Class Today. Print notes and integration examples Print composites examples Centroids. Example Problems Group Work Time. Defined Finding Centroids

Size: px

Start display at page:

Download "Class Today. Print notes and integration examples Print composites examples Centroids. Example Problems Group Work Time. Defined Finding Centroids"

Georgina Fletcher
7 years ago
Views:

1 Class Today Print notes and integration examples Print composites examples Centroids Defined Finding Centroids Using single integration Using double integration Example Problems Group Work Time 1

2 Recall working with distributed loads Distributed loads are sometimes reduced to a single resultant force at a particular location. The moment of a distributed load is calculated using the single, concentrated resultant force.

3 Recall working with distributed loads The moment calculated using the resultant force equals the summation of the moments for each differential area

4 Moments of The analysis of many engineering problems involves using the moments of quantities such as masses, forces, volumes, areas, or lines which, by nature, are not concentrated values.

5 The moment of an area 5

6 Center of Gravity / Mass Defined CENTER OF MASS locates the point in a system where the resultant mass can be concentrated so that the moment of the concentrated mass with respect to any axis equals the moment of the distributed mass with respect to the same axis. CENTER OF GRAVITY locates where the resultant, concentrated weight acts on a body. 6

7 Finding Centroids Calculate as a weighted average: 1. Compute the moment of each differential element [weight, mass, volume, area, length] about an axis 2. Divide by total [weight, mass, volume, area, length] 7

8 Centroids: Using Single Integration 1) DRAW a differential element on the graph. 2) Label the centroid (x, y) of the differential element. 3) Label the point where the element intersects the curve (x, y) 4) Write down the appropriate general equation to use. 5) Express each term in the general equation using the coordinates describing the curve or function. 6) Determine the limits of integration 7) Integrate 8

9 Centroids: Using Single Integration 1) DRAW a differential element on the graph. 2) Label the centroid (x, y) of the differential element. 3) Label the point where the element intersects the curve (x, y) 4) Write down the appropriate general equation to use. 5) Express each term in the general equation using the coordinates describing the curve or function. 6) Determine the limits of integration 7) Integrate 9

10 Centroids: Using Single Integration 1) DRAW a differential element on the graph. 2) Label the centroid (x, y) of the differential element. 3) Label the point where the element intersects the curve (x, y) 4) Write down the appropriate general equation to use. 5) Express each term in the general equation using the coordinates describing the curve or function. 6) Determine the limits of integration 7) Integrate 10

11 Centroids: Using Single Integration 1) DRAW a differential element on the graph. 2) Label the centroid (x, y) of the differential element. 3) Label the point where the element intersects the curve (x, y) 4) Write down the appropriate general equation to use. 5) Express each term in the general equation using the coordinates describing the curve or function. 6) Determine the limits of integration 7) Integrate 11

12 Using Double Integration 1) Determine whether you will integrate using dxdy or dydx. (This will make a difference in how you define your limits of integration.) 2) DRAW BOTH dx and dy elements on the graph 3) Label the centroid (x, y) 4) Write down the general equation 5) Define each term according to the problem statement 6) Determine limits of integration (be careful here) 7) Integrate 12

13 Finding Centroids of Composite Shapes 1) Divide the object into simple shapes. 2) Establish a coordinate axis system on the sketch 3) Label the centroid (x, y) of each simple shape 4) Set up a table as shown below to calculate values 5) Subtract empty areas instead of adding them. 6) Keep track of negative coordinates and carry signs through

14 Finding Centroids of Composite Shapes 1) Divide the object into simple shapes. 2) Establish a coordinate axis system on the sketch 3) Label the centroid (x, y) of each simple shape 4) Set up a table as shown below to calculate values 5) Subtract empty areas instead of adding them. y 2 6) Keep track of negative coordinates and carry signs through x

15 Finding Centroids of Composite Shapes 1) Divide the object into simple shapes. 2) Establish a coordinate axis system on the sketch 3) Label the centroid (x, y) of each simple shape 4) Set up a table as shown below to calculate values 5) Subtract empty areas instead of adding them. 2 6) Keep track of negative coordinates and carry signs through

16 Finding Centroids of Composite Shapes 1) Divide the object into simple shapes. 2) Establish a coordinate axis system on the sketch 3) Label the centroid (x, y) of each simple shape 4) Set up a table as shown below to calculate values 5) Subtract empty areas instead of adding them. 2 6) Keep track of negative coordinates and carry signs through

GRAPHING IN POLAR COORDINATES SYMMETRY

GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry - y-axis,