Problem 43 - University Physics with Modern Physics

Question

You want to find the moment of inertia of a complicated machine part about an axis through its center of mass. You suspend it from a wire along this axis. The wire has a torsion constant of 0.450 \(\mathrm{N} \cdot \mathrm{m} / \mathrm{rad}\) . You twist the part a small amount about this axis and let it go, timing 125 oscillations in 265 s. What is the moment of inertia you want to find?

Step-by-Step Solution

Answer

The moment of inertia is approximately \(0.0514 \, \text{kg} \cdot \text{m}^2\).

Step 1: Understand the System's Setup

The machine part is twisted on a wire with a known torsion constant, which causes it to oscillate like a torsional pendulum. We can use the oscillation data to determine the system's moment of inertia.

Step 2: Recall the Torsional Oscillation Formula

The angular frequency of torsional oscillations is given by the formula: \( \omega = \sqrt{ \frac{\kappa}{I} } \), where \( \omega \) is the angular frequency, \( \kappa \) is the torsion constant, and \( I \) is the moment of inertia.

Step 3: Calculate the Angular Frequency

First, find the period \( T \) of one oscillation by dividing the total time by the number of oscillations: \( T = \frac{265 \, \text{s}}{125} \). Then, calculate the angular frequency: \( \omega = \frac{2\pi}{T} \).

Step 4: Solve for the Moment of Inertia

Rearrange the torsional oscillation formula to solve for the moment of inertia, \( I \):\[ I = \frac{\kappa}{\omega^2} \]Substitute \( \omega \) and \( \kappa = 0.450 \, \text{N} \cdot \text{m} / \text{rad} \) into the equation to calculate \( I \).

Step 5: Perform the Calculations

Calculate the period: \( T = \frac{265}{125} = 2.12 \, \text{s} \).Calculate angular frequency: \( \omega = \frac{2\pi}{2.12} \approx 2.96 \, \text{rad/s} \).Calculate the moment of inertia: \( I = \frac{0.450}{(2.96)^2} = \frac{0.450}{8.76} \approx 0.0514 \, \text{kg} \cdot \text{m}^2 \).

Key Concepts

Torsional Oscillation

Torsional oscillation describes the twisting motion of an object suspended by a wire or rod. Think of it like twisting a pen in your fingers — when you let go, the pen unwinds back to its original position. This back-and-forth motion resembles how a pendulum swings, but instead of moving side to side, the object spins back and forth around the axis. Commonly, this setup is used to determine physical properties of objects, such as their moment of inertia, which tells us how hard it is to change their rotation.

Torsion Constant

The torsion constant, denoted as \( \kappa \), is a measure of the rigidity of the wire or rod from which an object is suspended. It tells us how much torque, or twisting force, is needed to twist the object by a certain angle. For example, a torsion constant of 0.450 N·m/rad means it takes 0.450 Newton-meters to twist the object by one radian. A large torsion constant indicates a stiffer wire, which is harder to twist. Knowing \( \kappa \) helps in calculating how the object will oscillate, and it's a crucial parameter in determining the system's angular frequency and moment of inertia.

Angular Frequency

Angular frequency, represented by \( \omega \), measures how fast an object rotates in a torsional oscillation setup. It's similar to the frequency of a pendulum swinging but in terms of rotation. Calculated using the formula \( \omega = \frac{2\pi}{T} \), where \( T \) is the period of one complete oscillation. The period tells us how long it takes for the object to return to its starting position after one full twist. A high angular frequency means the object oscillates back and forth quickly, indicating a strong tension in the wire. Angular frequency is critical for calculating other properties like the moment of inertia.

Oscillation Formula

The oscillation formula for a torsional system is an essential tool for determining the moment of inertia \( I \) of an object. The formula \( \omega = \sqrt{ \frac{\kappa}{I} } \) links the angular frequency \( \omega \) with the torsion constant \( \kappa \) and the moment of inertia \( I \). By rearranging it to \( I = \frac{\kappa}{\omega^2} \), we can find \( I \) using known values of \( \kappa \) and \( \omega \). This relationship shows how the moment of inertia affects the system's oscillation. A larger \( I \) means the object is more resistant to changes in its rotational motion, making it oscillate more slowly for a given torsion constant.