10.9.2 The Center of Mass of a Thin Plate

Question 1

The Center of Mass of a Thin Plate

Answer 1

• 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

Question 2

note

Answer 2

• 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.

Question 3

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?

Answer 3

(1/e−1,e^2−1/4(e−1))

Question 4

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?

Answer 4

∫baf(x)dx

Question 5

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?

Answer 5

(1/3, 1/3)

Question 6

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 ?

Answer 6

X=∫baxf(x)dx/∫baf(x)dx

Question 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 y-coordinate of the center of mass?

Answer 7

10/7

Question 8

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?

Answer 8

96/5

Question 9

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 ?

Answer 9

Y=∫ba1/2[f(x)]2dx/∫baf(x)dx

Question 10

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?

Answer 10

5

Question 11

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

Answer 11

(4/3π,4/3π)