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m = mass of object g = acceleration due to gravity θ = angle of elevation of the plane, measured from the horizontal. The frictionless plane is a concept from the writings of Galileo Galilei. In his 1638 The Two New Sciences, [1] Galileo presented a formula that predicted the motion of an object moving down an inclined plane.
The dimensionless added mass coefficient is the added mass divided by the displaced fluid mass – i.e. divided by the fluid density times the volume of the body. In general, the added mass is a second-order tensor, relating the fluid acceleration vector to the resulting force vector on the body. [1]
It also depends on the distribution of the mass: distributing the mass further from the center of rotation increases the moment of inertia by a greater degree. For a single particle of mass m {\displaystyle m} a distance r {\displaystyle r} from the axis of rotation, the moment of inertia is given by I = m r 2 . {\displaystyle I=mr^{2}.}
In classical mechanics, for a body with constant mass, the (vector) acceleration of the body's center of mass is proportional to the net force vector (i.e. sum of all forces) acting on it (Newton's second law): = =, where F is the net force acting on the body, m is the mass of the body, and a is the center-of-mass acceleration.
The gravitational acceleration vector depends only on how massive the field source is and on the distance 'r' to the sample mass . It does not depend on the magnitude of the small sample mass. This model represents the "far-field" gravitational acceleration associated with a massive body.
A space vehicle's flight is determined by application of Newton's second law of motion: =, where F is the vector sum of all forces exerted on the vehicle, m is its current mass, and a is the acceleration vector, the instantaneous rate of change of velocity (v), which in turn is the instantaneous rate of change of displacement.
The theorem tells us how different parts of the mass distribution affect the gravitational force measured at a point located a distance r 0 from the center of the mass distribution: [13] The portion of the mass that is located at radii r < r 0 causes the same force at the radius r 0 as if all of the mass enclosed within a sphere of radius r 0 ...
This includes Newton's law of universal gravitation, and the relation between gravitational potential and field acceleration. d 2 R / dt 2 and F / m are both equal to the gravitational acceleration g (equivalent to the inertial acceleration, so same mathematical form, but also defined as gravitational force per unit mass [8 ...