Any object, whether living or nonliving, can be considered as being composed of a large number of small particles, and for each particle with mass m and a vanishingly small volume dV, Newton's laws of motion hold. Linear momentum of a system of particles is defined as the sum of the products of the mass of each particle with its velocity:
in which mi is the mass of particle i and vi is its velocity. Newton characterized linear momentum as the quantity of motion. Writing Newton's second and third law for each particle in a system of particles and summing over all particles in the system, one can show that dL/dt = SF*
where SF* denotes the sum of all external forces acting on the system of particles. According to this equation, the time rate of change of linear mo mentum is equal to the sum of external forces. It is known as the equation for conservation of linear momentum. Gravitational force and the forces that arise as the result of contact of particles in the system with the particles outside the system are external forces. Newton's third law requires that forces that act between particles in the system under study do not contribute to the change of linear momentum. Such forces are called internal forces.
The center of mass of a system of particles is defined by the relation:
in which rc denotes the position of center of mass with respect to a Cartesian reference frame and ri is the position vector of particle i. The center of mass is not necessarily occupied by any particle in the system of particles. Using this definition in the equation for the conservation of linear momentum, one obtains an equation governing the position of the center of mass as a function of time:
2Fi = (2mi) ac in which ac is the acceleration of the center of mass.
The term moment of momentum about a point fixed on earth is defined by the following equation:
Ho = 2 ri/o X mivi in which ri/o denotes the position vector from the stationary point O to the particle i. The conservation of moment of momentum dictates that dHo/dt = 2ri/o X Fi
Again, in this equation, Fi represents the external force i acting on the ith particle of the system. Conservation of moment of momentum about the center of mass is governed by an equation of the same form:
Hc = 2 ri/c X mivi and ri/c is the position vector from the center of mass to the particle i.
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The use of dumbbells gives you a much more comprehensive strengthening effect because the workout engages your stabilizer muscles, in addition to the muscle you may be pin-pointing. Without all of the belts and artificial stabilizers of a machine, you also engage your core muscles, which are your body's natural stabilizers.