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The areal velocity magnitude (i.e., the areal speed) is this region's area divided by the time interval Δt in the limit that Δt becomes vanishingly small. The vector direction is postulated to be normal to the plane containing the position and velocity vectors of the particle, following a convention known as the right hand rule .
A set of equations describing the trajectories of objects subject to a constant gravitational force under normal Earth-bound conditions.Assuming constant acceleration g due to Earth's gravity, Newton's law of universal gravitation simplifies to F = mg, where F is the force exerted on a mass m by the Earth's gravitational field of strength g.
A common misconception occurs between centre of mass and centre of gravity.They are defined in similar ways but are not exactly the same quantity. Centre of mass is the mathematical description of placing all the mass in the region considered to one position, centre of gravity is a real physical quantity, the point of a body where the gravitational force acts.
Since the speed v is likewise unchanging, the areal velocity 1 ⁄ 2 vr ⊥ is a constant of motion; the particle sweeps out equal areas in equal times. The area A of a circular sector equals 1 ⁄ 2 r 2 φ = 1 ⁄ 2 r 2 ωt = 1 ⁄ 2 r v φ t. Hence, the areal velocity dA/dt equals 1 ⁄ 2 r v φ = 1 ⁄ 2 h.
The equation for universal gravitation thus takes the form: F = G m 1 m 2 r 2 , {\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}},} where F is the gravitational force acting between two objects, m 1 and m 2 are the masses of the objects, r is the distance between the centers of their masses , and G is the gravitational constant .
At a fixed point on the surface, the magnitude of Earth's gravity results from combined effect of gravitation and the centrifugal force from Earth's rotation. [2] [3] At different points on Earth's surface, the free fall acceleration ranges from 9.764 to 9.834 m/s 2 (32.03 to 32.26 ft/s 2), [4] depending on altitude, latitude, and longitude.
Newton illustrates his formula with three examples. In the first two, the central force is a power law, F(r) = r n−3, so C(r) is proportional to r n. The formula above indicates that the angular motion is multiplied by a factor k = 1/ √ n, so that the apsidal angle α equals 180°/ √ n.
This is the International Gravity Formula 1967, the 1967 Geodetic Reference System Formula, Helmert's equation or Clairaut's formula. [ 18 ] An alternative formula for g as a function of latitude is the WGS ( World Geodetic System ) 84 Ellipsoidal Gravity Formula : [ 19 ]