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Locating the centre of mass by mechanical means C J Sangwin School of Mathematics, University of Birmingham, Birmingham, B15 2TT, United Kingdom Telephone +44 121 414 6197, Fax +44 121 414 3389 Email: C.J.Sangwin@bham.ac.uk This article discusses moment planimeters , which are mechanical devices with which is it possible to locate the centre of mass of an irregular plane shape by mec hanical and graphical methods. They are a type of analogue computing device. In addition to this t hey may be used to ﬁnd the static moment (ﬁrst moment) and moment of inertia (second moment) o f a shape about a ﬁxed line. Moment planimeters, sometimes called integrometers or integrators , are direct developments of the planimeter which is a mechanical device used to directly measure the are a of a plane shape. While planimeters are reasonably well known, linear planimeters are less comm on than the polar planimeters of Amsler. Hence in this article we explain how planimeters work throug h the example of a linear planimeter, and then consider how these may be adapted to ﬁnd the centre of mass. More detailed comparisons between other types of area measuring planimeters may be fou nd in the comprehensive survey article of [2]. 1 Area and centre of mass Consider the region enclosed by the closed curve in Figure 1, through which we have drawn the axis. We consider the area to be split into two regions by this axis, and these regions are described by the the two functions, and . Since a general plane shape cannot be described in this way the assumption represents a considerable loss of genera lity, hence we shall provide alternative explanation in a moment. The area of this shape will be ) d x. In the linear planimeter a rigid straight line of length is constrained to move so that one end, traces around the boundary of the region. The other end, , is constrained to move along the -axis. This is shown in Figure 1. Note that ) = sin( (1) As is usual for a planimeter, we ﬁx a freely rotating disc usin g this line as an axel, an example of which is shown in Figure 3. In this arrangement the roll of the disc will be the component of the motion perpendicular to the line. If we consider an inﬁnites imally thin vertical strip of width and height , then during the horizontal motion from to + d , the wheel will record a roll of = sin( ) d x. (2)

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Figure 1: An irregular plane region, Hence ) d sin( )d w. If we denote the the total roll recorded on the wheel around th e boundary of the region from to along , and back along by δR we have that ) d δR w. This illustrates that multiplied by the total roll recorded while traces around the boundary will be equal to the area of the shape. This is the fundamental proper ty of planimeters. Next we turn attention to the centre of mass. Imagine a thin un iform strip of width , and hight The contribution this strip makes to the distance of the cent re of mass of the whole shape from the -axis will be x. And hence, , the distance of the centre of mass from the -axis will be given by RR RR (3) From this it apparent that it will be sufﬁcient to contrive a p lanimeter capable of being able to measure , since we are already capable of measuring the area. Let us assume that we can attach another wheel at which is at an angle of to the -axis. Then the roll recorded will be sin( ) = cos(2 ) = 1 2sin Considering the motion from to along the function we have, sin ) d x. If we the integrate back along , the terms in the two integrals cancel, so that δR x,

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= ( x,y B C ,y Figure 2: A small element and so δR By this procedure we have calculated and hence the line parallel to the -axis on which the centre of mass lies. We choose another line for the -axis, not parallel to the original, and repeat this procedu re. The intersection of the two lines thus obtained locates the c entre of mass. 2 Small elements In this section we take a slightly different approach and, in stead of considering integration of functions representing the boundary of the curve, we assume that the pl ane region has been decomposed into small curvy-parallelograms such as ABCD shown in Figure 2. Here, the line PQ is of ﬁxed length , the point moves around the boundary of the region and the other end runs along the -axis and so is constrained to move along ,y ) = ( 0) . We note that for ABCD , the area equals and the distance of the centre of mass of ABCD from the -axis is + d The point moves around the perimeter from , which has coordinates x,y , to and back to . In each portion of this movement we examine the roll recorde d by the two wheels considered in the previous Section and relate these to the area and centre o f mass. We consider ﬁrst a wheel using the line PQ as an axel. As this moves from to , the point is ﬁxed and the roll recorded AB is a pure roll proportional to the arc length . This is equal and opposite to that as the line moves from to , ie AB CD . As moves from to the angle PQ makes with the -axis is constant at + d with the horizontal and BC = sin( + d )d so that lw BC sin( + d )d = ( + d )d x. Similarly lw DA sin( )d x. If we deﬁne the roll around the perimeter of this small elemen t to be := AB BC CD DA then = d y.

