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An example is the calculation of the rotational kinetic energy of the Earth. As the Earth has a sidereal rotation period of 23.93 hours, it has an angular velocity of 7.29 × 10 −5 rad·s −1. [2] The Earth has a moment of inertia, I = 8.04 × 10 37 kg·m 2. [3] Therefore, it has a rotational kinetic energy of 2.14 × 10 29 J.
E t is the translational kinetic energy; E r is the rotational energy or angular kinetic energy in the rest frame; Thus the kinetic energy of a tennis ball in flight is the kinetic energy due to its rotation, plus the kinetic energy due to its translation.
kinetic energy: joule (J) wave vector: radian per meter (m −1) Boltzmann constant: joule per kelvin (J/K) wavenumber: radian per meter (m −1) stiffness: newton per meter (N⋅m −1) ^ Cartesian z-axis basis unit vector unitless angular momentum
The trivial case of the angular momentum of a body in an orbit is given by = where is the mass of the orbiting object, is the orbit's frequency and is the orbit's radius.. The angular momentum of a uniform rigid sphere rotating around its axis, instead, is given by = where is the sphere's mass, is the frequency of rotation and is the sphere's radius.
An arbitrarily shaped rigid rotor is a rigid body of arbitrary shape with its center of mass fixed (or in uniform rectilinear motion) in field-free space R 3, so that its energy consists only of rotational kinetic energy (and possibly constant translational energy that can be ignored).
It follows from the work–energy principle that W also represents the change in the rotational kinetic energy E r of the body, given by =, where I is the moment of inertia of the body and ω is its angular speed. [13] Power is the work per unit time, given by
The second term is the rotational term akin to the kinetic energy of the rigid rotor. Here P α {\displaystyle {\mathcal {P}}_{\alpha }} is the α component of the body-fixed rigid rotor angular momentum operator , see this article for its expression in terms of Euler angles .
Rotational or angular kinematics is the description of the rotation of an object. [21] In what follows, attention is restricted to simple rotation about an axis of fixed orientation. The z -axis has been chosen for convenience.