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In particle accelerators, velocity can be very high (close to the speed of light in vacuum) so the same rest mass now exerts greater inertia (relativistic mass) thereby requiring greater force for the same centripetal acceleration, so the equation becomes: [11] = where = is the Lorentz factor.
Velocity and acceleration in non-uniform circular motion. In non-uniform circular motion, an object moves in a circular path with varying speed. Since the speed is changing, there is tangential acceleration in addition to normal acceleration. The net acceleration is directed towards the interior of the circle (but does not pass through its center).
Transverse acceleration (perpendicular to velocity) causes a change in direction. If it is constant in magnitude and changing in direction with the velocity, circular motion ensues. Taking two derivatives of the particle's coordinates concerning time gives the centripetal acceleration
Newton illustrates his formula with three examples. In the first two, the central force is a power law, F(r) = r n−3, so C(r) is proportional to r n. The formula above indicates that the angular motion is multiplied by a factor k = 1/ √ n, so that the apsidal angle α equals 180°/ √ n.
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.
Consequently, the acceleration is the second derivative of position, [7] often written . Position, when thought of as a displacement from an origin point, is a vector: a quantity with both magnitude and direction. [9]: 1 Velocity and acceleration are vector quantities as well. The mathematical tools of vector algebra provide the means to ...
In a two-body rotation, such as a planet and moon rotating about their common center of mass or barycentre, the forces on both bodies are centripetal. In that case, the reaction to the centripetal force of the planet on the moon is the centripetal force of the moon on the planet. [6]
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]