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In this case, the three-acceleration vector is perpendicular to the three-velocity vector, = and the square of proper acceleration, expressed as a scalar invariant, the same in all reference frames, = + (), becomes the expression for circular motion, =. or, taking the positive square root and using the three-acceleration, we arrive at the ...
The formula for the acceleration A P can now be obtained as: = ˙ + + (), or = / + / +, where α is the angular acceleration vector obtained from the derivative of the angular velocity vector; / =, is the relative position vector (the position of P relative to the origin O of the moving frame M); and = ¨ is the acceleration of the origin of ...
are called the tangential acceleration and the normal or radial acceleration (or centripetal acceleration in circular motion, see also circular motion and centripetal force), respectively. Geometrical analysis of three-dimensional space curves, which explains tangent, (principal) normal and binormal, is described by the Frenet–Serret formulas ...
There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
The radial acceleration (perpendicular to direction of motion) is given by = =. It is directed towards the center of the rotational motion, and is often called the centripetal acceleration . The angular acceleration is caused by the torque , which can have a positive or negative value in accordance with the convention of positive and negative ...
The radial component can be observed due to the Doppler effect, the tangential component causes visible changes of the position of the object. In polar coordinates , a two-dimensional velocity is described by a radial velocity , defined as the component of velocity away from or toward the origin, and a transverse velocity , perpendicular to the ...
One radian is defined as the angle at the center of a circle in a plane that subtends an arc whose length equals the radius of the circle. [6] More generally, the magnitude in radians of a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, =, where θ is the magnitude in radians of the subtended angle, s is arc length, and r is radius.
For a constant mass, force equals mass times acceleration (=). For every action, there is an equal and opposite reaction. (In other words, whenever one body exerts a force F → {\displaystyle {\vec {F}}} onto a second body, (in some cases, which is standing still) the second body exerts the force − F → {\displaystyle -{\vec {F}}} back onto ...