TheTheory of Math Questions
TheTheory of Math Questions

Proof of the spherical version of the Sharogpath theorem
TheSharogpath theorem quoted in the story a modified version of theoriginal formula it is represented as:
Weightat speed, v= Rest weight × (1v^{2}/c^{2})
Where
v= velocity of the moving body
c= speed of lightness
Thevelocity of a body moving on the surface of a spherical body cannotbe equal the speed of lightness. Consider triangle ABC that isobtained from a spherical body as shown in the diagram below
Asphere has six lunes, and their area is the sum of the individuallunes. The triangle ABC has vertices that are represented by thelunes L_{a}, L_{b}, L_{c}.
Areaof lune = ¼(2R^{2}(2L_{a}+ 2L_{b}+2L_{c})–area of the sphere
=¼(2R^{2}(2L_{a}+ 2L_{b}+2L_{c})–4πR^{2}
=R^{2}(L_{a}+ L_{b}+L_{c})
Multiplyingeither side of the equal sign by 4 gives
=R^{2}(L_{a}+ L_{b}+L_{c}) πR^{2}
Factoringout R^{2}gives
Areaof lune, A = R^{2(}L_{a}+ L_{b}+L_{c}π)
L_{a}+ L_{b}+L_{c=}π+1/R^{2}A(This equation gives Sharogpaththeorem)
Where:
L_{a}+ L_{b}+L_{c}_{}arethe three sides of the triangle
Risthe radius of the sphere
Aisarea of the lunes

References on the theorem
Thethree sides of the triangle are represented by three lunes of thespherical body and they symbolized by L_{a}, L_{b}, L_{c}and they represent the three vertices of a triangle derived from thesphere. Object move along the lunes of a sphere at a very highvelocity, even though the velocity cannot exceed the speed oflightness. The body can therefore not reach a point where it isweightless during its motion. The theorem shows that as the velocityof the moving body, v approaches the speed of lightness, c it becomesdifficult to define the totalenergy of the moving body. The rest mass of the body remains the samethroughout the duration of motion its energy is equal to the energyof a particle that is at rest.
Accordingto Albert Einstein, a body that moves at a speed that is very closeto the speed of light has a total energy that is given by theequation mc^{2}/{(1sqrt. (v^{2}/c^{2})}.This equation is discussed in the article provided.
Theearth spherical in shape, it rotates about its axis at a very highspeed and the same time revolves around the sun. We move with theearth as it rotates at a speed that approaches that of light.However, we cannot notice that we rotate too since the world has agreater mass compared to our mass and therefore we act as a referenceframe to the each, the earth, therefore, does not move relative tous.