Find (i) the mass M, (ii) the moment Mo about the origin, and (iii) the center o

Question

Find (i) the mass M, (ii) the moment Mo about the origin, and (iii) the center of mass x for a wire that occupies the given interval of the x-axis. a. A wire that occupies the interval [0, [sqrt{pi}] ? ] whose linear density (g/m) is given by [ holeft(x ight)=xcdot sinx^2] ?(x)=x?sinx2. b. A wire that occupies the interval [?5,5] whose linear density (g/m) at any point x is proportional to the square of the distance from x to the origin. c. A wire that occupies the interval [1,6] whose linear density (g/m) at any point x is inversely proportional to the distance from x to the origin.

Explanation / Answer

a) (M = int_{0}^{sqrt{pi}} x sinx^{2} dx = 2g)

ii)

(M_{0} = int_{0}^{sqrt{pi}} x sinx^{2} ( x^{2})dx = pi/2 )

Mo = pi/2 g-m^2

iii) (C.M = int dm.x/M = int_{0}^{sqrt{pi}} x^2 sinx^{2} dx/2 =0.6088 m)

b) i) (M = int_{-5}^{5} x^2 dx = 250/3 =83.33 g)

ii) (Mo =int_{-5}^{5} x^2 dx *x^2 =int_{-5}^{5} x^4dx = 1350 g-m^2)

iii) (C.M = int_{-5}^{5} x^3 dx/M = 3.75 m)

c) i) (M= int_{1}^{6} 1/x dx = ln(6) = 1.791 g)

ii) (Mo =int_{1}^{6} 1/x *x^2dx =int_{1}^{6} x dx = 35/2 =17.5 g-m^2)

iii) (C.M = int_{1}^{6} 1/x *xdx/M =int_{1}^{6} dx/1.791 =2.790 m)

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