The **tennis racket theorem** or **intermediate axis theorem** is a result in classical mechanics describing the movement of a rigid body with three distinct principal moments of inertia. It is also dubbed the **Dzhanibekov effect**, after Russian cosmonaut Vladimir Dzhanibekov who noticed one of the theorem's logical consequences while in space in 1985^{ [1] } although the effect was already known for at least 150 years before that.^{ [2] }^{ [3] }

- Theory
- Stable rotation around the first and third principal axis
- Unstable rotation around the second principal axis
- See also
- References
- External links

The theorem describes the following effect: rotation of an object around its first and third principal axes is stable, while rotation around its second principal axis (or intermediate axis) is not.

This can be demonstrated with the following experiment: hold a tennis racket at its handle, with its face being horizontal, and try to throw it in the air so that it will perform a full rotation around the horizontal axis perpendicular to the handle, and try to catch the handle. In almost all cases, during that rotation the face will also have completed a half rotation, so that the other face is now up. By contrast, it is easy to throw the racket so that it will rotate around the handle axis (the third principal axis) without accompanying half-rotation around another axis; it is also possible to make it rotate around the vertical axis perpendicular to the handle (the first principal axis) without any accompanying half-rotation.

The experiment can be performed with any object that has three different moments of inertia, for instance with a book, remote control or smartphone. The effect occurs whenever the axis of rotation differs only slightly from the object's second principal axis; air resistance or gravity are not necessary.^{ [4] }

The tennis racket theorem can be qualitatively analysed with the help of Euler's equations. Under torque–free conditions, they take the following form:

Here denote the object's principal moments of inertia, and we assume . The angular velocities around the object's three principal axes are and their time derivatives are denoted by .

Consider the situation when the object is rotating around axis with moment of inertia . To determine the nature of equilibrium, assume small initial angular velocities along the other two axes. As a result, according to equation (1), is very small. Therefore, the time dependence of may be neglected.

Now, differentiating equation (2) and substituting from equation (3),

because and .

Note that is being opposed and so rotation around this axis is stable for the object.

Similar reasoning gives that rotation around axis with moment of inertia is also stable.

Now apply the same analysis to axis with moment of inertia This time is very small. Therefore, the time dependence of may be neglected.

Now, differentiating equation (1) and substituting from equation (3),

Note that is *not* opposed (and therefore will grow) and so rotation around the second axis is *unstable*. Therefore, even a small disturbance along other axes causes the object to 'flip'.

In physics, **angular momentum** is the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity—the total angular momentum of a closed system remains constant.

In physics, the **Coriolis force** is an inertial or fictitious force that acts on objects that are in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the left of the motion of the object. In one with anticlockwise rotation, the force acts to the right. Deflection of an object due to the Coriolis force is called the **Coriolis effect**. Though recognized previously by others, the mathematical expression for the Coriolis force appeared in an 1835 paper by French scientist Gaspard-Gustave de Coriolis, in connection with the theory of water wheels. Early in the 20th century, the term *Coriolis force* began to be used in connection with meteorology.

**Nutation** is a rocking, swaying, or nodding motion in the axis of rotation of a largely axially symmetric object, such as a gyroscope, planet, or bullet in flight, or as an intended behaviour of a mechanism. In an appropriate reference frame it can be defined as a change in the second Euler angle. If it is not caused by forces external to the body, it is called *free nutation* or *Euler nutation*. A *pure nutation* is a movement of a rotational axis such that the first Euler angle is constant. In spacecraft dynamics, precession is sometimes referred to as nutation.

**Precession** is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In other words, if the axis of rotation of a body is itself rotating about a second axis, that body is said to be precessing about the second axis. A motion in which the second Euler angle changes is called *nutation*. In physics, there are two types of precession: torque-free and torque-induced.

**Torque** is the rotational equivalent of linear force. It is also referred to as the **moment**, **moment of force**, **rotational force** or **turning effect**, depending on the field of study. The concept originated with the studies by Archimedes of the usage of levers. Just as a linear force is a push or a pull, a torque can be thought of as a twist to an object around a specific axis. Another definition of torque is the product of the magnitude of the force and the perpendicular distance of the line of action of force from the axis of rotation. The symbol for torque is typically , the lowercase Greek letter *tau*. When being referred to as moment of force, it is commonly denoted by *M*.

A **gyrocompass** is a type of non-magnetic compass which is based on a fast-spinning disc and the rotation of the Earth to find geographical direction automatically. The use of a gyrocompass is one of the seven fundamental ways to determine the heading of a vehicle. Although one important component of a gyrocompass is a gyroscope, these are not the same devices; a gyrocompass is built to use the effect of gyroscopic precession, which is a distinctive aspect of the general gyroscopic effect. Gyrocompasses are widely used for navigation on ships, because they have two significant advantages over magnetic compasses:

**Kinematics** is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics.

In physics, **angular velocity** refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angular position or orientation of an object changes with time. There are two types of angular velocity: orbital angular velocity and spin angular velocity. Spin angular velocity refers to how fast a rigid body rotates with respect to its centre of rotation. Orbital angular velocity refers to how fast a point object revolves about a fixed origin, i.e. the time rate of change of its angular position relative to the origin. Spin angular velocity is independent of the choice of origin, in contrast to orbital angular velocity which depends on the choice of origin.

