Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

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.

Step-by-Step Solution

Answer

The moment of the lamina with respect to any line through the origin is zero.

Step 1: Set Up the Problem

We are given a line represented by the equation \(L: ax + by = 0\), going through the origin. A lamina \(S\) has its center of mass at the origin, \((0, 0)\). We need to find the expression for the signed distance \(d\) from a point \((x, y)\) to this line \(L\).

Step 2: Derive the Expression for Distance

The general formula for the signed distance from a point \((x_0, y_0)\) to a line \(ax + by + c = 0\) is \(d = \frac{ax_0 + by_0 + c}{\sqrt{a^2 + b^2}}\). In our case, \(c = 0\), so the expression becomes \(d = \frac{ax + by}{\sqrt{a^2 + b^2}}\). This is the signed distance of the point \((x, y)\) from the line \(L\).

Step 3: Define the Moment of the Lamina with respect to the Line

The moment \(M\) of the lamina \(S\) with respect to line \(L\) is defined as the integral of the signed distance over the lamina: \(M = \int_S d \, dm\), where \(dm\) represents an infinitesimal mass element of the lamina.

Step 4: Apply the Condition of the Center of Mass

Since the center of mass of the lamina is at the origin \((0, 0)\), we know by symmetry that all contributions of the form \(ax_i + by_i\) over the entire lamina will sum to zero. Therefore, the moment \(M\) becomes zero because all signed distances, when integrated over the lamina, cancel each other out.

Step 5: Conclude the Moment Expression

The integral expression for the moment becomes zero due to symmetry around the origin. Therefore, the moment \(M = \int_S d \, dm = 0\), indicating that the lamina will balance on any line through its center of mass.

Key Concepts

Lamina

A lamina is a two-dimensional, flat object with negligible thickness. It can be thought of as a thin sheet or plate. In physics, we often study laminae to understand how mass is distributed across a plane. When considering the center of mass, this concept helps to determine balance and stability. For any lamina, the center of mass, if symmetrically distributed, gives insight into

• where the mass effectively acts
• how the object behaves under external forces
• its tendency to maintain orientation

These characteristics are crucial when analyzing how a lamina interacts with its environment, especially when subjected to gravitational forces.

Moment of a System

The moment of a system measures how much rotational force is present. In the case of a lamina, the moment with respect to a line is a way to evaluate how far the mass of the lamina shifts from this line. For a system, the moment would be calculated as:

• Moment = \ \( M = \int_S d \, dm \ \)
• where \ \( d \ \) is the signed distance from each infinitesimal mass element \ \( dm \ \) to the line

This concept proves useful in understanding the balance of the lamina. If the moment is zero when summing these distances, it indicates that the mass is evenly distributed around the line, thus achieving an equilibrium state.

Signed Distance

The signed distance from a point to a line helps us understand not just how far a point is from a line, but also on which side of the line the point lies. The formula for the signed distance is: \ \( d = \frac{ax + by}{\sqrt{a^2 + b^2}} \ \).This formula takes:

• coordinates \ \((x, y)\ \)
• coefficients \ \((a, b)\ \)

The square root in the denominator normalizes the expression, giving a direct distance measure, and the formula reveals the relationship between the position of a point and the line equation. This concept is foundational when assessing moments, as it helps define how elements of mass contribute to the overall balance of the lamina.

Symmetry in Physics

Symmetry is a powerful concept used to simplify problems. In physics, symmetry often suggests some form of invariance or consistency when an object is subject to certain operations, like rotations or reflections. When applied to the lamina:

• the symmetry about the center of mass indicates an equal distribution of mass on all sides of a point
• this implies that even if forces or moments are applied, they'll cancel out

With the exercise's context, the lamina's symmetry about the origin implies that any line through this point is a balancing line. This means any rotations or shifts occur equally in all directions, due to the equal spread of mass.