# The Steiner Formula and the Polar Moment of Inertia for the Closed Planar Homothetic Motions in Complex Plane

Authors: Ayhan Tutar; Onder Sener
Journal: Advances in Mathematical Physics
 (2015)

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## Abstract

The Steiner area formula and the polar moment of inertia were expressed during one-parameter closed planar homothetic motions in complex plane. The Steiner point or Steiner normal concepts were described according to whether rotation number was different from zero or equal to zero, respectively. The moving pole point was given with its components and its relation between Steiner point or Steiner normal was specified. The sagittal motion of a winch was considered as an example. This motion was described by a double hinge consisting of the fixed control panel of winch and the moving arm of winch. The results obtained in the second section of this study were applied for this motion.


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## Body

## 1. Introduction

For a geometrical object rolling on a line and making a complete turn, some properties of the area of a path of a point were given by [1]. The Steiner area formula and the Holditch theorem during one-parameter closed planar homothetic motions were expressed by [2]. We calculated the expression of the Steiner formula relative to the moving coordinate system under one-parameter closed planar homothetic motions in complex plane. If the points of the moving plane which enclose the same area lie on a circle, then the centre of this circle is called the Steiner point (h=1) [3, 4]. If these points lie on a line, we use Steiner normal instead of Steiner point. Then we obtained the moving pole point for the closed planar homothetic motions. We dealt with the polar moment of inertia of a path generated by a closed planar homothetic motion. Furthermore, we expressed the relation between the area enclosed by a path and the polar moment of inertia. As an example, the sagittal motion of a winch which is described by a double hinge being fixed and moving was considered. The Steiner area formula, the moving pole point, and the polar moment of inertia were calculated for this motion. Moreover, the relation between the Steiner formula and the polar moment of inertia was expressed.


## 2. Closed Homothetic Motions in Complex Plane

We consider one-parameter closed planar homothetic motion between two reference systems: the fixedE′ and the moving E, with their origins O,O′ and orientations in complex plane. Then, we take into account motion relative to the fixed coordinate system (direct motion).By taking displacement vectorsOO′=U and O′O=U′ and the total angle of rotation α(t), the motion defined by the transformation(1)X′(t)=h(t)Xeiα(t)+U′(t)is called one-parameter closed planar homothetic motion and denoted by E/E′, where h is a homothetic scale of the motion E/E′ and X and X′ are the position vectors with respect to the moving and fixed rectangular coordinate systems of a point X∈E, respectively. The homothetic scale h and the vectors X′ and U,U′ are continuously differentiable functions of a real parameter t.In (1), X′(t) is the trajectory with respect to the fixed system of a point X belonging to the moving system. If we replace U′=-Ueiα(t) in (1), the motion can be written as(2)X′t=htX-Uteiα(t).The coordinates of the above equation are(3)X′t=x1′t+ix2′t,U′t=u1′t+iu2′t,X=x1+ix2,Ut=u1t+iu2t.Using these coordinates, we can write(4)x1′t+ix2′t=htx1-u1+ihtx2-u2·(cos⁡α(t)+isinα(t)).From (4), the components of X′t may be given as (5)x1′t=cos⁡αthtx1-u1-sinαthtx2-u2,x2′t=sinαthtx1-u1+cos⁡αthtx2-u2.Using the coordinates of (2) as (6)X′t=x1′tx2′t,U′t=u1′tu2′t,X=x1x2,Ut=u1tu2tand rotation matrix(7)Rt=cos⁡⁡αt-sin⁡αtsin⁡αtcos⁡⁡αt,we can obtain(8)X′t=RthtX-Ut.If we differentiate (5), we have(9)dx1′=-sinαhx1-u1dα+cos⁡αdhx1-du1-cos⁡αhx2-u2dα-sinαdhx2-du2,dx2′=cos⁡αhx1-u1dα+sinαdhx1-du1-sinαhx2-u2dα+cos⁡αdhx2-du2.


