Problem 42 - University Physics with Modern Physics

Question

A thin metal disk with mass \(2.00 \times 10^{-3} \mathrm{kg}\) and radius 2.20 \(\mathrm{cm}\) is attached at its center to a long fiber (Fig. E14.42). The disk, when twisted and released, oscillates with a period of 1.00 s. Find the torsion constant of the fiber.

Step-by-Step Solution

Answer

The torsion constant is approximately \(2.14 \times 10^{-6} \mathrm{N}\cdot\mathrm{m}/\mathrm{rad}\).

Step 1: Understand the Physical System

We have a metal disk attached to a fiber which acts like a torsion pendulum. When twisted, it oscillates with a known period. Our task is to find the torsion constant of the fiber, which quantifies its stiffness or resistance to twisting.

Step 2: Identify the Key Equation

For a torsion pendulum, the period of oscillation \(T\) is related to the moment of inertia \(I\) and the torsion constant \(\kappa\) by the equation:\[T = 2\pi \sqrt{\frac{I}{\kappa}}.\] We rearrange this equation to solve for \(\kappa\): \[\kappa = \frac{4\pi^2 I}{T^2}.\]

Step 3: Calculate the Moment of Inertia

The moment of inertia \(I\) for a thin disk about an axis through its center perpendicular to its plane is given by:\[I = \frac{1}{2} m r^2,\] where \(m = 2.00 \times 10^{-3} \mathrm{kg}\) is the mass and \(r = 0.022 \mathrm{m}\) is the radius. Substituting these values:\[I = \frac{1}{2} \times 2.00 \times 10^{-3} \times (0.022)^2 \approx 5.41 \times 10^{-8} \mathrm{kg}\cdot\mathrm{m}^2.\]

Step 4: Compute the Torsion Constant

Using the calculated moment of inertia and given period \(T = 1.00 \mathrm{s}\), substitute these into the equation for \(\kappa\):\[\kappa = \frac{4\pi^2 \times 5.41 \times 10^{-8}}{(1.00)^2}.\]Calculating this gives:\[\kappa \approx 2.14 \times 10^{-6} \mathrm{N}\cdot\mathrm{m}/\mathrm{rad}.\]

Key Concepts

Torsion Constant

In the context of a torsion pendulum, the torsion constant is an important concept to grasp. It represents the stiffness or resistance a material has to twisting. Imagine twisting a fiber or a rubber band; the torsion constant measures how quickly it tries to return to its original state.

The greater the torsion constant, the stiffer the fiber and the harder it is to twist. For a pendulum, this impacts how quickly or slowly it will oscillate when released.

• A high torsion constant indicates less flexibility.
• A low torsion constant means the material bends or twists more easily.

Understanding the torsion constant helps in designing materials and systems where controlled twisting is essential. You calculate the torsion constant using the formula:\[\kappa = \frac{4\pi^2 I}{T^2},\]where \(I\) is the moment of inertia and \(T\) is the period of oscillation. This formula shows the relationship between how the mass distribution (moment of inertia) and the period of oscillation contribute to the torsion constant.

Moment of Inertia

Moment of inertia is a fundamental concept in physics, especially in rotational dynamics. It tells us how the mass of an object is distributed with respect to a given axis. Think of it as the rotational equivalent of mass in linear motion. For the disk in this example, its moment of inertia determines how easy or hard it is for the disk to start rotating.

More mass farther from the axis means a larger moment of inertia, making the object harder to spin. For a thin disk like the one in our problem, the formula to calculate moment of inertia is:\[I = \frac{1}{2} m r^2,\]where \(m\) is the mass of the disk and \(r\) is the radius.

• Use this formula to find \(I\) by plugging in the known values.
• In our exercise, \(m = 2.00 \times 10^{-3} \) kg and \(r = 0.022 \) m, resulting in \(I \approx 5.41 \times 10^{-8} \) kg·m².

The concept is crucial in understanding how various forces and torques affect rotating systems.

Oscillation Period

The oscillation period, represented as \(T\), is the time it takes for one complete cycle of the pendulum's motion. In other words, it's the time from one peak of the swing to the next. For a torsion pendulum, the period depends on both the moment of inertia and the torsion constant.

Calculating the period helps to understand how changes in the moment of inertia or torsion constant alter the motion. The equation used is:\[T = 2\pi \sqrt{\frac{I}{\kappa}},\]allowing us to see how these variables play off each other.

• If the torsion constant increases, making the fiber stiffer, the period decreases. This means faster oscillations.
• On the other hand, increasing the moment of inertia increases the period, leading to slower oscillations.

The oscillation period provides critical insight into the dynamics of a torsion pendulum and helps in designing systems with precise oscillatory behaviors.