Equilibrium of a Rigid Body - 1 

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

Introduction 

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The steps in the document can be repeated to solve similar problems. 

Problem Statement 

 

 

As shown in Figure 1, the L-shaped bar PQRS is attached to a vertical wall at P by a ball-and-socket joint and is supported by a thin rod, QU and two cables QT and RV. A large floral arrangement hangs from the bar at S and exerts a force .    Determine the components of the reaction force at P, the force in the rod QU, and the tension in the two cables.

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 weight of the floral arrangement, the force W.    Since the joint at P is a ball and socket joint, there will three unknowns acting on this point. These three unknowns are the three rectangular force components, , and . With these, write the vector .    To define W, 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 .    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 [X][_][P], then press the right arrow to move out of the subscript.	(3.1)     (3.2)
The system is in equilibrium. Therefore, the sum of all forces acting on the bar PQRS must be equal to zero and the sum of moments about any point must be equal to zero. To solve this problem, moments will be taken about P.    To obtain the vectors needed to formulate the equilibrium equations, specific unit vectors and direction vectors are needed. To derive the direction vectors, first define the position vectors of points , all given with respect to as the origin.
By examining Figure 1, define a position vector for the points P, Q, R, S, T, U, and V.    Use the Matrix palette to enter the vectors.    Press [Enter] to evaluate.		(3.3)

 

Using the position vectors defined in the previous step, find the direction vectors, and  

 

Use the Matrix palette to enter the vectors. 

 

Press [Enter] to evaluate. 

 

 

(3.4)

 

 

 

 

The unit vectors for the cables and the slender rod must now be found. 

 

When defining these unit vectors, it is assumed that they are in tension and pull on the bar PQRS.  

 

Find the unit vectors by normalizing the direction vectors. 

 

The notation for the magnitude of a vector is found in the common symbol palette ( ? ), or by typing two vertical bars from the keyboard. 

 

 

 

(3.5)

 

 

Equilibrium of forces is expressed by the vector equation . Equating the components to zero gives three equations. 

Enter the left-hand side of the force-equilibrium equation.  

 

Input the left-hand side of the equation and press [Enter].  

 

Right-click the output and select Conversions > To List.  

 

Then, right-click the list and select Conversions > To Set. 

 

 

 

In following steps, this set will be combined with the set of three scalar equations, obtained from the principle-of-moments equation written at point P.  

 

Determine the six unkowns in the system, namely, , by using the full set of six equations. 

 

 

(3.6)

 

 

(3.7)

 

 

(3.8)

 

 

The principle of moments state that for a system in equilibrium, the sum of moments about any point must be equal to zero. 

 

The sum of moments about point P can therefore be expressed algebraically as: 

 

 

 

From the equation three scalar equations can be obtained. 

Enter the expression for .  

 

Press [Enter] to evaluate. 

 

To calculate the cross-product of two vectors, use the times symbol from either the Operators or Common Symbols palette.  

 

Right-click the output and select Conversions > To List.  

 

Then, right-click the list and select Conversions > To Set. 

 

 

(3.9)

 

 

(3.10)

 

 

(3.11)

 

Combine the two sets of equations and solve to obtain the solution to the system.  

 

If a negative value is obtained for one of the reactions, the assumed direction of the force is opposite to the actual direction. 

 

Reference the two sets of equations via their equation labels. Press [Ctrl][L], then enter the appropriate reference equation number. 

 

Combine these sets of equations using the set-union symbol,, which can be obtained from the Common Symbols palette. 

 

Press [Enter] to display the resulting set. 

 

Right-click the resulting set and select Solve > Numerically Solve. 

 

 

 

(3.12)

 

 

(3.13)

 

 

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