eNotes: Mechanical Engineering
   



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3.2 VECTOR AND SCALAR FORCES


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definitions,



Recall that vectors can be added or subtracted using the paralleloram law. This is a variation or the triangle law. In both cases we are putting vectors head to tail. These methods favor drafting solutions to porblems that are not really necessary with calculators, but they are still very useful for understanding.



vectors can be added to get resultant forces in vector (rectangular component) form.



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



we could have also solved this problem using trigonometry.



3.2.1 Cartesian Vector Notation

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Remember from before,



3.2.2 Scalar Notation

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It can also be useful to keep the forces in scalar values, but the direction should still be defined on paper, instead of by convention, as is done with vectors. The most common method is to use x-y-z components, or forces relative to a given 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 tringle could be used to find the tensions in the cables.

egr20928.jpg
egr20927.jpg

Consider the example below,



3.2.3 3D Vectors

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We will use right-handed coordinates



Consider the following conditions,





To emphasize the main relations,



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









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







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.

egr20933.jpg

3.2.4 Dot (Scalar) Product

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



We can use a dot product to project one vector onto another vector.



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







3.2.5 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.

3.2.6 Practice Problems

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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. A force F acts at the end of a pipe. Determine the magnitudes of the components that act perpendicular to, and along the axis of the end of the pipe. (the pipe lies in the y-z plane)



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









6. An F-117A stealth fighter is supposed top be flying N20E, but a strong wind from West to East is pushing it off course. If the plane is pointed N20E, but is actually moving N23E, 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?



7. Given the system of vectors pictured, a) give the resultant force using cartesian notation b) find the magnitude of the resultant force in metric units. c) Then then using cosine angles, and finally d) projected onto the x-y plane.



3.2.7 References

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

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