91Ó°ÊÓ

The torsion constant, often denoted by \( \kappa \), is a measure of a material's resistance to twisting. It is an important factor in understanding how a disc or any other object behaves when subjected to a twisting force. In the context of the exercise,
the torsion constant is calculated using the relationship between torque and angular displacement. This relationship is given by the formula:

• \( \tau = \kappa \theta \)

where \( \tau \) is the torque applied, \( \theta \) is the angle of twist in radians, and \( \kappa \) is the torsion constant.
By rearranging the formula, you find the torsion constant:

• \( \kappa = \frac{\tau}{\theta} \)

In our example, a torque of 0.0600 Nâ‹…m causes an angular displacement of 2.50 radians, resulting in a torsion constant of 0.0240 Nâ‹…m/rad. This tells us how much torque is required to twist the wire by one radian.

Angular frequency, denoted by \( \omega \), represents how quickly an object oscillates in an angular motion, such as in the case of a torsion pendulum. It is a critical parameter in oscillatory systems.
The formula for calculating angular frequency for torsional oscillations is:

• \( \omega = \sqrt{\frac{\kappa}{I}} \)

Here, \( \kappa \) is the torsion constant, and \( I \) is the moment of inertia. Angular frequency is expressed in radians per second. In the exercise's given situation,
with a torsion constant \( \kappa = 0.0240 \text{ N} \cdot \text{m/rad} \) and a moment of inertia \( I = 0.735 \text{ kg} \cdot \text{m}^2 \), the angular frequency is found to be approximately \( 0.1805 \text{ rad/s} \). This indicates the rate at which the disk oscillates back and forth when twisted slightly and released.

A torsion pendulum is a type of oscillatory system where an object hangs suspended by a wire or rod and rotates back and forth around its axis. This simple device is a perfect demonstration of angular motion and provides insights into the fundamentals of rotational dynamics.
In the exercise, the disk acts as the pendulum, suspended horizontally by a wire at its center. When the disk is twisted and then released, it undergoes oscillations.
These oscillations, driven by the restoring torque of the wire, can be described by the torsion pendulum's angular frequency, covered earlier.
The behavior of the torsion pendulum can be further analyzed by considering its moment of inertia and the torsion constant. These parameters help in predicting the motion's frequency and understanding how variations in these values influence the pendulum's oscillations. In essence, the torsion pendulum is an excellent model for studying harmonic motion in rotational systems.