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From fig i want to know with what velocity the body strikes mass 4M  from neutral point?

Asked by venkat 11th December 2017, 11:34 AM
Answered by Expert
Answer:
There is a neutral point N in the line OC that connects the two spheres so that gravitational attractive force due to each other cancel and net zero force at that ponit. AS per the given figure let ON = r. The distance r is obtained by equating the two forces of attraction as given below
 
begin mathsize 12px style fraction numerator G M m over denominator open parentheses R plus r close parentheses squared end fraction space equals space fraction numerator G 4 M m over denominator begin display style open parentheses 6 r minus r close parentheses squared end style end fraction
end style
Solve the above equation to get r = 2R.
 
Now to project a projectile of mass m from first sphere to second sphere the required minimum velocity is obtained by calculating the workdone due to the movement of the projectile from surface of first sphere to the neutral point N against the net gravitational force due to the two spheres.
 
begin mathsize 12px style 1 half m v squared space equals space integral subscript 0 superscript R open parentheses fraction numerator G M m over denominator begin display style open parentheses R plus x close parentheses squared end style end fraction space space minus space fraction numerator G 4 M m over denominator begin display style open parentheses 5 R minus x close parentheses squared end style end fraction close parentheses d x space equals space 3 over 10 fraction numerator G M m over denominator R end fraction

H e n c e space v space equals space square root of fraction numerator 0.6 G M over denominator R end fraction end root end style

Answered by Expert 11th December 2017, 2:20 PM
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Tags: gravitation
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