15.10:
Physical Pendulum
Consider a watering can hanging from a hook. When displaced from its pivot point, it oscillates similar to a simple pendulum. The watering can is an example of a physical pendulum.
It can be modeled as its entire weight acting at its center of mass, which oscillates about the pivot point. Let the distance between the pivot point and the center of mass be L.
The oscillation is due to the restoring torque produced by gravity, which can be calculated. If the angle of oscillation is small, the torque is approximated.
It can also be written in terms of the pendulum's moment of inertia and angular acceleration.
The two expressions give an equation for simple harmonic motion, with the mass replaced by the moment of inertia and the force constant replaced by a product of three terms.
So, the angular frequency can be determined, and the time period can be derived.
For a simple pendulum, the moment of inertia gives the familiar expression for the time period.
15.10:
Physical Pendulum
When a rigid body is hanging freely from a fixed pivot point and is displaced, it oscillates similar to a simple pendulum and is known as a physical pendulum. The period and angular frequency of a physical pendulum are obtained by using the small-angle approximation and drawing parallels with a spring-mass system. The small-angle approximation (sinθ=θ) is valid up to about 14°.
When dealing with complicated systems, the mass moment of inertia is an important parameter, as it describes the mass distribution around the pivot point. The moment of inertia is a measure of an object's resistance to rotational motion. For a pendulum with a complicated mass distribution, calculating the moment of inertia can be a difficult and time-consuming task. However, by using the center of mass framework, we can simplify this calculation significantly. The complicated mass distribution of the body and resultant mass moment of inertia gets simplified into two terms: the moment of inertia about the pivot point and the distance between the pivot point and the center of mass. This simplification shows the remarkable power of using the center of mass framework.
The case of a real physical pendulum can be shown to reduce to the idealized simple pendulum by using the expression for a simple pendulum's moment of inertia.
Physical pendulums have useful applications. In extreme conditions, skyscrapers can sway up to two meters with a frequency of up to 20 Hz due to high winds or seismic activity. Several companies have developed physical pendulums that are placed on the top of skyscrapers. As the skyscraper sways to the right, the pendulum swings to the left, reducing the sway.
Physical pendulums can also be used to measure acceleration due to gravity.
Suggested Reading
- Young, H.D and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson: section 14.6; pages 455-456.
- 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 766-768.