91Ó°ÊÓ

The torsion constant, often denoted as \(\kappa\), is a fundamental property of a wire or rod involved in torsional oscillations. It measures how resistant the wire is to being twisted. In simple terms, it tells us the torque needed to twist the wire by a specific angle. This constant is crucial for understanding the stiffness of the wire when a force is applied tangentially.
The formula that relates torque and torsion constant is given by:

• \(\tau = \kappa \cdot \theta\)

where \(\tau\) is the torque (in Nm) and \(\theta\) is the angular displacement in radians.
To find \(\kappa\), you can rearrange the equation:

• \(\kappa = \frac{\tau}{\theta}\)

In this exercise, with a torque of \(0.5076 \, \text{Nm}\) and an angular displacement of \(0.0583 \, \text{radians}\), the torsion constant calculates to approximately \(8.70 \, \text{Nm/rad}\). This tells us the wire is relatively stiff, requiring substantial torque for a small angular twist.

The concept of moment of inertia, symbolized as \(I\), plays a vital role in torsional and rotational motion. It provides a measure of how much resistance an object exhibits to changes in its rotational motion. It's akin to mass in linear motion but applies to rotation.
For a solid disk, the moment of inertia can be computed using the formula:

• \(I = \frac{1}{2} M r^2\)

where \(M\) is the mass of the disk, and \(r\) is the radius. In our scenario:

• Mass, \(M = 6.50 \, \text{kg}\)
• Radius, \(r = 0.12 \, \text{m}\)
• Thus, \(I = \frac{1}{2} \times 6.50 \times (0.12)^2 = 0.0468 \, \text{kg} \cdot \text{m}^2\)

This moment of inertia is essential for predicting how the disk responds to applied torques and relates directly to the subsequent frequency of oscillations.

In oscillatory systems, specifically torsional oscillations, understanding frequency and period is key. The frequency \(f\) indicates how many cycles of oscillation occur per second and is given in hertz (Hz). Meanwhile, the period \(T\) is the time taken for one complete cycle of oscillation.
Frequency is calculated using the equation:

• \(f = \frac{1}{2\pi} \sqrt{\frac{\kappa}{I}}\)

where \(\kappa\) is the torsion constant and \(I\) the moment of inertia. For our disk:

• \(f \approx 2.16 \, \text{Hz}\)

On the other hand, the period \(T\) is simply the reciprocal of frequency:

• \(T = \frac{1}{f} \approx 0.463 \, \text{s}\)

These values illustrate how quickly the disk vibrates back and forth around its horizontal resting position.

The equation of motion for torsional oscillation describes how the angular position \(\theta(t)\) of the disk changes over time. It tells us the behavior of the oscillating system.
This equation typically takes the form:

• \(\theta(t) = \theta_0 \cos(\omega t + \phi)\)

where \(\theta_0\) is the initial angular displacement, \(\omega = 2\pi f\) is the angular frequency, and \(\phi\) represents the phase angle. Assuming the disk starts from rest:

• Phase angle \(\phi = 0\)
• Initial displacement \(\theta_0 = 0.0583 \, \text{radians}\)

Given our frequency calculation, \(\omega \approx 13.58 \, \text{rad/s}\), leading to:

• \(\theta(t) = 0.0583 \cos(13.58 t)\)

This equation encapsulates the dynamics of the disk's motion, oscillating harmonically about its equilibrium position.