Equilibrium of a Particle - Tension Forces 

? Maplesoft, a division of Waterloo Maple Inc., 2008 

Introduction 

This application is one of a collection of educational engineering examples using Maple. These applications use Clickable Engineering? methods to solve problems interactively. Steps are given at every stage of the solution, and many are illustrated using short video clips.  

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Problem Statement 

A giant disco ball weighing 50 kg is supported by three cables, as shown to the right in Figure 1.     Find the tension in each cable.     (Assume that m/.)

Figure 1

 

Solution 

Step	Result
To perform matrix computations, load the Student Linear Algebra package.    Tools > Load package > Student Linear Algebra	Loading Student:-LinearAlgebra
Define the position vectors, A, B and C.     To define A, use the assignment operator (a colon followed by an equal sign).    To enter the vector, use the Matrix palette. Set the number of rows to three and the number of columns to one and then press the Insert Vector[column] button. You could also use the Choose button and drag the mouse to select the matrix size.    Fill in an element, then press [Tab] to move to the next placeholder.     Press [Enter] to evaluate.     Repeat for B and C.	(3.1)       (3.2)       (3.3)
The weight of the disco ball is the product of its mass and the acceleration of gravity.     Compute this value.    Define variables for mass and gravity, then enter the expression for the weight and press [Enter].	(3.4)     (3.5)     (3.6)
Define w, the force associated with the weight.    Use the Matrix palette to enter the vector.    To refer to the weight, use the variable defined in the previous step.     Press [Enter] to evaluate.	(3.7)
The system is in equilibrium. Therefore, the sum of all the forces in the system (i.e., the weight plus the tension in each cable) must be equal to zero, as expressed by the vector equation        where are the forces in the directions A, B, and C, respectively. The are the magnitudes, and the are unit vectors.
Let be the matrix whose columns are the vectors , and let be the vector of magnitudes.  Then the equilibrium equation is with formal solution .
The unit vectors are found by dividing each position vector by its magnitude.     Normalize each direction vector.    Use the underscore ( _ ) to move the cursor to the subscript position, and the right arrow (→) to move back to the baseline. For example, to enter , type [u][_][a], then press the right arrow to move out of the subscript.    The notation for the magnitude of a vector is found in the Common Symbol palette ( ? ), or by typing two vertical bars from the keyboard.    Press [Ctrl][=] to evaluate the unit vectors inline.	=   =   =
Construct an augmented matrix, U, whose columns are the unit vectors .     Use standard angle brackets ( < and > ), and a vertical bar ( | ) to define the augmented matrix. These can be entered from the keyboard or from the Common Symbols and Operators palettes.      Press [Ctrl][=] to evaluate the augmented matrix inline.	=
Find the inverse of the augmented matrix U, namely , and calculate .     Let the result represent the vector T.    To obtain the inverse, raise the matrix to the power -1.     Multiply the inverse of the matrix by . Use a space for  multiplication    Right-click on the expression and select Approximate > 5.	(3.8)     (3.9)
The tension in each cable is given by the vector T, as shown    Use an equation label to reference previous  output. Press [Ctrl][L], then enter the appropriate reference equation number.	(3.10)

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