Chapter 9
Gravity
Gravitational Lens
A gravitational lens occurs when the gravitational pull of a very massive object, such as a galaxy, bends the light of an object behind it. In this photo, the bright yellowish object in the center of the image is a massive galaxy. Another galaxy is hidden behind it. The nearer, visible galaxy bends the light of the hidden galaxy and makes its light appear as blue streaks around the central galaxy.
Photo Researchers, Inc./Space Telescope Science Institute/NASA/Science Photo Library
Microsoft ® Encarta ® 2006. © 1993-2005 Microsoft Corporation. All rights reserved.
One arc second equals 1/3600th of a degree. One milli-arc second is 1000 times smaller than that.
Constant force (Newton) N
W= (ƒ cos ө)s kg-m/s2
Work – energy theorem wnet = KEƒ – KEI
Gravitational potential energy PE ≡ mgy
Average power/force on moving α =w/Δt =F¯V
Average angular speed ≡ө2-ө1/t2-t1 = Δө/Δt
Centripetal force fc=mac=m VE2/R
Law of gravitation F=G m1m2/r2
The gravitation force exerted by a spherical mass on a particle outside the sphere F=G mEm/RE2
The second condition for equilibrium asserts in an object is in rotational equilibrium, external torque ∑T=0, ∑F=0 (external force) ∑r=0, ∑F=0, ∑Fx=0, ∑Fy=0,∑Fz=0, up to 7 fields
The center of gravity Yeg = ∑miy1/∑mi
Angular momentum τ=1α=1(w-we/Δt)=1w-1we/Δt
Mass-energy equivalence E=m0C2/√1-v2/c2
Kinetic energy KE= mc2-m0c2
Average centripetal acceleration ac=v2/r
Centripetal force ac=vt2/r, FC=mac=vt2/r
Gravitational force g=G me/r2
Keplers third law, the ratio of the square of each (planet) T2=(4π2/Gms)r3=ksr3
Rotational kinetic energy ½ 1 w2
Gravitational potential energy PEg
Translational kinetic energy KEt
Rotational kinetic energy KEr (KE1+KEr+PEg)I = (KEt+KEr+PEg)g
Change in entropy Δs≡ΔQr/T, Δs=change between two equilibrium states, ΔQr = heat transferred, T = absolute temperature.
Gravitational force (Newton’s law) Fg=G memp/r2, Inertia Fnet=∑F = ma
Acceleration due to gravity g 09.81 ms-2
Gravity fields can bend light.
The factors of gravity are; Constant force, work energy theorem, gravitational potential energy, average force on moving wave, average angular speed, centripetal force and the laws of gravitation. The gravitational force exerted by a spherical mass on a particle out side the sphere is F=gMeM/Re2. The second condition for equilibrium asserts that an object is in rotational equilibrium, external torque up to 7 fields. Plus considering the center of gravity, angular momentum, the mass-energy equilance, the kinetic energy, keeping in mind Kepler’s third law which is based on the ratio of the square of each object. Also the translational kinetic energy, change in entropy, inertia and the acceleration due to the gravity have to be considered.
The impulse-momentum theorem states that the impulse of the force acting on an object equals the change in momentum of that object. The center of gravity of an object is the point where all the weight of the object can be considered to be concentrated.
Chandrasekhar limit is the maximum possible mass of an object that is prevented from collapsing under its own gravity by the degeneracy pressure of either electrons or neutrons. Spin- speed-pressure = gravity. The faster the spin, the faster the collapse, the faster it spins, the conservation of angular momentum.
Inverse square law force for short distance V(r) =-Gmm1/r - a Gmm1/r e-r/l. a = s strength of the Yukawa potential relative to the 1/r potential, for smaller the r = l, v(r) = -Gmm1/r (1+a). (1+a)G= gravitation constant. The strength of the Yukawa potential is restricted to 1a1< 10-6 for l =10-9cm. Gravitational mass distribution in a spherical p(r) =density. M(r) = mass enclosed within the radius r then g=Gm(r)/r2 and dg/dr=-2G M(r)/r3+G/r2 dm/dr = 2g/r+G/r2 4p r 2p(r). The value of G can be calculated from the measured values g and dg/dr provided we know the density p. g varies with depth for uniform-density sphere, g would decrease linearly with depth. Gravitation effects can not propagate with infinite speed; we expect that gravitational effects propagate in the form of waves at the speed of light.
