in \(\mathrm{N\cdot m/rad}\).
3. Click calculate to get the period, \(T\).
4. Check the result, which is reported in seconds and tells you how long one complete oscillation takes.

The formula explained

The formula \(T = 2\pi\sqrt{\frac{I}{\kappa}}\) computes the period of a torsional pendulum. It shows that a larger \(I\) makes the motion slower, while a larger \(\kappa\) makes it faster.

• T = period of the torsional pendulum, in seconds
• I = moment of inertia of the object, in \(\mathrm{kg\cdot m^2}\)
• \(\kappa\) = torsion constant of the wire or rod, in \(\mathrm{N\cdot m/rad}\)
• \(\pi\) = the constant pi, about 3.14159

Step by step method

1. Identify the moment of inertia \(I\) for the rotating object.
2. Identify the torsion constant \(\kappa\) for the wire or rod.
3. Substitute both values into \(T = 2\pi\sqrt{\frac{I}{\kappa}}\), then evaluate the square root and multiply by \(2\pi\).

Worked example

Problem. A disk has a moment of inertia of \(0.020\,\mathrm{kg\cdot m^2}\) and is suspended by a wire with torsion constant \(0.50\,\mathrm{N\cdot m/rad}\). Find the period.

1. Substitute the values into the formula: \(T = 2\pi\sqrt{\frac{0.020}{0.50}}\).
2. Compute the ratio: \(\frac{0.020}{0.50} = 0.040\), so \(T = 2\pi\sqrt{0.040}\).
3. Take the square root and multiply: \(\sqrt{0.040} = 0.20\), so \(T = 2\pi(0.20) \approx 1.26\,\mathrm{s}\).

Tips and common mistakes

• Make sure \(I\) and \(\kappa\) use consistent SI units, or the period will be wrong.
• The formula gives the time for one full oscillation, not the frequency. If you need frequency, use \(f = \frac{1}{T}\).

Frequently asked questions