### 2.1. The Steiner Formula for the Homothetic Motions

The formula for the areaF of a closed planar curve of the point X′ is given by (10)F=12∮x1′dx2′-x2′dx1′.If (5) and (9) are placed in (10), we have (11)2F=x12+x22∮h2dα+x1∮-2hu1dα-hdu2+u2dh+x2∮-2hu2dα+hdu1-u1dh+∮u12+u22dα+u1du2-u2du1.The following expressions are used in (11):(12)∮-2hu1dα-hdu2+u2dh=a*,∮-2hu2dα+hdu1-u1dh=b*,∮u12+u22dα+u1du2-u2du1=c.The scalar term c which is related to the trajectory of the origin of the moving system may be given as follows by taking Fo:=Fx1=0,x2=0: (13)2Fo=c.The coefficient m(14)m=∮h2dα=h2t0∮dα=h2t02πνwith the rotation number ν determines whether the lines with F = const. describe circles or straight lines. If ν≠0, then we have circles. If ν=0, the circles reduce to straight lines. If (12), (13), and (14) are substituted in (11), then (15)2F-Fo=x12+x22m+a*x1+b*x2can be obtained.


#### 2.1.1. A Different Parametrization for the Integral Coefficients

Equation (8) by differentiation with respect to t yields(16)dX′=dRhX-U+RdhX-dU.If X=P=p1p2 (the pole point) is taken, (17)0=dX′=dRhP-U+RdhP-dUcan be written. Then if U=u1u2 is solved from (17),(18)u1=hp1+p2dhdα-du2dα,u2=hp2-p1dhdα+du1dαare found.If (18) is placed in (12), (19)a*=∮-2h2p1dα+∮-2hdhp2+hdu2+u2dh,b*=∮-2h2p2dα+∮2hdhp1-hdu1-u1dhcan be rewritten. Also (19) can be expressed separately as(20)a:=∮-2h2p1dα,b:=∮-2h2p2dα,(21)μ1:=∮-2hdhp2+hdu2+u2dh,μ2≔∮2hdhp1-hdu1-u1dh,μ=μ1μ2.Using (20) and (21), the area formula (22)2F-Fo=x12+x22m+ax1+bx2+μ1x1+μ2x2is found.


### 2.2. Steiner Point or Steiner Normal for the Homothetic Motions

By takingm≠0, the Steiner point S=s1,s2 for the closed planar homothetic motion can be written (23)sj=∮h2pjdα∮h2dα,j=1,2.Then (24)∮h2p1dα=s1m,∮h2p2dα=s2mis found. If (24) is placed in (20) and by considering (22), (25)2F-Fo=mx12+x22-2s1x1-2s2x2+μ1x1+μ2x2is obtained. Equation (25) is called the Steiner area formula for the closed planar homothetic motion.By dividing this bym and by completing the squares, one obtains the equation of a circle(26)x1-s1-μ12m2+x2-s2-μ22m2-s1-μ12m2-s2-μ22m2=2F-F0m.All the fixed points of the moving plane which pass around equal orbit areas under the motion E/E′ lie on the same circle with the center(27)M=s1-μ12m,s2-μ22min the moving plane.In the case ofht=1, since μ1=μ2=0, the point M and the Steiner point S coincide [3]. Also by taking m=0, if it is replaced in (22), then we have(28)a+μ1x1+b+μ2x2-2F-F0=0.Equation (28) is a straight line. If no complete loop occurs, then η=0 and the circles are reduced to straight lines, in other words, to a circle whose center lies at infinity. The normal to the lines of equal areas in (28) is given by(29)n=a+μ1b+μ2which is called the Steiner normal [5].


### 2.3. The Moving Pole Point for the Homothetic Motions

Using (18), if P=p1p2 is solved, then the pole point P of the motion(30)p1=dh(du1-u2dα)+hdα(du2+u1dα)dh2+h2dα2,p2=dh(du2+u1dα)-hdα(du1-u2dα)dh2+h2dα2is obtained.Form≠0, using (14) and (23), we arrive at the relation in (24) between the Steiner point and the pole point.Form=0, using (20) and (29), we arrive at the relation between the Steiner normal and the pole point as follows:(31)ab=-2∮h2p1dα-2∮h2p2dα=n-μ.


