The operator sets the wheel turns the counter to zero and then traces the pointer around the perimeter of the shape Developments of the planimeter can establish the position of the first moment of area center of mass and even the second moment of area This area is also equal to the area of the parallelogram A ABB The measuring wheel now moves in the opposite direction subtracting this reading from the former The area of the shape is proportional to the number of turns through which the measuring wheel rotates Near the junction of the two links is a measuring wheel of calibrated diameter with a scale to show fine rotation and worm gearing for an auxiliary turns counter scale A planimeter also known as a platometer is a measuring instrument used to determine the area of an arbitrary two dimensional shape If by any chance you spot an inappropriate comment while navigating through our website please use this form to let us know and we ll take care of it shortly The other end of the linkage pivots freely on a weight that keeps it from moving We re doing our best to make sure our content is useful accurate and safe When the measuring wheel moves parallel to its axis the wheel skids without rolling so this movement is ignored Here x textstyle x is the воет собака причина perpendicular to the measuring arm and y textstyle y is parallel The net result is the measuring of the difference of the yellow and green areas which is the area of ABCD The measuring wheel measures the distance PQ perpendicular to EM where D textstyle D is the https://asklong.ru/ans/7 enclosed by the contour As the measuring arm traces the contour the device records the quantity x x5899 y textstyle x Delta y in small intervals by taking a product of the current perpendicular distance x textstyle x and the parallel displacement d y textstyle dy The operation of a linear planimeter can be justified by applying Green s theorem though the design of the major varieties predates the theorem s proof The precise way in which they are constructed varies with asklong.ru main types of mechanical planimeter being polar linear and Prytz or hatchet planimeters For a parametric equation in polar coordinates where both r and vary as a function of time this becomes x777B t 6 7 r t 7 d x58B8 t x777B t 6 7 r t 7 x58B8 x57D9 t d t The Amsler polar type consists of a two bar linkage For a polar planimeter the total rotation of the wheel is proportional to x777B t r t x58B8 x57D9 t d t textstyle int t r t dot theta t dt as the rotation is proportional to the distance traveled which at any point in time is proportional to radius and to change in angle as in the circumference of a circle x777B r d x58B8 7 x58C5 r textstyle int r d theta 7 pi r When the tracing is complete the scales at the measuring wheel show the shape s area Moving with the pointer from A to B the arm EM moves through the yellow parallelogram with area equal to PQ 765 EM The Swiss mathematician Jakob Amsler Laffon built the first modern planimeter in 6859 the concept having been pioneered by Johann Martin Hermann in 6868 When the planimeter s measuring wheel moves perpendicular to its axis it rolls and this movement is recorded 96 6 98 Many developments followed Amsler s famous planimeter including electronic versions The connection with Green s theorem can be understood in terms of integration in polar coordinates in polar coordinates area is computed by the integral x777B x58B8 6 7 r x58B8 7 d x58B8 textstyle int theta tfrac 6 7 r theta 7 d theta where the form being integrated is quadratic in r meaning that the rate at which area changes with respect to change in angle varies quadratically with the radius Moving from C to D the arm EM moves through the green parallelogram with area equal to the area of the rectangle D DCC As the area outline is traced this wheel rolls on the surface of the drawing There are several kinds of planimeters but all operate in a similar way By summing up x x5899 y textstyle x Delta y along the contour C textstyle C one approximates the integral x777E C x d y displaystyle oint C xdy through Riemann sum Note that any other vector field satisfying x7757 Q x7757 x x7767 x7757 P x7757 y 6 displaystyle begin aligned frac partial Q partial x frac partial P partial y 6 end aligned also suffices for example P Q x7767 y 5 textstyle P Q y 5 or x7767 6 7 y 6 7 x textstyle frac 6 7 y frac 6 7 x That means the planimeter measures the distance that its measuring wheel travels projected perpendicularly to the measuring wheel s axis of rotation The working of the linear planimeter may be explained by measuring the area of a rectangle ABCD see image At the end of one link is a pointer used to trace around the boundary of the shape to be measured The movements along BC and DA are the same but opposite so they cancel each other with no net effect on the reading of the wheel