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Ques 9

Asked by rbhatt16 11th November 2017, 7:43 AM
Answered by Expert
Answer:
begin mathsize 20px style the space have space length space equal space to space infinity space in space the space positive space straight x space direction comma
let space it space be space cut space at space straight x space equals space straight x subscript 0
It space is space balanced space at space space straight x space equals space straight x subscript 0 space plus space calligraphic l space space
the space density space of space stick space is space straight rho left parenthesis straight x right parenthesis

Torque space equals space densitiyx space length space xacceleration space due space to space gravity space xdisplacement
The space stick space need space to space be space balanced space that space is space no space torque space should space act space on space it space due space to space gravity
And space its space length space goes space straight x subscript 0 space to space infinite space therefore space we space have space limits space from space straight x subscript straight o space to space infinite
Torque space equals integral subscript straight x subscript 0 end subscript superscript infinity straight rho left parenthesis straight x right parenthesis left parenthesis straight x minus left parenthesis straight x subscript 0 plus straight l right parenthesis right parenthesis space straight g space dx equals 0
derivative space of space torque space straight w. straight r. straight t space straight x subscript 0 space should space be space zero space

dT over dx subscript 0 equals 0

straight g calligraphic l straight rho left parenthesis straight x subscript 0 right parenthesis minus straight g integral subscript straight x subscript 0 end subscript superscript infinity straight rho left parenthesis straight x right parenthesis dx equals 0
after space differentiation

calligraphic l fraction numerator dρ left parenthesis straight x subscript 0 right parenthesis over denominator dx subscript 0 end fraction equals space minus straight rho left parenthesis straight x subscript 0 right parenthesis
put space straight x subscript 0 space equals space straight x space and space solving space above space equation space we space get
straight rho left parenthesis straight x right parenthesis equals Ae to the power of negative straight x over calligraphic l end exponent space

density space decreases space exponentially space with space straight x






end style
Answered by Expert 6th December 2017, 4:24 PM
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