### 2.4. The Polar Moments of Inertia for the Homothetic Motions

The polar moments of inertia “T” symbolize a path for closed homothetic motions. We find a formula by using T, m, and n in this section and we arrive at the relation between the polar moments of inertia “T” and the formula of area “F” (see (37)). A relation between the Steiner formula and the polar moment of inertia around the pole for a moment was given by [6]. Müller [3] also demonstrated a relation to the polar moment of inertia around the origin, while Tölke [7] inspected the same relation for closed functions and Kuruoğlu et al. [8] generalized Müller’s results for homothetic motion.If we useα as a parameter, we need to calculate(32)T=∮x1′2+x2′2dαalong the path of X. Then, using (5),(33)T=x12+x22m+x1∮-2hu1dα+x2∮-2hu2dα+∮u12+u22dαis obtained.We need to calculate the polar moments of inertia of the origin of the moving system; thereforeTo=Tx1=0,x2=0; one obtains(34)To=∮u12+u22dα.If (34) is placed in (33),(35)T-To=x12+x22m+x1∮-2hu1dα+x2∮-2hu2dαcan be written. Also if (18) is placed in (35),(36)T-To=x12+x22m+x1∮-2h2p1dα-2hdhp2+2hdu2+x2∮-2h2p2dα+2hdhp1-2hdu1is obtained and by considering (22) and (36) together, we arrive at the relation between the polar moments of inertia and the formula for the area below:(37)T-To=2F-Fo+x1∮hdu2-u2dhhhhhhhh+x2∮-hdu1+u1dh.


## 2.1. The Steiner Formula for the Homothetic Motions

The formula for the areaF of a closed planar curve of the point X′ is given by (10)F=12∮x1′dx2′-x2′dx1′.If (5) and (9) are placed in (10), we have (11)2F=x12+x22∮h2dα+x1∮-2hu1dα-hdu2+u2dh+x2∮-2hu2dα+hdu1-u1dh+∮u12+u22dα+u1du2-u2du1.The following expressions are used in (11):(12)∮-2hu1dα-hdu2+u2dh=a*,∮-2hu2dα+hdu1-u1dh=b*,∮u12+u22dα+u1du2-u2du1=c.The scalar term c which is related to the trajectory of the origin of the moving system may be given as follows by taking Fo:=Fx1=0,x2=0: (13)2Fo=c.The coefficient m(14)m=∮h2dα=h2t0∮dα=h2t02πνwith the rotation number ν determines whether the lines with F = const. describe circles or straight lines. If ν≠0, then we have circles. If ν=0, the circles reduce to straight lines. If (12), (13), and (14) are substituted in (11), then (15)2F-Fo=x12+x22m+a*x1+b*x2can be obtained.


### 2.1.1. A Different Parametrization for the Integral Coefficients

Equation (8) by differentiation with respect to t yields(16)dX′=dRhX-U+RdhX-dU.If X=P=p1p2 (the pole point) is taken, (17)0=dX′=dRhP-U+RdhP-dUcan be written. Then if U=u1u2 is solved from (17),(18)u1=hp1+p2dhdα-du2dα,u2=hp2-p1dhdα+du1dαare found.If (18) is placed in (12), (19)a*=∮-2h2p1dα+∮-2hdhp2+hdu2+u2dh,b*=∮-2h2p2dα+∮2hdhp1-hdu1-u1dhcan be rewritten. Also (19) can be expressed separately as(20)a:=∮-2h2p1dα,b:=∮-2h2p2dα,(21)μ1:=∮-2hdhp2+hdu2+u2dh,μ2≔∮2hdhp1-hdu1-u1dh,μ=μ1μ2.Using (20) and (21), the area formula (22)2F-Fo=x12+x22m+ax1+bx2+μ1x1+μ2x2is found.


## 2.1.1. A Different Parametrization for the Integral Coefficients

Equation (8) by differentiation with respect to t yields(16)dX′=dRhX-U+RdhX-dU.If X=P=p1p2 (the pole point) is taken, (17)0=dX′=dRhP-U+RdhP-dUcan be written. Then if U=u1u2 is solved from (17),(18)u1=hp1+p2dhdα-du2dα,u2=hp2-p1dhdα+du1dαare found.If (18) is placed in (12), (19)a*=∮-2h2p1dα+∮-2hdhp2+hdu2+u2dh,b*=∮-2h2p2dα+∮2hdhp1-hdu1-u1dhcan be rewritten. Also (19) can be expressed separately as(20)a:=∮-2h2p1dα,b:=∮-2h2p2dα,(21)μ1:=∮-2hdhp2+hdu2+u2dh,μ2≔∮2hdhp1-hdu1-u1dh,μ=μ1μ2.Using (20) and (21), the area formula (22)2F-Fo=x12+x22m+ax1+bx2+μ1x1+μ2x2is found.


