{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Center of Mass Notes - Section 6.7 Moments and Centers of...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Section 6.7 Moments and Centers of Mass Goal: Some objects behave as if all their mass is concentrated at a single point called the center of mass. We will develop a way to find centers of mass for one and two dimensional systems. Point Masses: Look at a system of point masses along a line: The center of mass is the point at which we could place a fulcrum and have the line balance. mi xi is called the _____________________ of mi The moment about the origin of the system is: Moment measures the tendency of a system to twist. (if you multiply moment by gravity, you get torque) The system’s total mass is: The center of mass is: Example 1: Given two point masses, mA = 60 lb, mB = 100 lb on a line where A is located 3 feet from the fulcrum, where should B be located to have the two masses balanced? Point masses in 2 dimensions In a two dimensional system of point masses m1 , m2 , ..., mn at positions ( x1 , y1 ), ( x2 , y2 ), ..., ( xn , yn ) : n Total Mass: m = ∑ mi i =1 n Moment about y­axis: M y = ∑ mi xi i =1 n Moment about x­axis: M x = ∑ mi yi i =1 ⎛ My Mx ⎞ Center of Mass: ( x , y ) = ⎜ , ⎝m m⎟ ⎠ Example 2: m1 = 4, m2 = 8, m3 = 3, m4 = 2 P1 = (−2, 3), P2 = (2, −6 ), P3 = ( 7, −3), P4 = (5,1) Find the center of mass ( x , y ) . Center of Mass of Solids: The center of mass of a rectangle is at its center: Mass=(density)(area) in 2 dimensions, so we can use area to determine center of mass. Density is constant in our problems. Example 3: Find the center of mass of the object below. The density is 5. Center of mass of a triangle: x A + x B + xC yA + yB + yC x= ,y = 3 3 Example 4: Find the center of mass of the following region. Density = 1. Example 5: Find the center of mass. δ = 3 . Thin Plates Divide the region into n strips parallel to the y – axis. Assume the center of mass of each sub‐rectangle is at its geometric center. Then if δ ( xi ) is the density of the ith rectangle and ΔAi is the area of the ith rectangle, Δmi = In this course we will assume density is constant and write δ ( xi ) = δ 1b δ ∫ f ( x )2 − g( x )2 dx x = ab y = 2 ba δ ∫ ( f ( x ) − g( x )) dx δ ∫ ( f ( x ) − g( x ))dx b δ ∫ x( f ( x ) − g( x )) dx a a Suppose we integrate over the y‐axis. d 1d δ ∫ f ( y)2 − g( y)2 dy δ ∫ y( f ( y) − g( y))dy c c x=2 d y = d δ ∫ f ( y) − g( y)dy δ ∫ f ( y) − g( y)dy c c Example 6: Let R be the lamina with density 3 in the region in the xy plane bounded by x = e y , x = 1, and y = 2. Find the mass of the lamina. Example 7: Find the center of mass of the region bounded by y = 2 x and y = x 2 . Let δ = 1 . Using Symmetry Example 8: Find the centroid of the region bounded by y = 1 and y = x 2 . Let δ = 1 . Example 9: Find the center of mass of the region bounded by x = y 2 − y and y = x with constant density δ = 1 . ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern