Equilibrium of a Particle - Tension & Normal ForcesCopyright Maplesoft, a division of Waterloo Maple Inc., 2008 

Introduction 

This application is one of a collection of educational engineering examples with 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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The steps in the document can be repeated to solve similar problems. 

Problem Statement 

As shown in Figure 1 to the right, a smooth vertical pole has on it a cylinder of mass 7 kg.  This cylinder is held suspended above the ground by the rope PQ.      Determine T, the tension in the rope, and N, the normal force exerted by the pole on the cylinder.    (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
The weight of the cylinder 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].	(3.1)     (3.2)     (3.3)
Define W, the force associated with the weight.    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.    To refer to the weight, use the variable defined in the previous step.      Press [Enter] to evaluate.	(3.4)
The normal force is perpendicular to the vertical pole. Therefore, the -component of N must be zero. The first and third components of this vector are and , respectively.     Define the normal force, N.    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][_][n], then press the right arrow to move out of the subscript.    Use the Matrix palette to enter the force.    Press [Enter] to evaluate.	(3.5)
The tension T can be expressed as a product of its magnitude, and a unit vector along the rope.    Define the force of the tension, T.    Use the Matrix palette to enter the force.    Press [Enter] to evaluate.	(3.6)
The unit vector U is in the direction of where and are the position vectors defined to the right.    Define the vectors P and Q.    Use the Matrix palette to enter the force.    Press [Enter] to evaluate.	(3.7)       (3.8)
The unit vector is obtained by normalizing as shown to the right.    The notation for the magnitude of a vector is found in the Common Symbols palette,(?), or by typing two vertical bars on the keyboard.    Press [Enter] to evaluate.	(3.9)
The system is in equilibrium. Therefore the sum of all the forces must be equal to zero, that is, , or        The left-hand side of this vector equation is entered on the right, forming a vector in which the three unknowns appear. Equating to zero each component of this vector gives three equations to solve for the three unknowns.    Convert this vector to a list of expressions by right-clicking and selecting Conversions > To List     Solve the three component equations, right-click and select Solve > Solve.     Therefore, the magnitude of the tension is 133.47 N.	(3.10)     (3.11)     (3.12)
Substitute the values for and into N.     Use the template from the Expression palette to evaluate the expression at a point. Use an equation label to reference the desired point to evaluate at. Press [Ctrl][L], then enter the appropriate reference equation number.	(3.13)
The magnitude of the actual N is its normal vector. Compute this value.       Use an equation label to reference the vector N.	(3.14)

 

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