A non?uniform solid rectangle of height and width 2 cm and length 10 cm is place

ID: 2142196 • Letter: A

Question

A non?uniform solid rectangle of height and width 2 cm and length 10 cm is placed with one of its

corners at the origin of a standard Cartesian coordinate system. All of its mass points have x ? 0, y ? 0,

and z ? 0; one of its long sides is coincident with the plane y = 0, and another long side is coincident with

the plane z = 0. The mass of the solid rectangle is 3 kg. Its volume density is given by the function ?(x) =

Ax. (a) What is A? (b) What is the position vector of the CM of this solid?

Take a cross-section at a distance x from origin on x-axis

Mass = Ax * dx * (2*10^-2)^2

4*10^-4 *A*xdx

Total mass = integral(4*10^-4 *A*xdx), where x=0 to 0.1

= 4*10^-4*A * x^2 /2

= A*2*10^-6

Mass = 3 kg

So, A = 1.5*10^6 kg/m^4

(b) On x-axis, mass has to reach half at CM

So, 4*10^-4 *1.5*10^6 * x^2 /2 = 1.5

x = 0.0707 m = 7.07 cm

CM is at (7.07,1,1) cm

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