Question
Asked Dec 8, 2019
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Problem 8
x² + y° < 4, x20, and -1<z<land whose density is p(x, y,z)=x². Hint: Use and apply symmetry.
Find the center of mass of the half cylinder whose shape is described by
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Problem 8 x² + y° < 4, x20, and -1<z<land whose density is p(x, y,z)=x². Hint: Use and apply symmetry. Find the center of mass of the half cylinder whose shape is described by

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Expert Answer

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Step 1

Given:

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x? +y? <4; x>0 -1<:<1 Density, p = x? By symmetry, Z =0 and, = 0 i.e. their center of mass will be at the origin. But, since x 2 0, there will be no symmetry along x-axis. uupx ||| –,0,0 Thus, the coordinate of the center of mass will be

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Step 2

Cylindrical Coordinate

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In cylindrical coordinates: x =rcos e y =rsin e x² + y <4; =>(rcose)' +(rsin ) < 4; => r? <4 -0 <r<2 =>0<r<2 .. dV =rdrdedz and, Density, p = x' = r² cos? e

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Step 3

Calculation of ...

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т Also we know, p or, dm = pdV or, dm =r² cos² 0(rdrd0dz) or, dm =r° cos? Odrded A/2 2 1 I ||" cos? Odrd@dz or, m= -x/20-1 */2 r* or, m= [:L, | cos? Ode 4 Jo -x/2 /2 1+ cos 20 [16] de or, m = 4 -x/2 or, m 3D 4л

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Tagged in

Math

Calculus

Integration