Equilibrium of a Particle - Tension Forces 

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

Introduction 

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Problem Statement 

As per Figure 1, two collars and of masses 5 kg and 9 kg respectively, are connected by a cable and hang on a vertical frame composed of two smooth rods with the vertex angle shown.     Determine the equilibrium values for T, the tension in the cable, and , the angle the cable makes with the horizontal.    (Assume that g = 9.81 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
Treat collars P and Q as particles and construct their free-body diagrams.     Diagrams can be drawn by using the Canvas feature in Maple. To insert a Canvas into a worksheet, apply, Insert > Canvas.          Since the entire system is in equilibrium, the sum of forces acting on P or must equal zero. From the two free-body diagrams shown in Figure 2, obtain the two equilibrium equations:	Figure 2    In each case, a standard -coordinate system (with basis vectors i and j) is operative.
Define the gravitational constant , and the masses of the collars and .     Use the assignment operator (a colon followed by an equal sign) to define the following variables.    For subscript notation, 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 [m][_][p], then press the right arrow to move out of the subscript.    Press [Enter] to evaluate.	(3.1)
With the weight given by , obtain the weights of the two collars, and	(3.2)     (3.3)
Let be a unit vector in the direction of T, and let be the magnitude of T.      Let and be unit vectors respectively in the directions of the normal forces acting on and .      Let and be the respective magnitude of these normal forces. Thus, we introduce three unknowns, namely, , and .    Define and     Use the Matrix palette. Set the number of rows to two 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.    Obtain θ from the Greek palette.     Fill in an element, then press [Tab] to move to the next placeholder.     Press [Enter] to evaluate.	(3.4)
In addition to the tension , enter the vectors , , , and .     Use the Matrix palette to enter the vectors.    Press [Enter] to evaluate.
The two equilibrium equations represent four equations in the four unknows .  To obtain the first two of these equations, enter the left-hand side of the first equation.       Right-click and select Conversions > To List then again Conversions > To Set.	(3.5)     (3.6)     (3.7)
The remaining two equations are obtained likewise from the second equilibrium equation.    Perform the same operation as the step above.	(3.8)     (3.9)     (3.10)
A set of all four equations is obtained by forming a set union of the two subsets.     Solve this set.    Refer to the Common Symbols palette to obtain the union operator, .     Reference the subsets by their equation labels. Press [Ctrl][L], then enter the appropriate reference equation number.    Right-click the equation and select Solve > Solve.        There are two solutions; the meaningful one gives , the magnitude of the tension, as positive. The corresponding angle is negative, indicating that the cable tilts downward from to .	(3.11)     (3.12)

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