15.9:
Torsional Pendulum
A torsional pendulum is a rigid body, like a top, suspended from a string that is assumed to be massless — an assumption that is valid if the rigid body's mass is much larger than the string's mass.
When the top is twisted about the string's axis and released, it oscillates between two angles. The restoring torque is due to shearing of the string.
If the angular displacement is small, the restoring torque can be modeled as proportional to the angular displacement. The proportionality constant is called the string's torsion constant.
The torque can also be written in terms of the rigid body's moment of inertia and angular acceleration.
The two expressions give an equation for simple harmonic motion, with the independent variable being the angle of oscillation, the mass replaced by the moment of inertia, and the force constant replaced by the string's torsion constant.
The angular frequency of the oscillation is then determined, and from it, the time period is derived.
15.9:
Torsional Pendulum
A torsional pendulum involves the oscillation of a rigid body in which the restoring force is provided by the torsion in the string from which the rigid body is suspended. Ideally, the string should be massless; practically, its mass is much smaller than the rigid body's mass and is neglected.
As long as the rigid body's angular displacement is small, its oscillation can be modeled as a linear angular oscillation. The amplitude of the oscillation is an angle. The role of mass is played by the rigid body's moment of inertia about the point of suspension and the axis passing perpendicular to it.
Using the relationship between torque and angular acceleration, the equation is seen to mimic the equation of the simple harmonic motion of a simple pendulum. This observation allows for the easy determination of the angular frequency of the angular oscillation and its time period.
Suggested Reading
- Young, H.D and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson: section 14.4; page 451.
- OpenStax. (2019). University Physics Vol. 1. [Web version]. Retrieved from https://openstax.org/books/university-physics-volume-1/pages/1-introduction: section 15.4; pages 769–770.