## 2.2. Steiner Point or Steiner Normal for the Homothetic Motions

By takingm≠0, the Steiner point S=s1,s2 for the closed planar homothetic motion can be written (23)sj=∮h2pjdα∮h2dα,j=1,2.Then (24)∮h2p1dα=s1m,∮h2p2dα=s2mis found. If (24) is placed in (20) and by considering (22), (25)2F-Fo=mx12+x22-2s1x1-2s2x2+μ1x1+μ2x2is obtained. Equation (25) is called the Steiner area formula for the closed planar homothetic motion.By dividing this bym and by completing the squares, one obtains the equation of a circle(26)x1-s1-μ12m2+x2-s2-μ22m2-s1-μ12m2-s2-μ22m2=2F-F0m.All the fixed points of the moving plane which pass around equal orbit areas under the motion E/E′ lie on the same circle with the center(27)M=s1-μ12m,s2-μ22min the moving plane.In the case ofht=1, since μ1=μ2=0, the point M and the Steiner point S coincide [3]. Also by taking m=0, if it is replaced in (22), then we have(28)a+μ1x1+b+μ2x2-2F-F0=0.Equation (28) is a straight line. If no complete loop occurs, then η=0 and the circles are reduced to straight lines, in other words, to a circle whose center lies at infinity. The normal to the lines of equal areas in (28) is given by(29)n=a+μ1b+μ2which is called the Steiner normal [5].


## 2.3. The Moving Pole Point for the Homothetic Motions

Using (18), if P=p1p2 is solved, then the pole point P of the motion(30)p1=dh(du1-u2dα)+hdα(du2+u1dα)dh2+h2dα2,p2=dh(du2+u1dα)-hdα(du1-u2dα)dh2+h2dα2is obtained.Form≠0, using (14) and (23), we arrive at the relation in (24) between the Steiner point and the pole point.Form=0, using (20) and (29), we arrive at the relation between the Steiner normal and the pole point as follows:(31)ab=-2∮h2p1dα-2∮h2p2dα=n-μ.


## 2.4. The Polar Moments of Inertia for the Homothetic Motions

The polar moments of inertia “T” symbolize a path for closed homothetic motions. We find a formula by using T, m, and n in this section and we arrive at the relation between the polar moments of inertia “T” and the formula of area “F” (see (37)). A relation between the Steiner formula and the polar moment of inertia around the pole for a moment was given by [6]. Müller [3] also demonstrated a relation to the polar moment of inertia around the origin, while Tölke [7] inspected the same relation for closed functions and Kuruoğlu et al. [8] generalized Müller’s results for homothetic motion.If we useα as a parameter, we need to calculate(32)T=∮x1′2+x2′2dαalong the path of X. Then, using (5),(33)T=x12+x22m+x1∮-2hu1dα+x2∮-2hu2dα+∮u12+u22dαis obtained.We need to calculate the polar moments of inertia of the origin of the moving system; thereforeTo=Tx1=0,x2=0; one obtains(34)To=∮u12+u22dα.If (34) is placed in (33),(35)T-To=x12+x22m+x1∮-2hu1dα+x2∮-2hu2dαcan be written. Also if (18) is placed in (35),(36)T-To=x12+x22m+x1∮-2h2p1dα-2hdhp2+2hdu2+x2∮-2h2p2dα+2hdhp1-2hdu1is obtained and by considering (22) and (36) together, we arrive at the relation between the polar moments of inertia and the formula for the area below:(37)T-To=2F-Fo+x1∮hdu2-u2dhhhhhhhh+x2∮-hdu1+u1dh.


## 3. Application: The Motion of the Winch

In the previous sections we emphasized three concepts: geometrical objects as the Steiner point or the Steiner normal, the pole point, and the polar moments of inertia for closed homothetic motions in complex plane. In this section, we want to visualize the experimentally measured motion with these objects. Accordingly, we consider these characteristic directions for this motion.We will show how the kinematical objects which are used in the previous sections can be applied. In the study by Dathe and Gezzi [5], they considered human gait in planar motions. As an example, we have chosen the sagittal part of the movement of the winch at motion. We have chosen the winch, because the arm of winch can extend or retract during one-parameter closed planar homothetic motion. The motion of winch has a double hinge and “a double hinge” means that it has two systems, a fixed arm and a moving arm of winch (Figure 1). There is a control panel of winch at the origin of fixed system. “L” arm can extend or retract by h parameter.Figure 1
The arms of winch as a double hinge.


