Vector Mechanics & Statics
Illustrations of various concepts covered in a Statics Mechanics course.
Table of Contents
- Vector Mechanics
- Components, Magnitude and Angle of a vector.
- Vectors in 3 Dimensions
- Moment 2D
- Moment 3D
- More Moments 3D
- Moment Projection on an Axis
- 2D Couple
- Forces
- Force On Simply Supported Beam
- Equivalent Force Couple
- Centroids
- Centroid of Two Rectangles
- Area Moments
- Parallel Axis Theorem
- Two Rectangles; Second Moments
- Moments Under Rotation
Components, Magnitude and Angle of a vector.
By positioning the point A, the values for the Vector from the origin to point A are shown.[br]Both the components and ,magnitude and angle, are shown as well as the equations for translating between values.
Components, Magnitude and Angle of a vector.
Try moving the point to the various quadrants.[br]Look at the signs of the components.[br]Imagine a reference triangle between the vector and the x axis, what would sine and cosine of the origin angle be relative to the component signs?
Force On Simply Supported Beam
Description
Shown is a simple beam with supports at A and B and a force applied at point C. The supports A and B are 1 unit apart for simplicity. The location of the applied force can be moved with the slider. Note how the calculated reaction forces at the supports change with the location of the applied force.[br][br]The play buttons will show the equations used to calculate the reaction forces. There are several methods to calculate the forces and stepping through the equations with the play controls will show different possibilities. Each complete solution shows 4 equations.
Centroid of Two Rectangles
Centroid
First Move Point D to (0,1)[br][br]The centroid or geometric center of a plane figure is the arithmetic mean position of all the points in the figure. (https://en.wikipedia.org/wiki/Centroid)[br][br]If the area was a thin plate parallel to the earth surface, the centroid would be at the center of gravity. The center of gravity is the point where a shape would balance. The formula for the centroid can be expressed as a ratio of integrals, [math]\overline{X}=\frac{\int x\cdot dA}{\int dA}[/math] and [math]\overline{Y}=\frac{\int y\cdot dA}{\int dA}[/math] where [math]\left(\overline{X},\overline{Y}\right)[/math] is the centroid point and the integrals are over the area divided into differential area elements [math]dA[/math]. Tables of centroids of common shapes can be used. Because of the additive property of integrals, the centroid of the combination of several basic shapes can be calculated as [math]\overline{X}=\frac{\sum\overline{x_i}\cdot A_i}{\sum A_i}[/math] and [math]\overline{Y}=\frac{\sum\overline{y_i}\cdot A_i}{\sum A_i}[/math] where [math]\left(\overline{x_i},\overline{y_i}\right)[/math]is the centroid of each basic shape and [math]A_i[/math] is the area of each corresponding shape. Shapes can also be subtracted by using a negative area.[br][br]The best way to do these calculations is with a table or spreadsheet. Here a spreadsheet is used to calculate the centroid of two rectangles. The centroid of a rectangle is in its center. The sum (total) of the table columns of [math]Area[/math], [math]\overline{x}\cdot Area[/math] and [math]\overline{y}\cdot Area[/math] are respectively the terms [math]\sum A_i[/math], [math]\sum\overline{x_i}\cdot A_i[/math], and [math]\sum\overline{y_i}\cdot A_i[/math] in the formula above for [math]\overline{X}[/math] and [math]\overline{Y}[/math] .[br]
Description
This applet computes the centroid of two rectangles using a spreadsheet as described above. The two rectangles are defined by the points A and B for Area 1 and C and D for area 2. If area 2 overlaps area 1 it is converted into a negative area of the intersection of the rectangles. This way more interesting combined shapes can be formed. The centroids of each area are shown as pluses, + and the combined centroid is shown as a cross, x.
Activities
Move the points around and note how the centroids change. Note that the combined centroid is closer to the larger areas centroid.[br]Compare the centroids made by combining the rectangles with points A=(-3,1), B=(-1,4),C=(-1,1) and D=(0,2) with the centroids made by subtracting rectangles with points A=(-3,1),B=(0,4), C=(-1,2) and D=(0,4).
Parallel Axis Theorem
Description
This illustrates the parallel Axis Theorem for a rectangular area. The Moment of Inertia ( Second Moment of Area ) of a rectangle about an axis parallel to the x axis is shown. The graph on the right shows how the moment varies with location of the axis.[br]The rectangle width (a) and height (b) as well as the location of the axis can be varied with the sliders.[br]The radius of Gyration, k[sub]x[/sub] , is also shown for the axis. As the axis moves further from the centroid what happens to the radius of gyration? A good way to visualize the radius of gyration is to place the rotation axis at y=0 (Axis=0). Then note the length of [math]k_{_{^x}}[/math] relative the y position of the centroid.