Jacob Steiner established some properties of the area of a path of a point for a geometrical object rolling on a line and making a complete turn. In planar kinematics, the Steiner formula describes the dependence of the area of a curve, determined by a closed motion of one point, on the position of this point [1] . The Steiner area formula and the Holditch theorem during one-parameter closed planar homothetic motions were expressed by A. Tutar and N. Kuruoglu [2] . During one-parameter closed planar homothetic motions, the expression of the Steiner formula was calculated relative to fixed coordinate system. The points of the fixed plane which enclose the same area lie on a circle or a line in the fixed coordinate system. If it is a circle, then the center of this circle is called the Steiner point (h = 1) [3] [4] .

Dathe H. and Gezzi R. expressed the formula of kinetic energy for the closed planar kinematics [5] . In our previous paper [6] , we introduced a case distinction between movements with or without a vanishing winding number, which was important for practical purposes in mechanics. In this paper, we calculate the formula of kinetic energy for closed planar homothetic direct motions. In particular, we show the relation between the formula of kinetic energy and the Steiner formula.

As an example, Dathe H. and Gezzi R. have chosen the sagittal part of the movement of the human leg during walking for planar kinematics [7] . Since our intention is not simply to recapitulate some classical results, we consider their practical application. We wish to use them in order to characterize experimental data such as the ones related to the motion of winch. For elaboration, we consider the sagittal motion of a winch which is described by a double hinge being fixed and moving as an example. We obtain the formula of kinetic energy for the motion of winch.

We consider one parameter closed planar homothetic motion between two reference systems: the fixed and the moving, with their origins and orientations. Then, we take into account motion relative to the fixed coordinate system (direct motion).

By taking displacement vectors and, the total angle of rotation, the motion defined by the transformation

is called one-parameter closed planar homothetic motion and denoted by, where is a homothetic scale of the motion, and are the position vectors with respect to the moving and fixed rectangular coordinate systems of a point, respectively. The homothetic scale and the vectors and are continuously differentiable functions of a real parameter.

With the coordinates

and rotation matrice

Equation (1) reads components

From Equation (3), by differentiation with respect to t, we have

A moment with a first order in the time derivatives can be introduced by

which is the integral over the kinetic energy of a point with mass M = 1.

So, we can calculate this equation using Equation (4)

If Equation (6) is replaced in Equation (5),

is found.

If is taken, then, for the formula of kinetic energy of the origin point we have

If Equation (8) is replaced in Equation (7),

can be written.

Now we consider the case in which the motion is closed and naturally parametrized. Other cases will be dis-

cussed in an another publication. Then, it follows and With those as-

sumptions, we obtain

Equation (13) of [6] :

If Equation (13) of [6] is respectively replaced at coefficients of and in Equation (10) and by calculating necessary operations,

is found.

We consider Equations (10), (14), (15) and (18) of [6] , namely,

and

Finally, if Equations (10), (14), (15) and (18) of [6] are replaced in Equation (11),

is arrived at the relation between the formula of kinetic energy and the formula for the area.

The motion of winch has a double hinge and “a double hinge” is mean 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 moving system. “L” arm can extend or retract by h parameter. Also we define using the condition (Equation (10) of [6] ), the Steiner line and the total angle in relation to the double hinge. So we must use it for this section.

By considering Equation (36) of [6] ,

and if we calculate the time derivative of this,

are found.

We must calculate for the formula of kinetic energy in Equation (5)

Also we use Equation (35) of [6]

If we calculate the time derivative of this,

are found. Then if Equation (35) of [6] and Equation (15) are replaced in calculating data and in

Section 2, by using the parameter and:

If Equation (13) of [6] is replaced in Equation (17),

is found.

Now we can construct Equation (18) as the formula of area.

We consider Equations (40), (41) of [6] , namely,

If Equations (40) and (41) of [6] are replaced in Equation (19),

is arrived at the relation between the formula of kinetic energy and the area formula for application.