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Figure 3: Details of the roll recording wheel on a planimeter Every reasonable plane region can be decomposed into small elements consisting of such cur vy parallelograms. When doing this the rolls along internal ed ges of this decomposition cancel leaving only the roll around the outside perimeter to consider. Henc e we have that δR Z Z y, where δR is the total roll as moves around the (piecewise smooth) boundary of the region and the right hand side is nothing but the area. The second wheel is at on an axel at an angle to the horizonal. As before, AB CD Furthermore, BC sin( 2d )d 2( + d x, and DA sin( )d + 2 x, Deﬁne, as before, := AB BC CD DA then + d y. Again, + d = y. Hence, RR 3 Green’s Theorem for the plane A justiﬁcation of the polar planimeter of Amsler was given us ing Green’s Theorem in [1]. We justify the results of the informal arguments in the previous sectio ns using a similar approach. Assume we

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have a vector ﬁeld x,y ) = ( x,y ,V x,y )) . Green’s Theorem states that δR Z Z curl( ) d y, where δR is the line integral around the (piecewise smooth) boundary of the region . Imagine a vector ﬁeld of unit vectors in the plane, which we denote by . If a wheel is attached at which is constrained to always point in the direction of this ﬁeld, the roll of the wheel will record the total component of the vector ﬁeld in the direction of the motion, e ffectively measuring this integral. If we denote the roll of the wheel by we have δR δR Z Z curl( ) d y. For the linear planimeter we imagine a vector ﬁeld generated by attaching a unit vector perpendic- ular to the end of the line PQ , of ﬁxed length , at . It remains to ﬁnd this vector ﬁeld, and the corresponding curl. As before in Figure 2, assume that when is at it has coordinates x,y and the other end runs along the -axis and so is constrained to move along ,y ) = ( 0) . Then we have = ( + ( so that and furthermore, sin( ) = and cos( ) = The planimeter vector ﬁeld, which of course does not depend o n the -coordinate, is then x,y x,y sin( cos( Since curl( ) = ∂V ∂x ∂V ∂y (4) a trivial calculation shows that curl( ) = Hence δR Z Z y. Turning attention to the centre of mass we have the vector ﬁel d where the unit vector points points at an angle of to the horizontal. Hence, x,y ) = sin( ) = 2sin 1 = 2

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Figure 4: An example of a Koizumi linear roller planimeter and x,y ) = cos( ) = 2cos( )sin( ) = 2 By (4) we have that curl( ) = Hence, by Green’s Theorem we have that δR Z Z y, as required by (3) to ﬁnd the centre of mass. These are both rather trivial applications of Green’s Theor em. 4 Further generalizations In the previous sections we have considered how to calculate and . Further general- izations naturally occur with and so on. To calculate , for example, we note that sin ) = sin( sin(3 and so it will be sufﬁcient to have an instrument with wheels c apable of recording the motion of a wheel at an angle . Further generalizations are possible, and devices along t hese lines were indeed made and used. It is the practical considerations we turn to i n the next section. 5 Practical implementations The most popular practical implementation of a planimeter i s the polar planimeter of Amsler. The essential difference between this and the linear planimete r is that the point is constrained to move in