The **moment of inertia**, otherwise known as the **mass moment of inertia, angular mass** or **rotational inertia**, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis; similar to how mass determines the force needed for a desired acceleration. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rate of rotation.

In the physical science of dynamics, **rigid-body dynamics** studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are *rigid* simplifies analysis, by reducing the parameters that describe the configuration of the system to the translation and rotation of reference frames attached to each body. This excludes bodies that display fluid, highly elastic, and plastic behavior.

In physics, **circular motion** is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations of motion describe the movement of the center of mass of a body.

**Dynamical simulation**, in computational physics, is the simulation of systems of objects that are free to move, usually in three dimensions according to Newton's laws of dynamics, or approximations thereof. Dynamical simulation is used in computer animation to assist animators to produce realistic motion, in industrial design, and in video games. Body movement is calculated using time integration methods.

In classical mechanics, **Euler's rotation equations** are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with its axes fixed to the body and parallel to the body's principal axes of inertia. Their general form is:

In fluid mechanics, the **Taylor–Proudman theorem** states that when a solid body is moved slowly within a fluid that is steadily rotated with a high angular velocity , the fluid velocity will be uniform along any line parallel to the axis of rotation. must be large compared to the movement of the solid body in order to make the Coriolis force large compared to the acceleration terms.

The **rigid rotor** is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rigid rotor is the *linear rotor* requiring only two angles to describe, for example of a diatomic molecule. More general molecules are 3-dimensional, such as water, ammonia, or methane.

In applied mechanics, **bending** characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element.

**Rotation around a fixed axis** or **about a fixed axis of revolution** or **motion with respect to a fixed axis of rotation** is a special case of rotational motion. The fixed-axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible. If two rotations are forced at the same time, a new axis of rotation will appear.

In classical mechanics, **Poinsot's construction** is a geometrical method for visualizing the torque-free motion of a rotating rigid body, that is, the motion of a rigid body on which no external forces are acting. This motion has four constants: the kinetic energy of the body and the three components of the angular momentum, expressed with respect to an inertial laboratory frame. The angular velocity vector of the rigid rotor is *not constant*, but satisfies Euler's equations. Without explicitly solving these equations, Louis Poinsot was able to visualize the motion of the endpoint of the angular velocity vector. To this end he used the conservation of kinetic energy and angular momentum as constraints on the motion of the angular velocity vector . If the rigid rotor is symmetric, the vector describes a cone. This is the torque-free precession of the rotation axis of the rotor.

The details of a spinning body may impose restrictions on the motion of its angular velocity vector, **ω**. The curve produced by the angular velocity vector on the inertia ellipsoid, is known as the **polhode**, coined from Greek meaning "path of the pole". The surface created by the angular velocity vector is termed the **body cone**.

In Newtonian mechanics, the **centrifugal force** is an inertial force that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is parallel to the axis of rotation and passing through the coordinate system's origin. If the axis of rotation passes through the coordinate system's origin, the centrifugal force is directed radially outwards from that axis. The magnitude of centrifugal force *F* on an object of mass *m* at the distance *r* from the origin of a frame of reference rotating with angular velocity *ω* is:

- ↑ Эффект Джанибекова (гайка Джанибекова), 23 July 2009 (in Russian). The software can be downloaded from here
- ↑ Poinsot (1834)
*Theorie Nouvelle de la Rotation des Corps*, Bachelier, Paris - ↑ Derek Muller (September 19, 2019).
*The Bizarre Behavior of Rotating Bodies, Explained*. Veritasium. Retrieved February 16, 2020. - ↑ Levi, Mark (2014).
*Classical Mechanics with Calculus of Variations and Optimal Control: An Intuitive Introduction*. American Mathematical Society. pp. 151–152. ISBN 9781470414443.

- Dan Russell (5 March 2010). "Slow motion Dzhanibekov effect demonstration with table tennis rackets" . Retrieved 2 February 2017– via YouTube.
- zapadlovsky (16 June 2010). "Dzhanibekov effect demonstration" . Retrieved 2 February 2017– via YouTube. on Mir International Space Station
- Viacheslav Mezentsev (7 September 2011). "Djanibekov effect modeled in Mathcad 14" . Retrieved 2 February 2017– via YouTube.
- Louis Poinsot, Théorie nouvelle de la rotation des corps, Paris, Bachelier, 1834, 170 p. OCLC 457954839 : historically, the first mathematical description of this effect.
- "Ellipsoids and The Bizarre Behaviour of Rotating Bodies". - intuitive video explanation by Matt Parker

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