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In relativity, the COM frame exists for an isolated massive system.This is a consequence of Noether's theorem.In the COM frame the total energy of the system is the rest energy, and this quantity (when divided by the factor c 2, where c is the speed of light) gives the invariant mass of the system:
where u is the velocity of the ejected/accreted mass as seen in the object's rest frame. [17] This is distinct from v, which is the velocity of the object itself as seen in an inertial frame. This equation is derived by keeping track of both the momentum of the object as well as the momentum of the ejected/accreted mass (dm).
The moment of inertia of an object, symbolized by , is a measure of the object's resistance to changes to its rotation. The moment of inertia is measured in kilogram metre² (kg m 2). It depends on the object's mass: increasing the mass of an object increases the moment of inertia.
One may instead change to a coordinate frame fixed in the rotating body, in which the moment of inertia tensor is constant. Using a reference frame such as that at the center of mass, the frame's position drops out of the equations. In any rotating reference frame, the time derivative must be replaced so that the equation becomes
Since linear motion is a motion in a single dimension, the distance traveled by an object in particular direction is the same as displacement. [4] The SI unit of displacement is the metre . [ 5 ] [ 6 ] If x 1 {\displaystyle x_{1}} is the initial position of an object and x 2 {\displaystyle x_{2}} is the final position, then mathematically the ...
Proper acceleration at any speed is the physical acceleration experienced locally by an object. In spacetime it is a three-vector acceleration with respect to the object's instantaneously varying free-float frame. [13] Its magnitude α is the frame-invariant magnitude of that object's four-acceleration. Proper acceleration is also useful from ...
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 experimental method to locate the three-dimensional coordinates of the center of mass begins by supporting the object at three points and measuring the forces, F 1, F 2, and F 3 that resist the weight of the object, = ^ (^ is the unit vector in the vertical direction).