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W Figure 5: A moment planimeter schematic a circular arc rather than a straight line. Linear planimete rs were produced commercially, an example of which is shown in Figure 4. The point can be located in the circular magnifying glass, and is constrained to move in a vertical straight line by the trol ley, rather than along the -axis as in our examples. Notice that the wheel need not actually be at , but may be at any convenient position using an axel offset from, but parallel to, the line PQ Perhaps the simplest moment planimeter is an extension of th e linear planimeter, and a schematic of such a device is given in Figure 5. The point is constrained to move in a straight line, marked as the -axis, by an arm which is mounted upon a trolley. The wheel shown is used to measure the area of the shape around which traces. At the point , ﬁxed to the trolley is a gear wheel, which acts on the second gear wheel attached to the line PQ in such a way as to ensure that the angle of the recording wheel is at to the horizontal as required by the theory. A direct reading of the moment can be obtained if the wheel is calibrated to take account of the factor Such a device is shown in Figure 6. The whole instrument is sho wn to the left of the ﬁgure. The top of the ﬁgure comprises a trolley, constraining the device to move horizontally. The point is below the arm to the bottom right, and the ability to move this point effectively changes the length . Notice the three wheels, together with their Vernier scales from wh ich a reading is taken. One marked is for area, the other for centre of mass and the third for moments of inertia. The details of the gear wheels are shown in the ﬁgure to the right which shows the reverse of the instrument. Other conﬁgurations were possible, such as the Hele-Shaw Integra tor which employs three glass spheres upon which the roll recording wheels run, thus eliminating i naccuracies caused by inconsistencies of the contact of the paper with the wheels. In mechanical engineering, it is common to want to ﬁnd the wor k done in each stroke of an engine. It is relatively easy to measure the instantaneous pressure in the cylinder, and by ﬁnding the area under the graph of pressure against time the work done can be calcul ated. A linear planimeter speciﬁcally for this task is that of [4]. Finding the centre of mass was a pr oblem of particular importance to navel architects, who needed to ensure that the centre of mas s of a ship was below the water line. An interesting essay on this topic is given by Robb, A. M. in [3, p g 206–217].

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Figure 6: An Amsler moment planimeter References [1] R. W. Gatterdam. The planimeter as an example of Green’s t heorem. American Mathematical Monthly , 88(9):701–704, November 1981. [2] O. Henrici. Report on planimeters. British Association for the Advancement of Science, Report of the 64th Meeting , pages 496–523, 1894. [3] E. M. Horsburgh. Napier Tercentenay Celebration: Handbook of the Exhibitio n of Napier Relics and of Books, Instuments, and Devices for facilitating Calc ulation . The Royal Society of Edin- burgh, 1914. [4] L. T. Snow. Planimeter. United States Patent No. 718166, Jan 13 1903. [5] Fr. A. Willers. Practical Analysis: Graphical and Numerical Methods . Dover, New York, 1947. [6] Fr. A. Willers. Mathematische Maschinen und Instrumente . Akademie-Verlag, 2nd edition, 1951.

JSangwinbhamacuk This article discusses moment planimeters which are mechanical devices with which is it possible to locate the centre of mass of an irregular plane shape by mec hanical and graphical methods They are a type of analogue computing dev ID: 22770

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Locating the centre of mass by mechanical means C J Sangwin School of Mathematics, University of Birmingham, Birmingham, B15 2TT, United Kingdom Telephone +44 121 414 6197, Fax +44 121 414 3389 Email: C.J.Sangwin@bham.ac.uk This article discusses moment planimeters , which are mechanical devices with which is it possible to locate the centre of mass of an irregular plane shape by mec hanical and graphical methods. They are a type of analogue computing device. In addition to this t hey may be used to ﬁnd the static moment (ﬁrst moment) and moment of inertia (second moment) o f a shape about a ﬁxed line. Moment planimeters, sometimes called integrometers or integrators , are direct developments of the planimeter which is a mechanical device used to directly measure the are a of a plane shape. While planimeters are reasonably well known, linear planimeters are less comm on than the polar planimeters of Amsler. Hence in this article we explain how planimeters work throug h the example of a linear planimeter, and then consider how these may be adapted to ﬁnd the centre of mass. More detailed comparisons between other types of area measuring planimeters may be fou nd in the comprehensive survey article of [2]. 1 Area and centre of mass Consider the region enclosed by the closed curve in Figure 1, through which we have drawn the axis. We consider the area to be split into two regions by this axis, and these regions are described by the the two functions, and . Since a general plane shape cannot be described in this way the assumption represents a considerable loss of genera lity, hence we shall provide alternative explanation in a moment. The area of this shape will be ) d x. In the linear planimeter a rigid straight line of length is constrained to move so that one end, traces around the boundary of the region. The other end, , is constrained to move along the -axis. This is shown in Figure 1. Note that ) = sin( (1) As is usual for a planimeter, we ﬁx a freely rotating disc usin g this line as an axel, an example of which is shown in Figure 3. In this arrangement the roll of the disc will be the component of the motion perpendicular to the line. If we consider an inﬁnites imally thin vertical strip of width and height , then during the horizontal motion from to + d , the wheel will record a roll of = sin( ) d x. (2)