There are two forces; 1- gravitational mg by the mass, 2 – centrifugal pseudo-force ma produced by rotation, g=acceleration of gravity, a=centrifugal acceleration. For a known field strength E(r) the force and torques on a system (per unit charge) a system point charges e1,e2,e3…en located in a vacuum at position r1, r2,r3… rn the energy to bring the individual charges to their locations. No work is required to bring the first system into place. Now bring e2 to its place, distance r1-r2 = r12 from e, and because of Coulomb repulsion, work e 1e2/r12 is required. Next bring e3 for this we have to expend energy e1e3/r13 +e 2e3/r23 continues this way until all charges are in place. The total work of assembling w=e 1e2/r12 + e 1e3/r13 +e 2e3/r23 + …
Maxwell’s field theory = every volume – element dv in empty space, if the region of an electric field contains the energy udv, the energy density u=1/8pE2 accordingly an extended region of space should contain energy. U=1/8p∫ E2 dv the assemblage of h point charges, the expended work w is identical with the increase of field energy u. In a dipole electrostatic field fixed moment p in an electrostatic field E. Set e = e and e2=-e and r1=r+s, r2=r with es=p the E field produces placed point charges. We obtain energy contribution – pe when orient dipole parallel to the field, we must expend the work pe with the potential energy u of the dipole depends upon the angle q between the dipole and the field direction. The torque T=δu/δθ=-pe sinθ must exist, against which we must do work increases with θ for this turning moment. In a non uniform field to torque the field gradient is positive on T minimum is when it is perpendicular to the direction of the field.
Universal gravitation – every particle of matter in the universe attracts every other particle with a force which is directly proportional to the product of their masses and inversely proportional to the square of their distances apart. Gravitational force “attraction” exerted at an external point by a sphere of uniform density or a sphere composed of uniform concentric shells was the same as if it’s whole mass were concentrated at its centre. As for the space curvature, matter and energy and the pressure involved. The mass grips space-time telling it how to curve, space-time grips mass telling it how to move. Increases in gravity by shrinking will cause a release of energy, causing it to be hotter, causing it to expand; decreases in gravity by expanding will cause a reduction in gravity causing it to cool causing it to shrink.
Lorentz-Fitzgerald contraction – An electron has radius when at rest, its radius when moving at a speed is given by its radius. Since the velocity is bound to be less than the speed of light the ratio must be less than 1; the radius is less than the initial radius corresponding to a contraction. The mass of a particle (electron) would be inversely proportional to its radius, its mass when moving with the speed 99% of the speed of light the effective mass is 7.1 times the rest mass. If the rest mass is 99.9% the speed of light the effective mass is 22.4 times the rest mass.
M-Theory – The number of dimensions required for each of the gauge groups u(1) – represented by motions around a circle, su(2) epitomized by rotations along a spherical surface, su(3) exemplified by transformation in a space with several complex variables requires 4 dimensions adding 1,2,4=7. Combining this with the 4 dimensions of space-time produce the value of 11 for the minimum number, precisely the dimensionality of super gravity. Taken to the power of two (duality) to account for chirality distinctions. S-duality involves exchanging a week coupling constant for a strong one. T- Duality pertains to the replacement of a small complification radius with an enormous one. Gravitation limited to one-millimeter π radius bulk. Lowering the universal temperature would precipitate a phase transition. Infinite is relative and possible. Spherical!
Different forms of energy as a source of gravity. The types are the rest mass of protons and neutrons, the rest mass of electrons, electric fields in the nucleus, magnetic fields in a nucleus, strong fields in a nucleus, a weak field in nucleus, and the gravitational energy in the earth. Gravity gravitates, earths gravitational self-energy is 4.6 x 10-10 times its rest mass. If gravitational self-energy does not contribute in the normal way to the gravitational mass. Gravitational interactions of anti-matter inertial masses for particles and anti-particles correspond. The observed equality of gravitational and inertia for all forms of energy means that any system containing any combination of the different kinds of energy will have equal gravitational and inertial masses. Gravitational quadrupole radiation is electromagnetic dipole radiation.
Flight factors
There are four forces of flight: the drag, lift, thrust and gravity. The center of pressure is the line of balance between the 4 forces. An up angle of 5 degrees will create 25% lift; an up angle of 10% will create 30% lift, etc. The pressure differential is 2.5 ounces per square inch, which will produce a lift force of 20lbs per square foot.
Viscosity and Kinematic viscosity of air;
Viscosity: V=μ/p μ- (Pas x 105), Kinematic Viscosity V – (m2/5 x106)
Reynolds number (Re)
Average velocity u, diameter d, density p, viscosity μ, kinematic viscosity V.
Re = pud/μ = ud/V, compression Δp/p
Decreases in diameter; area A1 & A2, velocities u1 & u2, densities p1 & p2
P1A1u1=p2A2u2 the flow velocity is inversely proportional to the cross sectional area. When diameter is reduced flow velocity increase.
At speeds 2 times the speed of sound produces 400 degrees F on a ships exterior.
