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Newton showed that the force acting on the second particle equals the force F 1 (r) acting on the first particle, plus an inverse-cube central force [30] = + where L 1 is the magnitude of the first particle's angular momentum. If k 2 is greater than one, F 2 −F 1 is a negative number; thus, the added inverse-cube force is attractive.
When Newton's laws are applied to rotating extended bodies, they lead to new quantities that are analogous to those invoked in the original laws. The analogue of mass is the moment of inertia, the counterpart of momentum is angular momentum, and the counterpart of force is torque. Angular momentum is calculated with respect to a reference point ...
A single force acting at any point O′ of a rigid body can be replaced by an equal and parallel force F acting at any given point O and a couple with forces parallel to F whose moment is M = Fd, d being the separation of O and O′. Conversely, a couple and a force in the plane of the couple can be replaced by a single force, appropriately ...
F = total force acting on the center of mass m = mass of the body I 3 = the 3×3 identity matrix a cm = acceleration of the center of mass v cm = velocity of the center of mass τ = total torque acting about the center of mass I cm = moment of inertia about the center of mass ω = angular velocity of the body α = angular acceleration of the body
Example: A model airplane of mass 1 kg accelerates from rest to a velocity of 6 m/s due north in 2 s. The net force required to produce this acceleration is 3 newtons due north. The change in momentum is 6 kg⋅m/s due north. The rate of change of momentum is 3 (kg⋅m/s)/s due north which is numerically equivalent to 3 newtons.
The moment of force, or torque, is a first moment: =, or, more generally, .; Similarly, angular momentum is the 1st moment of momentum: =.Momentum itself is not a moment.; The electric dipole moment is also a 1st moment: = for two opposite point charges or () for a distributed charge with charge density ().
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Cauchy observed that the force across an imaginary surface was a linear function of its normal vector; and, moreover, that it must be a symmetric function (with zero total momentum). The understanding of stress in liquids started with Newton, who provided a differential formula for friction forces (shear stress) in parallel laminar flow.