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Euler's second law states that the rate of change of angular momentum L about a point that is fixed in an inertial reference frame (often the center of mass of the body), is equal to the sum of the external moments of force acting on that body M about that point: [1] [4] [5] =.
and so the cross product arises, see time derivative in rotating reference frame. The vector components of the torque in the inertial and the rotating frames are related by =, where is the rotation tensor (not rotation matrix), an orthogonal tensor related to the angular velocity vector by = ˙ for any vector u.
where x' is the position as seen by a reference frame that is moving at speed, v, in the "unprimed" (x) reference frame. [ note 3 ] Taking the differential of the first of the two equations above, we have, d x ′ = d x − v d t {\displaystyle dx'=dx-v\,dt} , and what may seem like the obvious [ note 4 ] statement that d t ′ = d t ...
where is the vector sum of the physical forces acting on the object, is the mass of the object, and is the acceleration of the object relative to the inertial reference frame. Transforming this equation to a reference frame rotating about a fixed axis through the origin with angular velocity ω {\displaystyle {\boldsymbol {\omega }}} having ...
In three dimensions, the torque is a pseudovector; for point particles, it is given by the cross product of the displacement vector and the force vector. The direction of the torque can be determined by using the right hand grip rule : if the fingers of the right hand are curled from the direction of the lever arm to the direction of the force ...
In physics and engineering, a resultant force is the single force and associated torque obtained by combining a system of forces and torques acting on a rigid body via vector addition. The defining feature of a resultant force, or resultant force-torque, is that it has the same effect on the rigid body as the original system of forces. [ 1 ]
Classical mechanics is the branch of physics used to describe the motion of macroscopic objects. [1] It is the most familiar of the theories of physics. The concepts it covers, such as mass, acceleration, and force, are commonly used and known. [2] The subject is based upon a three-dimensional Euclidean space with
The vector is the position vector of the force application point, and in this example it is drawn from the center of mass as the reference point of (see diagram). The straight line segment k {\displaystyle k} is the lever arm of the force F {\displaystyle \mathbf {F} } with respect to the center of mass.