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AP Calculus AB: 10.9.1 Center of Mass

Mathematics9 CardsCreated 3 months ago

This content explains the concept of the center of mass in a system of point masses, detailing how to calculate it using moments about the x- and y-axes. It includes definitions, formulas, and example problems to illustrate the calculation of total moments and the resulting center of mass.

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Center of Mass

  • The center of mass of an object is the point where you can assume all the mass is concentrated.
    • A moment can be thought of as a system’s tendency to rotate about an axis.
    • The moment about the y-axis is .

  • The moment about the x-axis is .

  • The center of mass, , can be found by dividing the moments by the total mass:

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Key Terms

Term
Definition

Center of Mass

  • The center of mass of an object is the point where you can assume all the mass is concentrated.
    • A moment can be thought of as a system...

note

  • The center of mass is the point where you can assume all the mass is concentrated. If you support an object at the center of mass, the obje...

Masses of 2 lb, 4 lb and 1 lb are located in a plane at points (3, 1), (0, 0), and (1, 4), respectively. What is the center of mass of the system?

(1, 6/7)

Suppose that m1, m2, …, mn is a collection of point masses situated in a plane at (x1, y1 ), (x2, y2 ), …, (xn, yn ), respectively. What are the total moments with respect to the x- and y-axes?

My=n∑k=1xkmk, Mx=n∑k=1ykmk

Masses of 2 lb, 4 lb and 1 lb are located in a plane at points (3, 1), (0, 0), and (1, 4), respectively. Which of the following is the total moment with respect to the y‑axis?

M y = 7

Masses of 2 lb, 4 lb and 1 lb are located in a plane at points (3, 1), (0, 0), and (1, 4), respectively. Which of the following is the total moment with respect to the x‑axis?

M x = 6

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AP Calculus AB: 10.9.1 Center of Mass
TermDefinition

Center of Mass

  • The center of mass of an object is the point where you can assume all the mass is concentrated.
    • A moment can be thought of as a system’s tendency to rotate about an axis.
    • The moment about the y-axis is .

  • The moment about the x-axis is .

  • The center of mass, , can be found by dividing the moments by the total mass:

note

  • The center of mass is the point where you can assume all the mass is concentrated. If you support an object at the center of mass, the object will balance.

  • When dealing with objects of uniform mass, the center of mass is located at the center of the object. But not all objects have uniform mass. The center of mass in these objects might seem strange.

  • Consider a system of point masses. A point mass is a point where you assume mass is concentrated.

  • What is the center of mass of the system?

  • Since all the mass is on the flat plane, the center of mass will be on the plane too.

  • Here are the formulas for finding the coordinates of the center of mass. The numerator of the x-coordinate is also known as the moment about the y-axis. The numerator of the y-coordinate is also known as the moment about the x-axis. A moment can be thought of as the tendency of an object to rotate.

Masses of 2 lb, 4 lb and 1 lb are located in a plane at points (3, 1), (0, 0), and (1, 4), respectively. What is the center of mass of the system?

(1, 6/7)

Suppose that m1, m2, …, mn is a collection of point masses situated in a plane at (x1, y1 ), (x2, y2 ), …, (xn, yn ), respectively. What are the total moments with respect to the x- and y-axes?

My=n∑k=1xkmk, Mx=n∑k=1ykmk

Masses of 2 lb, 4 lb and 1 lb are located in a plane at points (3, 1), (0, 0), and (1, 4), respectively. Which of the following is the total moment with respect to the y‑axis?

M y = 7

Masses of 2 lb, 4 lb and 1 lb are located in a plane at points (3, 1), (0, 0), and (1, 4), respectively. Which of the following is the total moment with respect to the x‑axis?

M x = 6

Which of the following does not accurately complete the sentence? The center of mass of an object is _____________.

the geometric center of the object

Suppose that m1, m2, …, mn is a collection of point masses situated in a plane at (x1, y1 ), (x2, y2 ), …, (xn, yn ), respectively. What are the coordinates of the center of mass of the system?

X=n∑k=1xkmkn∑k=1mk, Y=n∑k=1ykmkn∑k=1mk

Masses of 2 lb, 6 lb, 2 lb and 10 lb are located in a plane at points (0, 2), (3, 4), (0, 1), and (1, 1), respectively. Which of the following is the center of mass of the entire system?

(7/5, 2)