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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.
This illustrates that kinetic energy is also stored in rotational motion. Several mathematical descriptions of kinetic energy exist that describe it in the appropriate physical situation. For objects and processes in common human experience, the formula 1 / 2 mv 2 given by classical mechanics is suitable.
The kinetic and potential energies still change as the system evolves, but the motion of the system will be such that their sum, the total energy, is constant. This is a valuable simplification, since the energy E is a constant of integration that counts as an arbitrary constant for the problem, and it may be possible to integrate the ...
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 angular velocity ω whose axes are fixed to the body. They are named in honour of Leonhard Euler. Their general vector form is
Kinematics is used in astrophysics to describe the motion of celestial bodies and collections of such bodies. In mechanical engineering, robotics, and biomechanics, [7] kinematics is used to describe the motion of systems composed of joined parts (multi-link systems) such as an engine, a robotic arm or the human skeleton.
The previous equations for planar motion can be used here: corollaries of momentum, angular momentum etc. can immediately follow by applying the above definitions. For any object moving in any path in a plane, = = ^ the following general results apply to the particle.
which illustrates the kinetic energy is in general a function of the generalized velocities, coordinates, and time if the constraints also vary with time, so T = T(q, dq/dt, t). In the case the constraints on the particles are time-independent, then all partial derivatives with respect to time are zero, and the kinetic energy is a homogeneous ...
Here, the yellow ellipsoid is the angular momentum ellipsoid, and the expanding blue ellipsoid is the energy ellipsoid. During motion, both the energy and angular momentum-squared are conserved, thus we have two conserved quantities: {= = and so for any initial condition (), the trajectory of () must stay on the intersection curve between two ...