Physics Study Guide/Gravity

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Newtonian Gravity

Newtonian Gravity (simplified gravitation) is an apparent force (a.k.a. pseudoforce) that simulates the attraction of one mass to another mass. Unlike the three fundamental (real) forces of electromagnetism and the strong and weak nuclear forces, gravity is purely attractive. As a force it is measured in newtons. The distance between two objects is measured between their centers of mass.

$F=G\!\cdot \!{\frac {m_{1}m_{2}}{r^{2}}}$

Gravitational force is equal to the product of the universal gravitational constant and the masses of the two objects, divided by the square of the distance between their centers of mass.

$g=G\cdot {\frac {m_{1}}{r^{2}}}$

The value of the gravitational field which is equivalent to the acceleration due to gravity caused by an object at a point in space is equal to the first equation about gravitational force, with the effect of the second mass taken out.

$U=-G\cdot {\frac {Mm}{r}}$

Gravitational potential energy of a body to infinity is equal to the universal gravitational constant times the mass of a body from which the gravitational field is being created times the mass of the body whose potential energy is being measured over the distance between the two centers of mass. Therefore, the difference in potential energy between two points is the difference of the potential energy from the position of the center of mass to infinity at both points. Near the earth's surface, this approximates:

$\Delta U_{g}=mgh$

Potential energy due to gravity near the earth's surface is equal to the product of mass, acceleration due to gravity, and height (elevation) of the object.

If the potential energy from the body's center of mass to infinity is known, however, it is possible to calculate the escape velocity, or the velocity necessary to escape the gravitational field of an object. This can be derived based on utilizing the law of conservation of energy and the equation to calculate kinetic energy as follows:

${\begin{aligned}&{\boldsymbol {ke}}_{initial}=\Delta U\\\\&{\boldsymbol {ke}}_{initial}=U_{infinity}-U_{initial}\\\\&{\tfrac {1}{2}}mv^{2}=G\cdot {\frac {Mm}{r}}\\\\&v_{esc}={\sqrt {\frac {2GM}{r}}}\end{aligned}}$

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Variables

F: force (N) G: universal constant of gravitation, (6.67x10-11 N•m2/kg2) m1: mass of the first body m2: mass of the second body r: the distance between the point at which the force or field is being taken, and the center of mass of the first body g: acceleration due to gravity (on the earth’s surface, this is 9.8 m/s2) U: potential energy from the location of the center of mass to infinity (J) ΔUg: Change in potential energy (J) m and M: mass (kg) h: height of elevation (m) vesc: escape velocity (m/s)

Definition of terms

Universal constant of gravitation (G): This is a constant that is the same everywhere in the known universe and can be used to calculate gravitational attraction and acceleration due to gravity. 6.67x10-11 N·m2/kg2 Mass one (m1): One of two masses that are experiencing a mutual gravitational attraction. We can use this for the mass of the Earth (1023 kg). Mass two (m2): One of two masses that are experiencing a mutual gravitational attraction. This symbol can represent the mass of an object on or close to earth. Units: kilograms (kg) Acceleration due to gravity (g): This is nearly constant near the earth's surface because the mass and radius of the earth are essentially constant. At extreme altitudes the value can vary slightly, but it varies more significantly with latitude. This is also equal to the value of the gravitational field caused by a body at a particular point in space (9.8 m/s2) Escape velocity (vesc): The velocity necessary to completely escape the gravitational effects of a body.

A black hole is a geometrically defined region of space time exhibiting such large centripetal gravitational effects that nothing such as particles and electromagnetic radiation such as light may escape from inside of it. That is the escape velocity upon the event horizon is equivalent to the speed of light. General relativity is a metric theory of gravitation generalizing space time and Newton's law of universal gravitational attraction as a geometric property of space time.