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Find the specified quantities for the solids described below:

The center of mass of the region from Exercise 55, assuming that the density of the region is constant.

Short Answer

Expert verified

The center of mass is given by(x¯,y¯,z¯)=0,0,3R8(1+cosα).

Step by step solution

01

Given Information

The density of region is constant.

Region above sphere is given by equation ρ=Rand below the cone is given by equationϕ=α.

02

Evaluation of center of mass of x&y coordinate

For the given information, Center of ,mass if given by

x¯=MyzM=Exρdxdydzρdxdydz

y=MxM=EyρdxdydzEρdxdydz

As density of region is constant ρ=k, the axis of center lies above +zaxis and base lies in xyplane.

Hence,x¯=0,y¯=0

03

Evaluation of center of mass of z coordinate

It can be given by z¯=MxyMEzρdxdydzEρdxdydz

z¯=MxyM=EzρdVEρdV

=ϕ=0αθ=02πρ=0R(ρcosϕ)kρ2sinϕdρdθdϕa2πRkρ2sinϕdρdθdϕ

=sin2ϕ2ϕ=0ϕ=αkρ44ρ=0R(θ)θ=02πk(-cosϕ)ϕ=0α(θ)θ=02πρ33ρ=0R

Putting limits yields

z¯=sin2α2kR44(2π)k(1-cosα)(2π)R33

=sin2α×R4(2kπ)8×3(2kπ)R3(1-cosα)

=3R8(1+cosα)

Hence, required Center of Mass is given by(x¯,y¯,z¯)=0,0,3R8(1+cosα)

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