Moments of Inertia. Maplesoft, a division of Waterloo Maple Inc., 2008

Transcription

1 Introduction Moments of Inertia Maplesoft, a division of Waterloo Maple Inc., 2008 This application is one of a collection of educational engineering eamples using Maple. These ever stage of the solution, and man are illustrated using short video clips. Click on the buttons to watch the videos. The steps in the document can be repeated to solve similar problems. Problem Statement For the region shown in Figure 1, determine and the second moments about the - and -ais, respectivel. 60m R 100m Figure 1 Solution Step Result The region is bounded b a straight line, the - and -aes, and the vertical line. The equation of the line forming the upper boundar can be obtained from the two-point form of the straight line. The two points are and. Obtain the equation of the line forming the upper boundar. Click Tools > Tasks > Browse to open the Task browser. In this window, choose Algebra > Compute the Equation of the Line Passing through 2 Points Enter the first point: (1)

2 Equation of a Line > Two - Point. Click on Insert Minimal Content to insert the Task Template into the worksheet. Note that the template is inserted at the location of the cursor. Redefine the first and second points within the table, pressing [Enter] each time. Press [Enter] to eecute the last line and obtain the equation of the line. Enter the second point: Compute the equation of the line passing through the two points: (1) (2) (3) The second moments of with respect the - and -aes are defined as: To find the second moment of area about the -ais using onl a single integral, a differential element that is parallel to the -ais must be chosen. Due to the nature of the curve, two differential elements are needed as shown in Figure 2. Above the horizontal line, Figure 2 shows, the area element, in red. Below this line, is shown in green. 60 m d d 100 m Figure 2 Above the line the length of the differential element is less where is obtained from the equation of the line

3 Obtain the epression for the line. above Use an equation label to reference previous output. Press [Ctrl][L], then enter the appropriate reference equation number. To make the subject of the formula of the straight line, rightclick the epression and select Solve > Isolate Epression for >. Right-click the output and select Right-hand Side, then use equation labels to implement the subtraction. Below the line, the element has constant length 140. Above this line, the length of the element is given b. isolate for right hand side (4) (5) (6) (7) (8) Hence, is given b the sum of integrals implemented to the right. For subscript notation, use the underscore ( _ ) to move the cursor to the subscript position, and the the baseline. For eample, to enter, tpe [I][_][][], then press the right arrow to move out of the subscript. To calculate a definite integral, click on the definite integral template from the Epression palette. Overwrite and with the upper and lower limits and overwrite with the appropriate epression. Press [Enter] to evaluate.

4 The differential element which is parallel to the -ais is shown in Figure 3. Therefore, can be epressed as: Therefore, 60 m 100 m Since and the region d is bounded b and the integral giving given to the right. is Figure 3 (9) Alternativel, the moments of inertia can be calculated b the iterated double integrals shown to the right. To enter and evaluate a definite double integral, click on the definite integration template from the Epression palette. Press [Tab] twice to move the highlighted field over f. Click on the definite integral template a second time. The integral is now an iterated double integral. Overwrite all the fields b pressing [Tab] to move from field to field. [Shift][Tab] will move through the template fields in the opposite direction. (10) (11) Press [Enter] to evaluate the double integral. Legal Notice: The copright for this application is owned b Maplesoft. The application is intended to

5 demonstrate the use of Maple to solve a particular problem. It has been made available for product evaluation purposes onl and ma not be used in an other contet without the epress permission of Maplesoft.