Pirouette effect

In figure skating, the pirouette effect is used to achieve a quick rotation around the body axis.

The pirouette effect is the increase or decrease in the speed of rotation caused by the Coriolis effect , which occurs when the mass of a rotating object is pulled closer to or further away from the axis of rotation . The effect can be experienced in everyday life in the eponymous pirouette in figure skating . Figure skaters first set themselves in rotation with their arms outstretched to the side. Keeping the arms close to the body reduces the runners' moment of inertia . Since the angular momentum is retained, the angular velocity of the rotation increases. Conversely, the speed of rotation decreases when the arms are stretched out. ${\displaystyle \Theta }$ ${\displaystyle L=\Theta \cdot \omega }$ ${\displaystyle \omega }$

Gymnasts and high divers use the same principle when doing somersaults or twists . In the air, arms and legs are drawn in, in order to gain the fastest possible turn from the angular momentum obtained when jumping. Opening the posture before hitting the ground reduces the turning speed and allows a standing landing.

The pirouette effect also occurs with other rotary movements, for example with a tornado in the development phase. In the case of a supernova , the interior of the star collapses, the resulting neutron star then has rotation times in the millisecond range.

Physical basics of the pirouette effect

Physical size	Formula symbol	SI units
Angular momentum	${\displaystyle L}$	N · m · s, kg · m 2 / s
speed	${\displaystyle v}$	m / s
Dimensions	${\displaystyle m}$	kg
Angular velocity	${\displaystyle {\omega }}$	rad / s
Radius of gyration	${\displaystyle i}$	m
Rotational energy	${\displaystyle W}$	N m, kg m 2 / s 2
Lifting work	${\displaystyle W_{\mathrm {Diff} }}$	N m, kg m 2 / s 2

Mass point that changes from a circular path with a radius to a path with a larger radius . ${\displaystyle i_{1}}$ ${\displaystyle i_{2}}$

The angular momentum can be expressed as the product of the radius of gyration , mass and angular velocity : ${\displaystyle {\vec {L}}}$ ${\displaystyle i}$ ${\displaystyle m}$ ${\displaystyle {\vec {\omega }}}$

${\displaystyle {\vec {L}}=i^{2}\,m\,{\vec {\omega }}}$

Due to the conservation of angular momentum, the following applies to a system without external influences and unchanged mass, where the indices and denote two states of the system: ${\displaystyle 1}$ ${\displaystyle 2}$

${\displaystyle L_{1}=m\cdot i_{1}^{2}\cdot \omega _{1}=m\cdot i_{2}^{2}\cdot \omega _{2}=L_{2}}$

This means that the angular velocities are inversely proportional to the squares of the radii of gyration:

${\displaystyle {\frac {i_{1}^{2}}{i_{2}^{2}}}={\frac {\omega _{2}}{\omega _{1}}}}$

For point masses at a distance , the circumferential speed can be used instead of the angular speed, so that: ${\displaystyle i}$ ${\displaystyle v_{1,2}=i_{1,2}\cdot \omega _{1,2}}$

${\displaystyle {\frac {i_{1}}{i_{2}}}={\frac {v_{2}}{v_{1}}}.}$

Since the rotational energies is, applies ${\displaystyle \textstyle W={\frac {m}{2}}\cdot v^{2}}$

${\displaystyle {\frac {W_{1}}{W_{2}}}={\frac {\omega _{1}}{\omega _{2}}}={\frac {i_{2}^{2}}{i_{1}^{2}}}.}$

With constant mass , if, for example, the radii of inertia and one of the angular velocities are known, the other angular velocity, the rotational energies and the lifting work can be calculated using the above formula . ${\displaystyle m_{1}=m_{2}}$ ${\displaystyle W_{\mathrm {Diff} }=W_{1}-W_{2}}$

Determination of the lifting work as the product of force and distance. The centrifugal force is integrated by changing the radius . ${\displaystyle W_{\mathrm {Diff} }}$ ${\displaystyle F_{\mathrm {Zf} }=m\omega ^{2}i}$ ${\displaystyle i}$