Propagation of a sonic wave is a change in pressure (sound) that propagates at sonic velocity (a) from the source in all directions known as calm, subsonic(m<1), sonic(m=1) and supersonic(m>1), mach cone(2ά) sin ά=a/u=1/m called a mach angle. Sonic velocity is proportional to the square root of absolute temperature. A sudden change in pressure is adiabatic (a=√ΚRT) 5 times the speed of sound is hypersonic. Mach numbers are the ratio of flow velocity to sonic velocity, a pressure increase brings about an increase in density (Δρ=Δρ/a2).
Shock waves are created when air undergoes large and rapid compression. A thin wave of large pressure change is produced. The wave face at the rear of the compression wave, being at a higher temperature propagates faster than the wave at the front. The rear wave gradually catches up with the front wave increasing the pressure discontinuously, which is called a shock wave. The shock wave is only associated with an increase in pressure in the flow direction. Since a shock wave is different from a sound wave, because of a large change in pressure the propagation velocity of a shock is larger the larger the pressure rise, the greater the propagation velocity. A Rankine-Hugoniont equation shows the relationships between the pressure, density, and temperature ahead and behind a shock wave.
Turbine flow is descried as follows; If a turbine is placed in the course of a flow (air/water) the turbine rotation is owing to the velocity of energy. The flow velocity is obtainable from the rotational velocity of the turbine, while the integrated volume is calculated by the number of revolutions.
Thrust is the mass and time rather than density speed cross section, we can express thrust at the rate which it inserts momentum into the flow. Thrust happens when V2 is greater than V1. The trust time, distance and power is worked per time power becomes force times velocity. The efficiency is called Froude propulsion efficiency moving the largest possible mass-per-time (m/t)
The tension is T=Δpt and the stress is σn=Δpr/t. Photon power is propulsion derived from light.
Electric propulsion is charged particles discharged backwards to give forward thrust. 1 electro thermal, arc jets, and resistojet. 2 electrostatic, ion accelerators, 3 electromagnetic, coaxial magnetic plasma. Electric propulsion has a low thrust but a very low flow rate giving a 1sp of some 4000s (specific impulse) Stockes law for drag for a sphere is D=6πμau, a= the radius of the sphere. For terminal velocities mass equal gravity mg, sphere densities po, and fluid medium pf.
V=mg/6πua = 2a²g (po-pf)/qu. The mass of 1 kilogram exerts a force of 9.8 newtons at the surface. The acceleration due to gravity is 9.8m-². F=ma F= force in the product of mass m and the acceleration a.
The required escape velocity Ve for vertical flight is Ve=√2go re where go is the acceleration of gravity at the earths surface 32.174 ft/s² and re is the radius of earth 3959 miles or 6371km. Escape velocity is around 36,700ft/s or 25,000mi/h or 11.2km/s.
Space hazards are cosmic radiation, meteorites, weightlessness, light deprivation, and temperature which is -69.7ºF and -56.5ºC.
Time is taken as 10km/second =36,000km/hr = mach30 or 30 times the speed of sound, 1/30,000 of the speed of light. 1 parsecs = 3,262 light years or 3.086 x 10¹³km. Light travels 300 million meters a second or 300,000 km a second. 1 light year = 10¹³km. The moon is 1.3 light seconds away; the sun is 8 light minutes away.
Gravity
Humans tolerate 15 g gravity for a few minutes – when perpendicular, but only 6 g when in the direction of acceleration.
Humans can take 1 g (32' per second per second the speed of light)
Acceleration due to gravity
g/cm/s2=9.780356(1+0.0052885 sin2 ø – 0.0000059 sinz Zø)
-0.003086 H ø latitude H height
Ø H=0 H=1Km H=5Km H=10Km
0 9.78036 9.77727 9.76493 9.74950
5 9.78075 9.77760 9.76532 9.74989
10 9.78191 9.77882 9.76648 9.75105
15 9.78381 9.78072 9.76838 9.75295
20 9.78638 9.78330 9.77095 9.7552
25 9.78956 9.78647 9.77413 9.75870
30 9.79324 9.79016 9.77781 9.76238
35 9.79732 9.79424 9.78189 9.76646
40 9.80167 9.79858 9.78624 9.77081
45 9.80616 9.80307 9.79073 9.77530
50 9.81065 9.80757 9.79522 9.77979
55 9.81501 9.81193 9.79958 9.78415
60 9.81911 9.81602 9.80368 9.78825
65 9.82281 9.81972 9.80738 9.79195
70 9.82601 9.82292 9.81058 9.79515
75 9.82860 9.82551 9.81317 9.79774
80 9.83051 9.82743 9.81508 9.79965
85 9.83168 9.82860 9.81625 9.80082
90 9.83208 9.82899 9.81665 9.80122
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