### 3.1. The Mathematical Model

We start by writing the equations of the double hinge in Cartesian coordinates. Then we define, using the conditionm=0, the Steiner normal and the total angle in relation to the double hinge.By taking displacement vectorsOO′=U and O′O=U′ and the total angle of rotation l-k=α, the motion can be defined by the transformation(38)X′(t)=h(t)Xei(l(t)-k(t))+U′(t).By taking (39)Rt=cos⁡⁡lt-kt-sin⁡lt-ktsin⁡lt-ktcos⁡⁡lt-kt,U′t=Lcos⁡ltLsinlt,we have(40)X′t=htRtX+U′t.Also we know that U′=-RU. Therefore, (41)Ut=u1tu2t=-Lcos⁡kt-Lsinktcan be written. So the double hinge may be written as(42)x1′t=cos⁡l(t)-k(t)htx1+Lcos⁡k-sinl(t)-k(t)htx2+Lsink,x2′t=sinl(t)-k(t)htx1+Lcos⁡k+cos⁡l(t)-k(t)htx2+Lsink.We begin by calculating the time derivative of (42). In this way, we obtain the velocities x1′˙(t),x2′˙(t), which have to be inserted into (10):(43)x1′x2′˙-x2′x1′˙=h2x12+x22+L2(l˙(t)-k˙(t))+x12hLcos⁡ktl˙t-k˙thhhhhhhhhh+hLcos⁡(k(t))k˙(t)-Ldhsin(k(t))+x22hLsinktl˙t-k˙thhhhhhhhhl+hLsin(k(t))k˙(t)+Ldhcos⁡(k(t))+L2k˙(t).We now integrate the previous equation using periodic boundary conditions by assuming the integrands as periodic functions. The periodicity of f implies that integrals of the following types vanish, ∮df=∫1Ff˙dt=f1F=0. As a result of this, some of the integrals of (43) are not equal to zero and we finally obtain a simplified expression for the area (44)2F=x1∫t1t22Lhcos⁡kl˙-k˙dthhhhh+∫t1t2Lhcos⁡k·k˙-dhsinkdt+x2∫t1t22Lhsinkl˙-k˙dthhhhhh+∫t1t2Lhsink·k˙+dhcos⁡kdt.We may have the following expressions from (44): (45)∫t1t22Lhcos⁡kl˙-k˙dt+∫t1t2Lhcos⁡k·k˙-dhsinkdt=a*,∫t1t22Lhsinkl˙-k˙dt+∫t1t2Lhsink·k˙+dhcos⁡kdt=b*.Differentiating (41) with respect to t and then using the result in (45), we obtain (12) for application.In Section2.1.1, using (18),(46)a*=∫t1t2-2h2p1dα︸a+∫t1t2-2hdhp2+hdu2+u2dh︸μ1,b*=∫t1t2-2h2p2dα︸b+∫t1t2-2hdhp1+hdu1+u1dh︸μ2are found and we have a straight line below:(47)2F=a+μ1x1+b+μ2x2.In this case, we have the Steiner normal(48)n=a+μ1b+μ2=L∫t1t22hcos⁡k(l˙-k˙)+hcos⁡k·k˙-dhsinkdt∫t1t22hsink(l˙-k˙)+hsink·k˙+dhcos⁡kdt.


### 3.2. The Moving Pole Point of the Winch Motion

If (41) is replaced in (30), the pole point P=p1p2 with the components (49)p1=dhLsink·l˙-hl˙-k˙Lcos⁡k·l˙dh2+h2l˙-k˙2,p2=dh-Lcos⁡k·l˙-hl˙-k˙Lsink·l˙dh2+h2l˙-k˙2is obtained and(50)P=p1p2=Ll˙dh2+h2l˙-k˙2dhsink-hl˙-k˙cos⁡k-dhcos⁡k-hl˙-k˙sinkcan be written. Also using (46) and (48), we reach the relation between the Steiner normal and the pole point (31).


### 3.3. The Polar Moments of Inertia of the Winch Motion

Using (32) and (42), if (41) is replaced in (33),(51)T=x1∮2hLcos⁡k(l˙-k˙)dt+x2∮2hLsink(l˙-k˙)dtis obtained. By considering (46), (47), and (51) together, we arrive at the relation between the polar moments of inertia and the formula for the area below:(52)T=2F+x1L∮-hcos⁡k·k˙+dhsink-x2L∮hsink·k˙+dhcos⁡k.