The lifting work can also be determined directly:

${\displaystyle {\begin{aligned}W_{\mathrm {Diff} }&={\frac {m}{2}}\cdot \left(\omega _{1}^{2}\cdot i_{1}^{2}-\omega _{2}^{2}\cdot i_{2}^{2}\right)\\&={\frac {m}{2}}\cdot \omega _{1}^{2}\cdot i_{1}^{2}\left(1-{\frac {i_{1}^{2}}{i_{2}^{2}}}\right)\end{aligned}}}$

The pirouette effect is an interplay between lifting energy and rotational energy. The difference between the rotational energies is the lifting work that can be converted back into rotational energy when changing to a smaller radius; d. that is, reducing the radius requires increasing effort over the distance . When the radius is increased, the energy bound in the rotation is released. ${\displaystyle i_{2}-i_{1}}$

The radius of gyration of a mass point is its distance from the axis of rotation. If there are several mass points, this effective distance is determined by adding up the contributions of all masses with their respective radii : ${\displaystyle m_{l}}$ ${\displaystyle r_{l}}$

${\displaystyle i^{2}\sum _{l}m_{l}=\sum _{l}r_{l}^{2}\cdot m_{l}}$

For rigid bodies that do not rotate about a main axis of inertia , which is generally the case for mass points that interact with one another and are not in a plane perpendicular to the axis of rotation, conservation of angular momentum must be

${\displaystyle {\begin{aligned}{\vec {L}}_{1}&={\vec {L}}_{2}\\\mathbf {\Theta } _{1}{\vec {\omega }}_{1}&=\mathbf {\Theta } _{2}{\vec {\omega }}_{2}\end{aligned}}}$

with the inertia tensors and are assumed. ${\displaystyle \mathbf {\Theta } _{1}}$ ${\displaystyle \mathbf {\Theta } _{2}}$

Trigonometric explanation

Speed ​​and energy calculation with trigonometric functions

The rotating mass point is offset from the radius to the radius , for example by extending the connection with the pivot point. The mass moves tangentially in a straight line to the outer path. It takes the speed and the rotational energy from the inner radius as kinetic energy up to the radius : ${\displaystyle i_{1}}$ ${\displaystyle i_{2}}$ ${\displaystyle v_{1}}$ ${\displaystyle i_{2}}$

${\displaystyle W_{1}={\frac {1}{2}}m\,v_{1}^{2}}$

On the outer radius, the speed can be broken down into the components and . is the new peripheral speed, and is the imaginary radial speed, which, however, because it is on the new radius = zero, can be converted into lifting work. ${\displaystyle v_{1}}$ ${\displaystyle v_{2}}$ ${\displaystyle v_{\mathrm {R} }}$ ${\displaystyle v_{2}}$ ${\displaystyle v_{\mathrm {R} }}$

The tags used in the picture

Physical size	formula
	${\displaystyle \cos(\alpha )={\sqrt {1-(i_{1}/i_{2})^{2}}}}$
Peripheral speed 2	${\displaystyle v_{2}=\sin(\alpha )\,v_{1}}$
Radial vector	${\displaystyle v_{\mathrm {R} }=\cos(\alpha )\,v_{1}}$
Rotational energy difference	${\displaystyle W_{\mathrm {Diff} }={\frac {1}{2}}m\,v_{\mathrm {R} }^{2}}$

The type of transition to another radius is irrelevant for the final state. In practice the movement will be spiral, but the result is that the values ​​for energy and speed correspond to the simplified example.

Individual evidence

1. ↑ Sports mechanics, section “Angular momentum and conservation of angular momentum”, Fig. 70 on p. 78.

literature

• G. Bäumler: Sports Mechanics: Basics for Study and Practice . BLV Verlagsgesellschaft, Munich Vienna Zurich 1981, ISBN 3-405-12435-2 .
• D. Halliday, R. Resnick, J. Walker: Physics . 2nd Edition. WILEY-VCH Verlag GmbH & co. KGaA, Berlin 2009, ISBN 978-3-527-40645-6 .