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In this paper, the kinetic energy formula was expressed during one-parameter closed planar homothetic inverse motions in complex plane. Then the relation between the kinetic energy formula and the Steiner formula was given. As an example the sagittal motion of a telescopic crane was considered. This motion was described by a double hinge consisting of the fixed control panel of telescopic crane and the moving arm of telescopic crane. The results were applied to experimentally measured motion.

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 [

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

is called one-parameter closed planar homothetic direct motion in complex plane.

By taking displacement vector

is called one-parameter closed planar homothetic inverse motion in complex plane and denoted by

In Equation (1),

If we consider the below coordinates of Equation (1)

we can write

From Equation (2), the components of

If we show the coordinates of the Equation (1)

and the rotation matrice

we can obtain

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

Using Equation (7) we can calculate the equation

If Equation (9) is replaced in Equation (8),

is obtained.

If

If Equation (11) is replaced in Equation (10),

can be written.

For

If Equation (14) of [

is obtained.

Now we consider the case in which the motion is closed and naturally parametrized. Then, it follows

If we consider the equations

and

and Equations (9), (11) and (12) of [

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

The motion of telescopic crane has a double hinge and “a double hinge” means that it has two systems a fixed arm and a moving arm of telescopic crane (

If we calculate the time derivative of the equation

we obtain

We must calculate

periodic functions. The periodicity of f implies that integrals of the following types vanish

If we calculate the time derivative of the equation

we have

Then if Equation (36) of [

is obtained.

If we consider the equations

and

and Equations (41) and (42) of [

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