Page 2

Figure 1: An irregular plane region, Hence ) d sin( )d w. If we denote the the total roll recorded on the wheel around th e boundary of the region from to along , and back along by δR we have that ) d δR w. This illustrates that multiplied by the total roll recorded while traces around the boundary will be equal to the area of the shape. This is the fundamental proper ty of planimeters. Next we turn attention to the centre of mass. Imagine a thin un iform strip of width , and hight The contribution this strip makes to the distance of the cent re of mass of the whole shape from the -axis will be x. And hence, , the distance of the centre of mass from the -axis will be given by RR RR (3) From this it apparent that it will be sufﬁcient to contrive a p lanimeter capable of being able to measure , since we are already capable of measuring the area. Let us assume that we can attach another wheel at which is at an angle of to the -axis. Then the roll recorded will be sin( ) = cos(2 ) = 1 2sin Considering the motion from to along the function we have, sin ) d x. If we the integrate back along , the terms in the two integrals cancel, so that δR x,

Page 3

= ( x,y B C ,y Figure 2: A small element and so δR By this procedure we have calculated and hence the line parallel to the -axis on which the centre of mass lies. We choose another line for the -axis, not parallel to the original, and repeat this procedu re. The intersection of the two lines thus obtained locates the c entre of mass. 2 Small elements In this section we take a slightly different approach and, in stead of considering integration of functions representing the boundary of the curve, we assume that the pl ane region has been decomposed into small curvy-parallelograms such as ABCD shown in Figure 2. Here, the line PQ is of ﬁxed length , the point moves around the boundary of the region and the other end runs along the -axis and so is constrained to move along ,y ) = ( 0) . We note that for ABCD , the area equals and the distance of the centre of mass of ABCD from the -axis is + d The point moves around the perimeter from , which has coordinates x,y , to and back to . In each portion of this movement we examine the roll recorde d by the two wheels considered in the previous Section and relate these to the area and centre o f mass. We consider ﬁrst a wheel using the line PQ as an axel. As this moves from to , the point is ﬁxed and the roll recorded AB is a pure roll proportional to the arc length . This is equal and opposite to that as the line moves from to , ie AB CD . As moves from to the angle PQ makes with the -axis is constant at + d with the horizontal and BC = sin( + d )d so that lw BC sin( + d )d = ( + d )d x. Similarly lw DA sin( )d x. If we deﬁne the roll around the perimeter of this small elemen t to be := AB BC CD DA then = d y.

Page 4

Figure 3: Details of the roll recording wheel on a planimeter Every reasonable plane region can be decomposed into small elements consisting of such cur vy parallelograms. When doing this the rolls along internal ed ges of this decomposition cancel leaving only the roll around the outside perimeter to consider. Henc e we have that δR Z Z y, where δR is the total roll as moves around the (piecewise smooth) boundary of the region and the right hand side is nothing but the area. The second wheel is at on an axel at an angle to the horizonal. As before, AB CD Furthermore, BC sin( 2d )d 2( + d x, and DA sin( )d + 2 x, Deﬁne, as before, := AB BC CD DA then + d y. Again, + d = y. Hence, RR 3 Green’s Theorem for the plane A justiﬁcation of the polar planimeter of Amsler was given us ing Green’s Theorem in [1]. We justify the results of the informal arguments in the previous sectio ns using a similar approach. Assume we

Page 5

have a vector ﬁeld x,y ) = ( x,y ,V x,y )) . Green’s Theorem states that δR Z Z curl( ) d y, where δR is the line integral around the (piecewise smooth) boundary of the region . Imagine a vector ﬁeld of unit vectors in the plane, which we denote by . If a wheel is attached at which is constrained to always point in the direction of this ﬁeld, the roll of the wheel will record the total component of the vector ﬁeld in the direction of the motion, e ffectively measuring this integral. If we denote the roll of the wheel by we have δR δR Z Z curl( ) d y. For the linear planimeter we imagine a vector ﬁeld generated by attaching a unit vector perpendic- ular to the end of the line PQ , of ﬁxed length , at . It remains to ﬁnd this vector ﬁeld, and the corresponding curl. As before in Figure 2, assume that when is at it has coordinates x,y and the other end runs along the -axis and so is constrained to move along ,y ) = ( 0) . Then we have = ( + ( so that and furthermore, sin( ) = and cos( ) = The planimeter vector ﬁeld, which of course does not depend o n the -coordinate, is then x,y x,y sin( cos( Since curl( ) = ∂V ∂x ∂V ∂y (4) a trivial calculation shows that curl( ) = Hence δR Z Z y. Turning attention to the centre of mass we have the vector ﬁel d where the unit vector points points at an angle of to the horizontal. Hence, x,y ) = sin( ) = 2sin 1 = 2

