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In the simplest case the speed, mass, and radius are constant. Consider a body of one kilogram, moving in a circle of radius one metre, with an angular velocity of one radian per second. The speed is 1 metre per second. The inward acceleration is 1 metre per square second, v 2 /r.
Planck units modified so that 8 π G = 1 are known as reduced Planck units, because the Planck mass is divided by √ 8 π. Also, the Bekenstein–Hawking formula for the entropy of a black hole simplifies to S BH = ( m BH ) 2 /2 = 2 π A BH .
Therefore, as the body accumulates matter at a given fixed density (in this example, 997 kg/m 3, the density of water), its Schwarzschild radius will increase more quickly than its physical radius. When a body of this density has grown to around 136 million solar masses (1.36 × 10 8 M ☉ ), its physical radius would be overtaken by its ...
The γ factor approaches infinity as v approaches c, and it would take an infinite amount of energy to accelerate an object with mass to the speed of light. The speed of light is the upper limit for the speeds of objects with positive rest mass, and individual photons cannot travel faster than the speed of light. [39]
For astronomical bodies other than Earth, and for short distances of fall at other than "ground" level, g in the above equations may be replaced by (+) where G is the gravitational constant, M is the mass of the astronomical body, m is the mass of the falling body, and r is the radius from the falling object to the center of the astronomical body.
If a first body of mass m A is placed at a distance r (center of mass to center of mass) from a second body of mass m B, each body is subject to an attractive force F g = Gm A m B /r 2, where G = 6.67 × 10 −11 N⋅kg −2 ⋅m 2 is the "universal gravitational constant". This is sometimes referred to as gravitational mass.
The formula defines the energy E of a particle in its rest frame as the product of mass (m) with the speed of light squared (c 2). Because the speed of light is a large number in everyday units (approximately 300 000 km/s or 186 000 mi/s), the formula implies that a small amount of mass corresponds to an enormous amount of energy.
These proportionalities may be expressed by the formula: where g is the surface gravity of an object, expressed as a multiple of the Earth's, m is its mass, expressed as a multiple of the Earth's mass (5.976 × 10 24 kg) and r its radius, expressed as a multiple of the Earth's (mean) radius (6,371 km). [9]