Question

Let \(S\) be a lamina in the \(x y\) -plane with center of mass at the origin, and let \(L\) be the line \(a x+b y=0\), which goes through the origin. Show that the (signed) distance \(d\) of a point \((x, y)\) from \(L\) is \(d=(a x+b y) / \sqrt{a^{2}+b^{2}}\), and use this to conclude that the moment of \(S\) with respect to \(L\) is \(0 .\) Note: This shows that a lamina will balance on any line through its center of mass.

Short answer

The distance to the line is \(\frac{ax + by}{\sqrt{a^2 + b^2}}\), resulting in a zero moment for any line through the center of mass.

Step-by-step solution

Identify the Line Equation

We have a line equation given by \(L: ax + by = 0\). This equation represents a line through the origin with direction vector \((a, b)\).

Formula for Distance from a Point to a Line

The formula to find the signed distance from a point \((x, y)\) to a line \(ax + by + c = 0\) is given by \(d = \frac{ax + by + c}{\sqrt{a^2 + b^2}}\). In our case, \(c = 0\), so the formula simplifies to \(d = \frac{ax + by}{\sqrt{a^2 + b^2}}\).

Conclude the Line's Distance Formula

Using the above formula, for a point \((x, y)\) on the line, the signed distance \(d\) from the line is \(d = \frac{ax + by}{\sqrt{a^2 + b^2}}\). This is the distance from the point \((x, y)\) to the line \(L\).

Moment of the Lamina with Respect to Line L

The moment of a lamina about a line is the integral of the distance of each mass element from the line, weighted by the mass. Since the center of mass is at the origin, symmetric about the origin, any mass element on one side of the line will have a corresponding element on the opposite side, canceling each other's contributions.

Conclusion about Moment Being Zero

Because the lamina is symmetric around its center of mass at the origin, and each point \((x, y)\) has an associated \((-x, -y)\), their moments about the line \(L\) cancel. Thus, the integral of the distance times mass contribution on each side of the line results in a total moment of zero.

Key concepts

Lamina

A lamina is a flat, two-dimensional surface with uniform thickness, often considered in physics and engineering for analyzing systems of mass and balance. It can be thought of as a weighted plane which distributes its mass over the entire surface that it occupies.
The utility of understanding a lamina comes when analyzing balance and equilibrium. When calculating the center of mass, it helps you determine how weight is distributed. For a symmetrical lamina centered on the origin, as in this exercise, the mass on one side fully balances the mass on the other. This symmetry means the center of mass is a pivotal point for calculations involving balance about any axis through it.
In practical applications, a lamina is crucial for studying how materials will behave when placed in specific orientations, predicting their stability, and creating systems that will balance under various forces.

Signed Distance Formula

The signed distance formula is a mathematical tool used to find how far a point is from a line in a two-dimensional coordinate system. It's especially useful when the line in question goes through the origin, simplifying calculations.
For a line described by the equation \( ax + by = 0 \), the formula for the signed distance \( d \) from a point \( (x, y) \) to this line is \( d = \frac{ax + by}{\sqrt{a^2 + b^2}} \). Here, the denominator \( \sqrt{a^2 + b^2} \) ensures that the distance is normalized according to the line's orientation, providing a consistent measure of perpendicular distance.
The "signed" part indicates that the position of the point relative to the line can be given a direction—positive or negative—depending on which side of the line the point falls. This directional aspect is useful in calculating the moments of objects like laminas, where the direction of offset contributes to the equilibrium analysis.

Moment of Inertia

The moment of inertia for a lamina is crucial in understanding how its mass is distributed in relation to an axis or a line. This exercise shows that for a lamina with a center of mass at the origin, the moment of inertia about any line through the origin is zero.
Why is this the case? The moment of inertia involves integrating the products of mass elements with their squared distances from the line—in this setup, from one side of each line segment to the opposite one. With perfect symmetry, the contributions from both sides completely cancel each other out.
This result, a moment of zero, strongly indicates balance. Intuitively, imagine placing a perfectly symmetrical piece of cardboard on a straight-edged rail. It stays balanced without tipping, thanks to this zero moment property.

Line Equation

Understanding line equations is foundational in geometry and algebra. A line equation typically takes the form \( ax + by = 0 \) when the line passes through the origin. This scenario simplifies because there is no constant \( c \), making operations like finding distances and calculating other attributes straightforward.
The line equation provides not just a boundary but also an essential tool for conversions such as polar to Cartesian and vice versa. In exercises like these, knowing the line equation allows one to employ various formulas like the signed distance or find intersections with other geometrical entities.
In practical contexts, comprehending the line equation enables designing efficient solutions in fields ranging from computer graphics to robotics, ensuring precise navigation and representation in physical or digital spaces.