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AP Calculus AB: 10.9.2 The Center of Mass of a Thin Plate

Mathematics11 CardsCreated 8 months ago

This content explains how to find the center of mass of a thin plate of uniform density using integration. It covers the difference from point mass systems, introduces area and moment calculations with definite integrals, and provides example problems involving geometric regions bounded by curves.

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The Center of Mass of a Thin Plate

  • The center of mass of an object is the point where you can assume all the mass is concentrated.

  • The center of mass of a thin plate (planar lamina) of uniform density is located

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Key Terms

Term
Definition

The Center of Mass of a Thin Plate

  • The center of mass of an object is the point where you can assume all the mass is concentrated.

  • The center of mass of a thin...

note

  • Finding the center of mass of a continuous region or a thin plate is different from a system of point masses because there are an infinite ...

Given a thin plate on the xy-plane bounded by the x‑axis, the curve y = f (x) = e x, and the lines x = 0 and x = 1, what is the center of mass?

(1/e−1,e^2−1/4(e−1))

Consider a thin region of uniform density bounded by a function f (x) and the x‑axis with a ≤ x ≤ b. Which of the following formulas produces the area of the region?

∫baf(x)dx

Given a thin plate on the xy-plane bounded by the x‑axis, the line y = f (x) = 1 − x, and the y‑axis, what is the center of mass?

(1/3, 1/3)

Consider a thin region of uniform density bounded by a function f (x) and the x‑axis with a ≤ x ≤ b. Let (X, Y ) be the center of mass of the region. Which of the following formulas produces the x-coordinate X ?

X=∫baxf(x)dx/∫baf(x)dx

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AP Calculus AB: 10.9.2 The Center of Mass of a Thin Plate
TermDefinition

The Center of Mass of a Thin Plate

  • The center of mass of an object is the point where you can assume all the mass is concentrated.

  • The center of mass of a thin plate (planar lamina) of uniform density is located

note

  • Finding the center of mass of a continuous region or a thin plate is different from a system of point masses because there are an infinite number of points to consider. But you can find the center of mass using a definite integral.

  • Start by considering an arbitrary rectangle in the region. The rectangle has a very thin base and its height is defined by the curve of f.

  • The height times the width tells you the area of the rectangle. Multiplying by the distance x weights it, since that value is the center of the rectangle. Integrating and dividing by the area gives you the x-coordinate of the center.

  • A similar process works for the y-coordinate. The height times the width tells you the area. Half the height, or f(x)/2, weights the rectangle.

  • Consider a semicircle of radius one. Where is the center of mass?

  • The x-coordinate is easy to find. By symmetry, half of the mass is to the left of the y-axis and the other half is to the right. So the x-coordinate must be at the axis, or at x = 0.

  • You know that the y-coordinate will be lower than half the height because there is more mass toward the bottom of the figure than there is toward the top.

  • Start by finding the area. The area of a semicircle is half the area of a circle.

  • Now use the formula and plug in the equation of a semicircle.

  • Evaluate the integral to find the y-coordinate.

  • Put the two pieces together and you have the location of the center of mass.

Given a thin plate on the xy-plane bounded by the x‑axis, the curve y = f (x) = e x, and the lines x = 0 and x = 1, what is the center of mass?

(1/e−1,e^2−1/4(e−1))

Consider a thin region of uniform density bounded by a function f (x) and the x‑axis with a ≤ x ≤ b. Which of the following formulas produces the area of the region?

∫baf(x)dx

Given a thin plate on the xy-plane bounded by the x‑axis, the line y = f (x) = 1 − x, and the y‑axis, what is the center of mass?

(1/3, 1/3)

Consider a thin region of uniform density bounded by a function f (x) and the x‑axis with a ≤ x ≤ b. Let (X, Y ) be the center of mass of the region. Which of the following formulas produces the x-coordinate X ?

X=∫baxf(x)dx/∫baf(x)dx

Given a thin plate on the xy-plane bounded by the x‑axis, the line x = 8, and the curve of f (x) = x ^2/3, what is the y-coordinate of the center of mass?

10/7

Given a thin plate on the xy-plane bounded by the x‑axis, the line x = 8, and the curve of f (x) = x ^2/3, what is the area of the region?

96/5

Consider a thin region of uniform density bounded by a function f (x) and the x‑axis with a ≤ x ≤ b. Let (X, Y ) be the center of mass of the region. Which of the following formulas produces the y-coordinate Y ?

Y=∫ba1/2[f(x)]2dx/∫baf(x)dx

Given a thin plate on the xy-plane bounded by the x‑axis, the line x = 8, and the curve of f (x) = x^ 2/3, what is the x-coordinate of the center of mass?

5

Find the center of mass of a thin plate in the shape of a quarter-circle of radius 1 as shown below.

(4/3π,4/3π)