Mathematical Methods for Physics and Engineering : A Comprehensive Guide

6.5 EXERCISES

This is an example of the general result for planar bodies that the moment of

inertia of the body about an axis perpendicular to the plane is equal to the sum

of the moments of inertia about two perpendicular axes lying in the plane; in an

obvious notation

Iz=

∫

r^2 dm=

∫

(x^2 +y^2 )dm=

∫

x^2 dm+

∫

y^2 dm=Iy+Ix.

6.11 In some applications in mechanics the moment of inertia of a body about a
single point (as opposed to about an axis) is needed. The moment of inertia,I,
about the origin of a uniform solid body of densityρis given by the volume
integral

I=

∫

V

(x^2 +y^2 +z^2 )ρdV.

Show that the moment of inertia of a right circular cylinder of radiusa,length

2 band massMabout its centre is

M

(

a^2

2

+

b^2

3

)

.

6.12 The shape of an axially symmetric hard-boiled egg, of uniform densityρ 0 ,is
given in spherical polar coordinates byr=a(2−cosθ), whereθis measured
from the axis of symmetry.

(a) Prove that the massMof the egg isM=^403 πρ 0 a^3.

(b) Prove that the egg’s moment of inertia about its axis of symmetry is^342175 Ma^2.

6.13 In spherical polar coordinatesr, θ, φthe element of volume for a body that
is symmetrical about the polar axis isdV=2πr^2 sinθdrdθ, whilst its element
of surface area is 2πrsinθ[(dr)^2 +r^2 (dθ)^2 ]^1 /^2. A particular surface is defined by
r=2acosθ,whereais a constant and 0≤θ≤π/2. Find its total surface area
and the volume it encloses, and hence identify the surface.
6.14 By expressing both the integrand and the surface element in spherical polar
coordinates, show that the surface integral
∫
x^2
x^2 +y^2

dS

over the surfacex^2 +y^2 =z^2 ,0≤z≤1, has the valueπ/

√

2.

6.15 By transforming to cylindrical polar coordinates, evaluate the integral

I=

∫∫∫

ln(x^2 +y^2 )dx dy dz

over the interior of the conical regionx^2 +y^2 ≤z^2 ,0≤z≤1.
6.16 Sketch the two families of curves

y^2 =4u(u−x),y^2 =4v(v+x),

whereuandvare parameters.
By transforming to theuv-plane, evaluate the integral ofy/(x^2 +y^2 )^1 /^2 over
the part of the quadrantx>0,y>0 that is bounded by the linesx=0,y=0
and the curvey^2 =4a(a−x).
6.17 By making two successive simple changes of variables, evaluate

I=

∫∫∫

x^2 dx dy dz