Lecture 13-6

# Lecture 13-6 - Side note: To do #29 of page 846 you need...

This preview shows pages 1–5. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

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

Unformatted text preview: Side note: To do #29 of page 846 you need the following deﬁnition: Defn. Average Value of f (x, y, z ) The average value of f (x, y, z ) over a closed, bounded region R is 13.6 Moments and Centers of Mass In R2 : Find the center of mass (C.O.M.) of a thin plate of density δ (x) over the region bounded by y = f (x) on top and y = g (x) on bottom from x = a to x = b. C.O.M. = (¯, y ), x¯ where x = ¯ My Mx ,y= ¯ M M b M = total mass = (area)(density) = a δ (x) f (x) − g (x) dx Mx and My are moments. Recall moment = (directed distance)(mass) b Mx = moment about the x-axis = a y δ (x) f (x) − g (x) dx ˜ b My = moment about the y -axis = a xδ (x) f (x) − g (x) dx ˜ 1 In R3 : There is no need to worry about mixing x’s and y ’s. We can also ﬁnd the center of mass of two kinds of objects since we have a 3rd dimension. Center of Mass of a thin plate of density δ (x, y ) over a region R Center of Mass: Mass: Moments: 2 Center of Mass of a solid D of density δ (x, y, z ) Center of Mass: Mass: Moments: 3 Example 1 - Thin Plate Find the center of mass of a thin plate of density δ (x, y ) = y over the region bounded by y = x2 and y = 2x. 4 Example 2 - Solid SETUP ONLY. Find the center of mass of a solid with density δ (x, y, z ) = x + z bounded below by the plane z = 0, above by the paraboloid z = 4 − x2 − y 2 and by the cylinder x2 + y 2 ≤ 1. 5 ...
View Full Document

## This note was uploaded on 03/03/2010 for the course MATH 2224 taught by Professor Mecothren during the Spring '03 term at Virginia Tech.

### Page1 / 5

Lecture 13-6 - Side note: To do #29 of page 846 you need...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online