- Find the balance point (center of mass) of straight objects and flat surfaces
- Utilize a shape’s symmetry to find the centroid, or geometric center, of flat objects
- Use Pappus’s theorem to calculate the volume of an object
Center of Mass and Moments
The Main Idea
- Center of Mass:
- The balancing point of an object or system
- Point where the total mass can be concentrated without changing the object’s behavior
- Moments:
-
- Measure of the rotational force applied to an object
- First moment with respect to origin: [latex]M = \sum_{i=1}^n m_i x_i[/latex]
-
- One-Dimensional Systems:
- Center of mass: [latex]\overline{x} = \frac{\sum_{i=1}^n m_i x_i}{\sum_{i=1}^n m_i}[/latex]
- Two-Dimensional Systems:
- Moments: [latex]M_x = \sum_{i=1}^n m_i y_i[/latex], [latex]M_y = \sum_{i=1}^n m_i x_i[/latex]
- Center of mass: [latex]\overline{x} = \frac{M_y}{m}[/latex], [latex]\overline{y} = \frac{M_x}{m}[/latex]
Problem-Solving Strategy
- Identify the system (point masses or continuous object)
- Choose an appropriate coordinate system
- Calculate the total mass of the system
- Compute moments with respect to axes
- Use formulas to find the center of mass coordinates
Find the center of mass for three point masses:
- [latex]m_1 = 2\text{ kg}[/latex] at [latex](-1, 3)[/latex]
- [latex]m_2 = 6\text{ kg}[/latex] at [latex](1, 1)[/latex]
- [latex]m_3 = 4\text{ kg}[/latex] at [latex](2, -2)[/latex]
Suppose four point masses are placed on a number line as follows:
Find the moment of the system with respect to the origin and find the center of mass of the system.
Suppose three point masses are placed on a number line as follows (assume coordinates are given in meters):
Find the center of mass of the system.
Center of Mass of Thin Plates
The Main Idea
- Lamina:
- Thin sheet with uniform density
- Mass distributed continuously across a 2D region
- Centroid:
- Geometric center of a 2D shape
- Coincides with center of mass for uniform density
- Key Formulas:
- Mass: [latex]m = \rho \int_a^b f(x) dx[/latex]
- [latex]x[/latex]-axis Moment: [latex]M_x = \rho \int_a^b \frac{[f(x)]^2}{2} dx[/latex]
- [latex]y[/latex]-axis Moment: [latex]M_y = \rho \int_a^b x f(x) dx[/latex]
- Center of Mass: [latex]\overline{x} = \frac{M_y}{m}, \overline{y} = \frac{M_x}{m}[/latex]
- Symmetry Principle:
- If a region is symmetric about a line, its centroid lies on that line
Problem-Solving Strategy
- Identify the region [latex]R[/latex] and its bounding function [latex]f(x)[/latex]
- Set up integrals for mass and moments ([latex]M_x[/latex] and [latex]M_y[/latex])
- Evaluate the integrals
- Calculate center of mass coordinates using [latex]M_x[/latex], [latex]M_y[/latex], and [latex]m[/latex]
- Simplify and interpret the results
Let [latex]R[/latex] be the region bounded above by the graph of the function [latex]f(x)={x}^{2}[/latex] and below by the [latex]x[/latex]-axis over the interval [latex]\left[0,2\right].[/latex] Find the centroid of the region.
Let R be the region bounded above by the graph of the function [latex]f(x)=6-{x}^{2}[/latex] and below by the graph of the function [latex]g(x)=3-2x.[/latex] Find the centroid of the region.
Find the centroid of the region bounded by [latex]y = \sqrt{x}[/latex] and the [latex]x[/latex]-axis from [latex]x = 0[/latex] to [latex]x = 4[/latex].
Center of Mass of a Region Bounded by Two Functions
The Main Idea
- Region Definition:
- Bounded above by function [latex]f(x)[/latex]
- Bounded below by function [latex]g(x)[/latex]
- Bounded on sides by [latex]x = a[/latex] and [latex]x = b[/latex]
- Key Formulas:
- Mass: [latex]m = \rho \int_a^b [f(x) - g(x)] dx[/latex]
- [latex]x[/latex]-axis Moment: [latex]M_x = \rho \int_a^b \frac{1}{2}([f(x)]^2 - [g(x)]^2) dx[/latex]
- [latex]y[/latex]-axis Moment: [latex]M_y = \rho \int_a^b x[f(x) - g(x)] dx[/latex]
- Center of Mass: [latex]\overline{x} = \frac{M_y}{m}, \overline{y} = \frac{M_x}{m}[/latex]
- Centroid of Rectangle:
- Located at [latex](x_i^, \frac{f(x_i^) + g(x_i^*)}{2})[/latex]
Problem-Solving Strategy
- Identify bounding functions [latex]f(x)[/latex] and [latex]g(x)[/latex] and integration limits
- Set up integrals for mass and moments ([latex]M_x[/latex] and [latex]M_y[/latex])
- Evaluate the integrals, often requiring integration by parts or substitution
- Calculate center of mass coordinates using [latex]M_x[/latex], [latex]M_y[/latex], and [latex]m[/latex]
- Simplify and interpret the results
The Symmetry Principle
The Main Idea
- Symmetry Principle:
- If a region is symmetric about a line, its centroid lies on that line
- Simplifies calculations for symmetric shapes
- Theorem of Pappus for Volume:
- Volume = Area of region × Distance traveled by centroid
- Applies to solids of revolution
Problem-Solving Strategy
- Identify symmetry in the given region
- Use symmetry to simplify centroid calculations
- For solids of revolution:
- Calculate the area of the region
- Determine the path of the centroid
- Apply the Theorem of Pappus
Let R be the region bounded above by the graph of the function [latex]f(x)=1-{x}^{2}[/latex] and below by [latex]x[/latex]-axis. Find the centroid of the region.
Let [latex]R[/latex] be a circle of radius [latex]1[/latex] centered at [latex](3,0).[/latex] Use the theorem of Pappus for volume to find the volume of the torus generated by revolving [latex]R[/latex] around the [latex]y[/latex]-axis.
Find the volume of a washer-shaped solid formed by revolving the region between concentric circles of radii [latex]3[/latex] and [latex]5[/latex] centered at [latex](4,0)[/latex] around the [latex]y[/latex]-axis.