Problem 38 - Thomas Calculus

Question

Moment of inertia of wire hoop A circular wire hoop of constant density \(\delta\) lies along the circle \(x^{2}+y^{2}=a^{2}\) in the \(x y\) -plane. Find the hoop's moment of inertia about the \(z\) -axis.

Step-by-Step Solution

Answer

The moment of inertia about the z-axis is \(I_z = 2\pi a^3 \delta\).

Step 1: Identify the Key Components

The wire hoop lies on the circle given by the equation \(x^2 + y^2 = a^2\) on the \(xy\)-plane. We are asked to find the moment of inertia about the \(z\)-axis. The line mass density is constant at \(\delta\).

Step 2: Use Definition of Moment of Inertia

The moment of inertia, \(I_z\), about the \(z\)-axis for a continuous mass distribution is defined as \(I_z = \int r^2 \, dm\), where \(r\) is the perpendicular distance from the \(z\)-axis to the element \(dm\).

Step 3: Parameterize the Circle

The circle can be parameterized using \(x = a\cos(\theta)\) and \(y = a\sin(\theta)\), where \(\theta\) is the angle around the circle. The distance \(r\) from the \(z\)-axis is \(\sqrt{x^2 + y^2} = a\). Since \(r = a\), we have a constant distance for all mass elements of the hoop.

Step 4: Express Mass Element in Terms of Parameter

The mass element \(dm\) can be expressed as \(dm = \delta \, ds\), where \(ds\) is the arc length differential. The arc length element in terms of \(\theta\) is \(ds = a \, d\theta\). Thus, \(dm = \delta a \, d\theta\).

Step 5: Integrate to Find Moment of Inertia

We substitute into the inertia formula: \(I_z = \int r^2 \, dm\). Since \(r = a\), we have \(I_z = \int a^2 \, dm = \int_0^{2\pi} a^2 \, \delta a \, d\theta = a^3 \delta \int_0^{2\pi} \, d\theta = 2\pi a^3 \delta\).

Step 6: Final Expression

The moment of inertia of the wire hoop about the \(z\)-axis is \(I_z = 2\pi a^3 \delta\). This expression accounts for the entire constant density of the wire distributed along the circle of radius \(a\).

Key Concepts

Circular Hoop

A circular hoop is essentially a loop shaped like a circle. It is a key object in understanding rotational dynamics, especially when discussing the moment of inertia. Imagine a circular ring made from wire. It's typically defined in 2D on the Cartesian coordinate system, as you see often in physics or engineering.The circle for our hoop is defined by the equation \(x^2 + y^2 = a^2\), meaning every point \((x, y)\) on this circle is at a consistent distance, \(a\), from the center, which is located at the origin \((0,0)\). This distance \(a\) is the radius of the circle. A hoop is unique because it is hollow in the center, unlike a disc.In the problem, this hoop lies on the \(xy\)-plane, which is a flat 2D surface. When you analyze a circular hoop for its moment of inertia about an axis like the \(z\)-axis, it is pivotal as it has direct implications on how the hoop rotates under various forces applied to it.

Constant Density

Density is about how much mass is contained in a given volume. For a wire hoop, however, we're interested in linear density, which tells us how much mass is present along its length. The problem describes the hoop as having a constant density, denoted by \(\delta\). Constant density is simplified as each segment of the hoop has the same mass per unit length, making calculations straightforward. It means we do not need to worry about any variations in density along the curve of the hoop. This simplification is quite handy when calculating physical properties like moment of inertia.Knowing the density is constant helps to express the mass element, \(dm\), for a small element of the hoop. For hoops or wires, you'll often see this as \(dm = \delta \, ds\), where \(ds\) is the arc length differential. This means you can multiply the density by a small length of the hoop to get a small mass, which greatly helps in integral calculations.

Parameterization of Circle

Parameterization is a technique used to express a geometric shape using parameters. For a circle, this usually involves using trigonometric functions because they naturally represent circular motion or shapes. In our problem, the circle is parameterized by

• \(x = a \cos(\theta)\)
• \(y = a \sin(\theta)\)

where \(\theta\) is the parameter, typically the angle in radians.Parameterization is incredibly useful as it transforms a 2D problem into a 1D approach, using \(\theta\) to describe every point on the circle. It provides an elegant way to "walk" around the circle by varying \(\theta\) from \(0\) to \(2\pi\), which covers the whole circle once.By parameterizing, you simplify integration and other calculations involving the circle, as it lets you express elements like the radius or length differentials in terms of a single variable, which is crucial for solving for the moment of inertia in this scenario.