9.6 POLAR FORM OF VECTORS

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We can also represent the same forces as scalar magnitudes, and direction,

9.6.1 Cartesian Vector Notation

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This is like the rectangular form, except that i, j & k are used as placeholders.

9.6.2 Scalar Notation

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We can represent forces as simple scalar values (not vectors). But, we must still remember that theses are still vectors. For scalar values we need to take care to define their direction for the problem, or for specific values. One common way to do this is to define positive `x-y' axes, and then refer to `x' and `y' components. From this a positive `x' component implies one direction - a negative component implies the opposite direction.

For example,

Scalar notation is often made obvious by using `x', and `y', or similar subscripts.

direction, location, signs, etc. are all defined by convention, and very compact mathematical methods can be used.

These problems can also be solved using cosine and sine law force additions on force triangles. Considering the last example,

Consider the large pendulum below as an example where a force triangle could be used to find the tensions in the cables.

[picture]

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NOTE: all of the vectors added are position vectors. If we are to consider rotation vectors, they cannot be simply added. You must consider alternatives such as Euler angles, etc.

Consider the example below,

9.6.3 Unit Vector Representation

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The unit vector representation follows.

The unit vector representation can be developed from basics.

9.6.4 3D Vectors

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We will use right-handed coordinates

Consider the following example,

To emphasize the main relations,

**************** Solve previous problem with Mathcad *******

An example to illustrate this technique is, ([Hibbeler, 1992], prob. 2-56, pg. 49)

Consider the example below,

******************* Solve using Mathcad ********************

Consider the case below, where we know positions, and forces, but we want to find the resultant force,

********************** Mathcad example *********************

As a practical example of where 3D vectors might be required, consider the power line pole. It uses a tension cable anchored in the ground to resist the forces exerted by the power lines.

[picture]

9.6.5 Dot (Scalar) Product

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The definition of the dot product is,

Evaluating the dot product.

We can use a dot product to find the angle between two vectors

Projecting vectors.

Consider the example below where we find the component of one vector that lies in the direction of the other vector.

The use for the dot product will become obvious in later sections.

Consider the example below,

********************** Solve Using Mathcad ************************

9.6.6 Summary

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the basics of statics as a topic were covered

engineering units and calculations

representations covered in this section were,

- scalar values

- vector values

rectangular

polar

cartesian

direction cosines

vector projection

direction vectors

The dot product was shown as a way to project one vector onto another, or final angles between them.

9.6.7 Practice Problems

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0.

1. Four forces act on bolt A as shown. Determine the resultant of the forces on the bolt.

2. Find the tension in cable A and B if the tension in cable C is 100N.

3. A disabled automobile is pulled by two ropes as shown below. If the resultant of the two forces must be 300lb, parallel to the forward roll of the car, find (a) the tension in each of the ropes, knowing that a = 30°, (b) the value of a such that the tension in rope 2 is minimum.

4. Convert between the representations given on the left, and the results requested on the right.

5. An F-117A stealth fighter is supposed top be flying N20°E, but a strong wind from East to West is pushing it off course. If the plane is pointed N20°E, but is actually moving N23°E, and its 22,000 lb engine is at full thrust, a) what force is the wind exerting on the plane? b) What is the answer in newtons?

9.6.8 References

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Hibbeler, R.C., Engineering Mechanics: Statics and Dynamics, 6th edition, MacMillan Publishing Co., New York, USA, 1992.

Soustas-Little, R.W. and Inman, D.J., Engineering Mechanics Statics, Prentice-Hall, 1997.