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In this paper, during one-parameter closed planar homothetic direct motions, the formula of kinetic energy is expressed. Then we show the relation between the formula of kinetic energy and the Steiner formula. We investigate some properties of closed planar homothetic motions. These motions appear between two coordinate systems, fixed and moving (direct motion). Finally, we show how the results can be applied to experimentally measured motions. As an example, we consider a motion of winch in the sagittal direction. We obtain the formula of kinetic energy for the motion of winch during one-parameter closed planar homothetic direct motions.

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 [

Dathe H. and Gezzi R. expressed the formula of kinetic energy for the closed planar kinematics [

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 [

We consider one parameter closed planar homothetic motion between two reference systems: the fixed

By taking displacement vectors

is called one-parameter closed planar homothetic motion and denoted by

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

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

sumptions, we obtain

Equation (13) of [

If Equation (13) of [

is found.

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

and

Finally, if Equations (10), (14), (15) and (18) of [

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 (

By considering Equation (36) of [

and if we calculate the time derivative of this,

are found.

We must calculate

Also we use Equation (35) of [

If we calculate the time derivative of this,

are found. Then if Equation (35) of [

Section 2, by using the parameter

If Equation (13) of [

is found.

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

We consider Equations (40), (41) of [

If Equations (40) and (41) of [

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