The power P exerted by a force F on a particle of mass m moving with velocity v is defined as the dot product of the force and the velocity. Power is a scalar quantity. The power of the resultant force acting on a particle can be shown to be equal to the time rate of change of kinetic energy of the particle:
P = F • v = dT/dt in which T = (1/2) mv2 is the kinetic energy and v is the speed of the particle. Kinetic energy of a rigid body in planar motion is given by the expression:
in which vc is the velocity of the center of mass, Ic is the mass moment of inertia about the center of mass, and w is the angular speed. When a rigid body rotates around a fixed point O, this equation reduces to the form:
in which Io is the mass moment of inertia with respect to point O.
The power of a force acting on a rigid body is equal to the dot product of the force F and the velocity v of the point of its application. If the point has zero velocity or its velocity is perpendicular to the force, then the force creates no power. The power of the resultant force and the resultant couple acting on a rigid body is equal to the time rate of change of kinetic energy:
P = (2F) • vc + (2MC) • m = dT/dt where 2F denotes the resultant force acting on the object and (2Mc) is the resultant moment acting on the body about the center of mass.
The change of kinetic energy of a rigid body is equal to the work done on it by the external forces and moments:
in which T1 and T2 denote the kinetic energy of the body at times t1 and t2 and W is the work done on the body by external forces and couples.
The change of kinetic energy of a tree of rigid bodies is equal to the work done on it by the external forces and moments plus the work done by the joint moments.
in which P is as usual the power produced by the external forces and couples acting on the tree of rigid bodies, Mi-j denotes the moment exerted by body j on body i at their joint, and Mj is the angular velocity of body j.
The work done by gravity on a body B in going from configuration 1 to configuration 2 can be written as
in which h2 and h1 represent the vertical distance from the center of mass to some arbitrarily chosen datum plane in configurations 2 and 1, respectively.
The work done by a spring in going from configuration 1 to configuration 2 can be written as
in which k is the stiffness of the spring, and x2 and x1 are the extensions of the spring at configurations 2 and 1, respectively.
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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.