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Figure 4: An example of a Koizumi linear roller planimeter and x,y ) = cos( ) = 2cos( )sin( ) = 2 By (4) we have that curl( ) = Hence, by Green’s Theorem we have that δR Z Z y, as required by (3) to ﬁnd the centre of mass. These are both rather trivial applications of Green’s Theor em. 4 Further generalizations In the previous sections we have considered how to calculate and . Further general- izations naturally occur with and so on. To calculate , for example, we note that sin ) = sin( sin(3 and so it will be sufﬁcient to have an instrument with wheels c apable of recording the motion of a wheel at an angle . Further generalizations are possible, and devices along t hese lines were indeed made and used. It is the practical considerations we turn to i n the next section. 5 Practical implementations The most popular practical implementation of a planimeter i s the polar planimeter of Amsler. The essential difference between this and the linear planimete r is that the point is constrained to move in

Page 7

W Figure 5: A moment planimeter schematic a circular arc rather than a straight line. Linear planimete rs were produced commercially, an example of which is shown in Figure 4. The point can be located in the circular magnifying glass, and is constrained to move in a vertical straight line by the trol ley, rather than along the -axis as in our examples. Notice that the wheel need not actually be at , but may be at any convenient position using an axel offset from, but parallel to, the line PQ Perhaps the simplest moment planimeter is an extension of th e linear planimeter, and a schematic of such a device is given in Figure 5. The point is constrained to move in a straight line, marked as the -axis, by an arm which is mounted upon a trolley. The wheel shown is used to measure the area of the shape around which traces. At the point , ﬁxed to the trolley is a gear wheel, which acts on the second gear wheel attached to the line PQ in such a way as to ensure that the angle of the recording wheel is at to the horizontal as required by the theory. A direct reading of the moment can be obtained if the wheel is calibrated to take account of the factor Such a device is shown in Figure 6. The whole instrument is sho wn to the left of the ﬁgure. The top of the ﬁgure comprises a trolley, constraining the device to move horizontally. The point is below the arm to the bottom right, and the ability to move this point effectively changes the length . Notice the three wheels, together with their Vernier scales from wh ich a reading is taken. One marked is for area, the other for centre of mass and the third for moments of inertia. The details of the gear wheels are shown in the ﬁgure to the right which shows the reverse of the instrument. Other conﬁgurations were possible, such as the Hele-Shaw Integra tor which employs three glass spheres upon which the roll recording wheels run, thus eliminating i naccuracies caused by inconsistencies of the contact of the paper with the wheels. In mechanical engineering, it is common to want to ﬁnd the wor k done in each stroke of an engine. It is relatively easy to measure the instantaneous pressure in the cylinder, and by ﬁnding the area under the graph of pressure against time the work done can be calcul ated. A linear planimeter speciﬁcally for this task is that of [4]. Finding the centre of mass was a pr oblem of particular importance to navel architects, who needed to ensure that the centre of mas s of a ship was below the water line. An interesting essay on this topic is given by Robb, A. M. in [3, p g 206–217].

Page 8

Figure 6: An Amsler moment planimeter References [1] R. W. Gatterdam. The planimeter as an example of Green’s t heorem. American Mathematical Monthly , 88(9):701–704, November 1981. [2] O. Henrici. Report on planimeters. British Association for the Advancement of Science, Report of the 64th Meeting , pages 496–523, 1894. [3] E. M. Horsburgh. Napier Tercentenay Celebration: Handbook of the Exhibitio n of Napier Relics and of Books, Instuments, and Devices for facilitating Calc ulation . The Royal Society of Edin- burgh, 1914. [4] L. T. Snow. Planimeter. United States Patent No. 718166, Jan 13 1903. [5] Fr. A. Willers. Practical Analysis: Graphical and Numerical Methods . Dover, New York, 1947. [6] Fr. A. Willers. Mathematische Maschinen und Instrumente . Akademie-Verlag, 2nd edition, 1951.

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