Expert: Jayendra Upadhye - 5/10/2004

Question
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Followup To
Question -
Jayen,it's obvious you are well educated.Thank you. I think I understood the black hole,you're saying there's nothing special about the event horizon except the boundry where light cannot escape.that was a very good explanation.I knew there had to be something very cool about it "a dimensionless point that drops out of 3D space"!Regarding the mass of objects, I'm sure all the formulas are there,I understood all the components of them except the V (earth's velocity,rotating or revolving?) and T(orb.period in sec,rotating or revolving?)and r(radius,but Earth's or Sun's?) I lost you when you put all the formulas close together, I didn't understand which formula was for which part of the equation. Let me try- Sun's mass is: Earth's velocity squared(in KPS?)*radius*Grav.Constant,is that right? then you wrote earth's mass is g=G*Me/r*r. I didn't get the small g,but anyway Me=9.81*(r*r)/G. Where did the 9.81 come from? Sorry,I hope I don't seem dense,but I've always been fascinated by the cosmos & everything in it,especially the exotic stuff like neutron stars,quasars & wormholes(maybe)and it's even better when you understand more about it. Thank you.
Answer -
Oh! No problem, I will dilate on that..
Actually Mike, I am sometimes vary of making my answer look too long winded and elaborate. Some of the people asking me questions are quite unforgiving about that! as will be apparent if you will see my ratings! Others have been very supportive and generous! My reward here is in the feeling that one managed to "connect" afterall..(communication being the devil that it truly is!)..
Ok..I will go point by point..
1 - V is velocity of the earth in its orbit around the sun.
   this can be found right away by dividing the
   circumference of it's orbit by the time taken to do so
   (go around once which is "period" T).
   hence V = 2*pi*r/T in secs.
   r is mean radius of the slightly elliptical orbit.
   So V = 2*pi*r/(365.25 days * 24 hrs/day * 3600 secs /hr)
   meters/sec.
2 - The formula for the sun's mass: (M is mass of sun)
   G*M/(r*r) = (V*V)/r this leads to
   M = (V*V)*r/G where G is universal constant of
       gravitaion.
3 - The 9.81 met/sec squared is gravitational acceleration
   at the earth's surface.
   Me = 9.81 * (r*r)/G where r is earth's average radius.
   (for the earth has an equatorial bulge and weights for
   the same object are yield slightly different values as
   one moves from the poles to the equator, as the
   equatorial bulge takes one away from the earth's
   center!)
4 - Pls note that in all gravitating systems where one mass
   is very large compared to the other (earth and most man
   made objects on it), one may imagine "all" mass of
   earth to be concentrated in an imaginary "point mass"
   at its center and the object being attracted as
   residing on a massless sphere (the earth's surface) r
   units asway. (this analogy of point masses can be
   extended to earth and moon as they are sufficiently far
   away from each other, but would fail if they were to be
   very near to each other).
Jayen.
    
    Hi,Jayen-I'm starting to wonder if you are too advanced in the
formula area to explain this plain enough for me. I also believe in knowledge for the sake of it,and I've gone this far with you, it would be a shame to give up when I'm getting closer! Now,I understand velocity.As far as the Sun's mass,when you write the simpler part which is          M=(V*V)*r/G,the V is still Earth's velocity,right? And is r the Earth's radius,also,in this formula about the Sun? OK,now Me=9.81*(r*r)/G looks pretty easy,I guess I just have to look up the gravitational constant and multiply.The only other thing I don't quite understand is why the two formulas are different when they are both calculating mass,and lastly,I never quite understood the expression "9.81 met/sec squared" How can you square something with 2 different parameters in it? Is it just the meters that are squared,or what? Thank you again,Jayen,I guess sometimes it's challenging when 2 people are not face-to-face and there's a time lag in communication.I'm just trying to visualize all this stuff,and I'm sure with your help,I'll get it.  Mike  

Answer
Mike..
1 - M = (V*V)*r/G where
   M= solar mass in Kilograms
   r = radius of earth's orbit in Meters
     = approx. 93 million miles * 1.6 km/mile * 1000 meters
     = 148800 million meters (approximate only as we start
       with 93 million miles as orbital radius for earth.
   V = earth's velocity in it's orbit
     = 2*pi*r/T
     = (2*3.1415946 * 148800 * 10^6 )/(365.25 days / year
       * 24 hours /day * 3600 sec/hour
     = 29626.415 met/sec = 29.626 km/sec
2 - The two formulae are different because in the sun related formula we equate gravitational acceleration as derived by the gravitational constatnt with that value as derived from uniform circular motion (centripetal acceleration).
   
3 - 9.81 met/sec squared means just that..
   9.81 met / (sec)^2 ie with the denominator having
   seconds term squared.
   acceleration is velocity/time = velocity/sec
   but velocity is itself in met /sec.
   so acceleration (met/sec)*1/sec which yields above
   units with seconds squared in the denominator.
Mike hope this suffices
Why dont you mail me your queries directly at jupadhye@hotmail.com? As else we will have a followup follwed by another and then another..
Jayen