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CBSE Class 11-science Answered

4 identicle bodies of mass m are located at vertices of square of side l they revolve under the effect of mutual gravitational force in circle circumscribing the square then the speed of each body will be
Asked by shivanipanchal6883 | 05 Jan, 2023, 02:44: PM
Expert Answer
Figure shows system that consists of 4 partcles of mass m positioned at the vertex of square of side l .
 
Each mass is attracted along sides by force FS and also along diagonal by force FD .
 
Force FS = begin mathsize 14px style G space m squared over l squared end style
Force FD = begin mathsize 14px style G space m squared over open parentheses square root of 2 l close parentheses squared space equals 1 half space cross times fraction numerator G m squared over denominator space l squared end fraction end style
where G is universal gravitational constant.
 
Forces acting on sides gives resultant along diagonal .
 
Hence net resultant force Fnet acting on each mass along diagonal is
 
Fnet = √2 FS + FD = begin mathsize 14px style G space m squared over l squared open parentheses square root of 2 space plus space 1 half close parentheses space equals space 1.914 space G space m squared over l squared end style
If above force is acting as centripetal force for the masses to rotate along circle
that is circumscribing the square , then speed v of each mass is determined as follows
 
begin mathsize 14px style 1.914 space G space m squared over l squared space equals space m space v squared over R end style
where R = begin mathsize 14px style 1 half open parentheses square root of 2 space l close parentheses space equals space fraction numerator l over denominator square root of 2 end fraction end style  is radius of circle

begin mathsize 14px style 1.914 space G space m squared over l squared space equals space m space v squared over l cross times square root of 2 end style
After simplification of above expression, we get  begin mathsize 14px style v space equals space square root of 1.354 space fraction numerator G space m over denominator l end fraction end root end style
 
Answered by Thiyagarajan K | 05 Jan, 2023, 05:32: PM

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