In vector format, the moment can be defined as the cross product between the radius vector, "'r "'( the vector from point O to the line of action ), and the force vector, "'F " ':
12.
If the torque is defined as the z-component of this cross product, then the torque is simply " rF " perp where " F " perp is the force perpendicular to the radius vector, or tangential to the instantaneous velocity of the blade ( See figure below)
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In Newton's Philosophi?Naturalis Principia Mathematica, he demonstrated that the gravitational force between two mass points is inversely proportional to the square of the distance between the points, and he fully solved the corresponding " two-body problem " demonstrating that the radius vector between the two points would describe an ellipse.
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"' Problem 2 "'explores the case of an ellipse, where the center of attraction is at its center, and finds that the centripetal force to produce motion in that configuration would be directly proportional to the radius vector . ( This material becomes Proposition 10, Problem 5 in the " Principia " .)
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Johannes Kepler had proved by an elaborate series of measurements that each planet revolves in an foci of the orbit, that the radius vector of each planet drawn from the sun sweeps out equal areas in equal times, and that the squares of the periodic times of the planets are in the same proportion as the cubes of their mean distances from the sun.
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At the apapsis and periapsis, the positions of closest and furthest distance from the attracting center, the velocity and radius vectors are perpendicular; therefore, the angular momentum " L 1 " per mass " m " of the particle ( written as " h " 1 ) can be related to the rate of sweeping out areas
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M ( r ) is the mass contained within a distance r from the center of the stellar system; R is the projected distance of a star from the apparent center; V _ R and V _ T are the components of a star's velocity parallel to, and perpendicular to, the apparent radius vector; and G is the gravitational constant.
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Which equals the speed " v " times " r " ?", the component of the radius vector perpendicular to the velocity . " h " is the magnitude of the specific angular momentum because it equals the magnitude " L " of the angular momentum divided by the mass " m " of the particle.
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NOTE : This page uses common physics notation for spherical coordinates, in which \ theta is the angle between the " z " axis and the radius vector connecting the origin to the point in question, while \ phi is the angle between the projection of the radius vector onto the " x-y " plane and the " x " axis.
20.
NOTE : This page uses common physics notation for spherical coordinates, in which \ theta is the angle between the " z " axis and the radius vector connecting the origin to the point in question, while \ phi is the angle between the projection of the radius vector onto the " x-y " plane and the " x " axis.
