PreviouS Problem List Next 1 point) Let C be the counter-clockwise planar circle
ID: 2887295 • Letter: P
Question
PreviouS Problem List Next 1 point) Let C be the counter-clockwise planar circle with center at the origin and radius r > 0. Without computing them, determine for the following vector fields F whether the line integrals F. dr are positive, negative, or zero and type P, N, or Z as appropriate. helds F whethertheinentegrasLF drareposme negamorzeno and ve A. F = the radial vector field = x1+ . B. F the circulating vector field--i zj C. F the circulating vector field - yi -rj: D. F the constant vector field-i+jExplanation / Answer
Solution :- The sign on a line integral ?F?dr can be determined by looking at the curve in the vector field. As there are a number of free and accessible graphing utilities available online, it is much easier to make these predictions now than it used to be. We will use the graph of the counter-clockwise planar circle with radius r>0 in a number of different vector fields in order to determine the sign of ?F?dr.
A ) No matter the size of the circle, a particle moving around the circle encounters a net force of 0 (all the directions and magnitudes of the arrows cancel out). The line integral is 0. Z
B) Since the field moves things counterclockwise, the motion at all times is in the same direction as the field. The line integral is positive. P
C) Since the the field moves things clockwise, the motion at all times is in the opposite direction as the field. The line integral is negative. N
D) As we move counterclockwise, the field is in our direction in the first half of quadrant I, then opposite all the way until the middle of quadrant III, then in our direction again until the start. So all together we balance out. The line integral is 0 . Z
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