## 3.1. The Mathematical Model

We start by writing the equations of the double hinge in Cartesian coordinates. Then we define, using the conditionm=0, the Steiner normal and the total angle in relation to the double hinge.By taking displacement vectorsOO′=U and O′O=U′ and the total angle of rotation l-k=α, the motion can be defined by the transformation(38)X′(t)=h(t)Xei(l(t)-k(t))+U′(t).By taking (39)Rt=cos⁡⁡lt-kt-sin⁡lt-ktsin⁡lt-ktcos⁡⁡lt-kt,U′t=Lcos⁡ltLsinlt,we have(40)X′t=htRtX+U′t.Also we know that U′=-RU. Therefore, (41)Ut=u1tu2t=-Lcos⁡kt-Lsinktcan be written. So the double hinge may be written as(42)x1′t=cos⁡l(t)-k(t)htx1+Lcos⁡k-sinl(t)-k(t)htx2+Lsink,x2′t=sinl(t)-k(t)htx1+Lcos⁡k+cos⁡l(t)-k(t)htx2+Lsink.We begin by calculating the time derivative of (42). In this way, we obtain the velocities x1′˙(t),x2′˙(t), which have to be inserted into (10):(43)x1′x2′˙-x2′x1′˙=h2x12+x22+L2(l˙(t)-k˙(t))+x12hLcos⁡ktl˙t-k˙thhhhhhhhhh+hLcos⁡(k(t))k˙(t)-Ldhsin(k(t))+x22hLsinktl˙t-k˙thhhhhhhhhl+hLsin(k(t))k˙(t)+Ldhcos⁡(k(t))+L2k˙(t).We now integrate the previous equation using periodic boundary conditions by assuming the integrands as periodic functions. The periodicity of f implies that integrals of the following types vanish, ∮df=∫1Ff˙dt=f1F=0. As a result of this, some of the integrals of (43) are not equal to zero and we finally obtain a simplified expression for the area (44)2F=x1∫t1t22Lhcos⁡kl˙-k˙dthhhhh+∫t1t2Lhcos⁡k·k˙-dhsinkdt+x2∫t1t22Lhsinkl˙-k˙dthhhhhh+∫t1t2Lhsink·k˙+dhcos⁡kdt.We may have the following expressions from (44): (45)∫t1t22Lhcos⁡kl˙-k˙dt+∫t1t2Lhcos⁡k·k˙-dhsinkdt=a*,∫t1t22Lhsinkl˙-k˙dt+∫t1t2Lhsink·k˙+dhcos⁡kdt=b*.Differentiating (41) with respect to t and then using the result in (45), we obtain (12) for application.In Section2.1.1, using (18),(46)a*=∫t1t2-2h2p1dα︸a+∫t1t2-2hdhp2+hdu2+u2dh︸μ1,b*=∫t1t2-2h2p2dα︸b+∫t1t2-2hdhp1+hdu1+u1dh︸μ2are found and we have a straight line below:(47)2F=a+μ1x1+b+μ2x2.In this case, we have the Steiner normal(48)n=a+μ1b+μ2=L∫t1t22hcos⁡k(l˙-k˙)+hcos⁡k·k˙-dhsinkdt∫t1t22hsink(l˙-k˙)+hsink·k˙+dhcos⁡kdt.


## 3.2. The Moving Pole Point of the Winch Motion

If (41) is replaced in (30), the pole point P=p1p2 with the components (49)p1=dhLsink·l˙-hl˙-k˙Lcos⁡k·l˙dh2+h2l˙-k˙2,p2=dh-Lcos⁡k·l˙-hl˙-k˙Lsink·l˙dh2+h2l˙-k˙2is obtained and(50)P=p1p2=Ll˙dh2+h2l˙-k˙2dhsink-hl˙-k˙cos⁡k-dhcos⁡k-hl˙-k˙sinkcan be written. Also using (46) and (48), we reach the relation between the Steiner normal and the pole point (31).


## 3.3. The Polar Moments of Inertia of the Winch Motion

Using (32) and (42), if (41) is replaced in (33),(51)T=x1∮2hLcos⁡k(l˙-k˙)dt+x2∮2hLsink(l˙-k˙)dtis obtained. By considering (46), (47), and (51) together, we arrive at the relation between the polar moments of inertia and the formula for the area below:(52)T=2F+x1L∮-hcos⁡k·k˙+dhsink-x2L∮hsink·k˙+dhcos⁡k.



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*Source: 978294-2015-09-01.xml*