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A body is projected with a speed u at an angle of 30 with the horizontal at t=0 the angular momentum of a body about the point of projection is x√3mu^3/64g at t= 3u/4g find value of x

Asked by harshitj291 14th March 2019, 8:27 PM
Answered by Expert
Answer:
begin mathsize 12px style a n g u l a r space m o m e n t u m space equals space r with rightwards arrow on top space cross times space p with rightwards arrow on top space space space............... left parenthesis 1 right parenthesis

w h e r e space space r with rightwards arrow on top space equals space left parenthesis x space i with hat on top space plus space y space j with hat on top space plus space z space k with hat on top space right parenthesis space space i s space t h e space p o s i t i o n space v e c t o r space o f space o b j e c t space a t space t h e space g i v e n space t i m e
a n d space space p with rightwards arrow on top space equals space m space left parenthesis space v subscript x space i with hat on top space plus space v subscript y space space j with hat on top space plus space v subscript z space k with hat on top space right parenthesis space i s space t h e space m o m e n t u m space v e c t o r

w e space h a v e comma space space space space x space equals space space u space cos 30 space open parentheses fraction numerator 3 u over denominator 4 g end fraction close parentheses equals space fraction numerator 3 square root of 3 over denominator 8 end fraction u squared over g space space space space space space..................... left parenthesis 2 right parenthesis
a n d space space y space equals space u space sin 30 space open parentheses fraction numerator 3 u over denominator 4 g end fraction close parentheses space minus space 1 half g open parentheses fraction numerator 3 u over denominator 4 g end fraction close parentheses squared space equals space 3 over 32 u squared over g space space................ left parenthesis 3 right parenthesis

v subscript x space equals space u space cos 30 space equals space fraction numerator square root of 3 over denominator 2 end fraction u space space space.......................... space space left parenthesis 4 right parenthesis
v subscript y space equals space u space sin 30 space space minus space g open parentheses fraction numerator 3 u over denominator 4 g end fraction close parentheses space equals space minus u over 4
u sin g space e q n s. left parenthesis 2 right parenthesis comma space left parenthesis 3 right parenthesis comma space left parenthesis 4 right parenthesis space a n d space left parenthesis 5 right parenthesis space comma space w e space c a n space c a l c u l a t e space t h e space a n g u l a r space m o m e n t u m space space i v e n space i n space e q n. left parenthesis 1 right parenthesis comma space a s space f o l l o w s

r with rightwards arrow on top space cross times space p with rightwards arrow on top space equals space open vertical bar table row cell space i with hat on top end cell cell space j with hat on top end cell cell space k with hat on top end cell row cell fraction numerator 3 square root of 3 over denominator 8 end fraction u squared over g end cell cell 3 over 32 u squared over g end cell 0 row cell fraction numerator square root of 3 over denominator 2 end fraction m u end cell cell negative u over 4 end cell 0 end table close vertical bar space space equals space fraction numerator negative 3 over denominator 32 end fraction open parentheses square root of 3 space plus space 1 close parentheses space space m space u cubed over g space space......... left parenthesis 6 right parenthesis end style
 
if we compare the angular momentum as given by eqn.(6) with the given expression   (x√3/64) m u3 /g  ,  we get x = -2√3 (√3 + 1 )
Answered by Expert 15th March 2019, 9:55 AM
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