Procedure to produce discretized path composed by semi-circle and straigth sub-paths

In some problems such as mechanics [1], thermodynamics [2], fluids [3], and electrostatics [4], information about path is required, but it is sometimes avoided by introducing other higher level and more sophysticated concepts. The reason is simply to avoid mathematical difficulties. Illustration of a path is given in Figure 1. Suppose that an object moves along this path which has friction and the total work done by the friction is to be found. Without information about the path and how the object orientation in every point, the work can not be calculated. It is only an illustration the need to have information about path.

A path s can be discretized into ds , which consists of dx and dy as in the formula

which can be expressed also in the form of

with θ is the incline angle of ds measured from x axis as illustrated in Figure 2. Next step is how to discretize s into N segments with width ds . Other example to discretize s is using parametric equation, but it does not give equal ds as reported in disretizing self-siphon path [4]. A procedure to produce equal ds is described in this article.

A simple way to have ds is through

or it can be found that

for large number of N . Equation ( 1)-( 3) can be expressed in vector form which is

The incline angle θ can be used as parameter to generate s (explicitely, x and y ) based on value of s ∆ and iteration index i . It can be written that

Then, discretization of incline angle θ into θ

with initial angle a θ and final angle b θ . For a straight line it will be found that 0 = ∆θ . In Equation ( 9) number of segment M is used instead of N as in Equation ( 4). In Equation ( 4) value of N is set and s ∆ is obtained. But in Equation (9) value of M is determined throung chosen value of s ∆ . The later scheme is preferred in the implementation of discritizing the path s . For a semi-circle path with radius R it can be found that ( )

Using θ ∆ it can written that

which holds for several types of sub-path, such as semi-circle and straight path

As an example a path consists of three types of sub-path is presented. Suppose there is a path which consists of a horizontal path with length 1 L , a semi-circle with radius R from

Other way than Equation ( 4) and (10), is first by defining s ∆ to find N or M . This approach will be used instead of the first one. Then, the procedure can be described as follow.

Step 01. Start.

Step 02. Define value of s ∆ .

Step 03. Define number of sub-path 3 = M .

Step 04. Start with 1 = j .

Step 05. Define initial angle ja θ , final angle jb θ , and radius j R of path j . (*)

Step 06. Define length j L of path j .

Step 07. Calculate number of segment z L N j j ∆ = of path j .

Step 08. Calculate discretized incline angle R z j ∆ = ∆θ of path j . (**)

Step 09. Increment j value with 1.

Step 10. If J j < then goto Step 05.

Step 11. Define initial position 0

x and 0 y , also initial angle 0 θ .

Step 12. Calculate number of segment of all sub-paths

Step 13. Start with 0 = i .

Step 14. Calculate

Step 15. Calculate

Step 16. Determine incline angle group

. (***)

Step 17. Assign discretized incline angle

for sub-path j .

Step 18. Calculate

Step 19. Increment i value with 1.

Step 20. If N i < then goto Step 14.

Step 21. Finish. Result of the procedure for given problem is as illustrated in Figure 3. It can be seen from the figure that smaller value of s ∆ gives better curve of path s . Back to paragraphs in the beginning of this article, since the path is already discritized then the work done by friction can be calculated numerically. In a segment i with width s ∆ with incline angle i θ the friction force would be

then the work should be

By summing work done by friction for all segments

total work done by friction foce can be calculated. But if the centripetal force is also considered than a correction must be introduced into Equation ( 12)

with t ∆ is time needed to elapse the segment i with width s ∆ . For a straigth line then the second term inside the rectangular bracket in the right side of Equation ( 15) would be zero and return to Equation (12). Further detail of implementation of Equation ( 15) is out of the scope of this article.

Procedure to produce dicritized path composed by straight and semi-circle sub-paths has been presented and the result has been shown. Smaller value of s ∆ gives better result to form of the produced path.

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