VDU_Physics_2_005  
Solved UniversityPhysics ProblemsPhysics 2. NonUniform Mass DensityJavier Montenegro Joo jmj@VirtualDynamics.Org


NonUniform Mass Density 

(1) Determine the mass of a compact ball of radius R whose mass density varies linearly with the radius according to is D(r) = D_{o} r, where D_{o }is a constant. Solution. At a radial distance r consider a differential spherical shell of radius r and width dr, its differential volume will be dv = S dr:




(2)
Determine the mass of a spherical shell of outer radius R and thickness
b, if its mass density is D(r) = D_{o} r, where D_{o
}is a constant. Solution.


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(3) A compact cylindrical shell of length ‘L’ has inner radius ‘a’ and, outer radius ‘b’, and it is known that its mass density varies radially according to D = k_{ }r, where k_{ }= Constant. Determine the mass of the shell. Solution. In this problem, an integral over the entire volume of the shell is needed, to accomplish this it is necessary to consider a differential cylindrical shell ring of radius ‘r’ and width ‘dr’ and length ‘L’:


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(4)
A compact block of
height H has a square base of side L. The density of the material of the
block linearly varies with altitude, from “Do” at the bottom; up to “2
Do” at its top. (a) Construct a mathematical expression for the density
of the block (b) Making use of the constructed expression for the
density of the block, calculate its density at half its height. (c)
Determine the total mass of the block.
Solution.


