Module MA1132 (Frolov), Advanced Calculus
Homework Sheet 9
Due: at the beginning of the tutorial session Thursday/Friday, 31/32 March 2016
You may use Mathematica to sketch the integration regions and solids, and to check the
results of integration.
Moment of inertia: The tendency of a solid to resist a change in rotational motion about
an axis is measured by its moment of inertia about that axis. If the solid occupies a region G
in an xyz-coordinate system, and if its density function δ(x, y, z) is continuous on G, then the
moments of inertia about the x-axis, the y-axis, and the z-axis are denoted by Ix , Iy , and Iz ,
respectively, and are defined by
ZZZ
(y 2 + z 2 )δ(x, y, z)dV ,
Ix =
Z Z ZG
(x2 + z 2 )δ(x, y, z)dV ,
Iy =
(1)
G
ZZZ
Iz =
(x2 + y 2 )δ(x, y, z)dV .
G
Newton’s law of gravitation: Let a solid occupy a region G in an xyz-coordinate system,
and let its density function δ(x, y, z) be continuous on G. Then the gravitational force F =
Fx i + Fy j + Fz k exerted by the solid on a point particle of mass m located at (ξ, η, ζ) is given
by
ZZZ
x−ξ
δ(x, y, z)dV ,
Fx (ξ, η, ζ) = Gm
r3
G
ZZZ
y−η
Fy (ξ, η, ζ) = Gm
δ(x, y, z)dV ,
r3
(2)
G
ZZZ
z−ζ
Fz (ξ, η, ζ) = Gm
δ(x, y, z)dV ,
3
G r
p
r = (x − ξ)2 + (y − η)2 + (z − ζ)2 ,
where G is the gravitational constant.
The force can be obtained from the gravitational potential field U (ξ, η, ζ) as follows
ZZZ
1
U (ξ, η, ζ) = −G
δ(x, y, z)dV ,
G r
∂U (ξ, η, ζ)
∂U (ξ, η, ζ)
∂U (ξ, η, ζ)
Fx (ξ, η, ζ) = −m
, Fy (ξ, η, ζ) = −m
, Fz (ξ, η, ζ) = −m
.
∂ξ
∂η
∂ζ
(3)
In what follows we set m = 1, G = 1, and consider homogeneous solids with δ(x, y, z) = 1.
1. Considerpthe solid G bounded above by the surface z = h and below by the surface
az 2 = h2 x2 + y 2 .
1
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
What is the surface z = h?
p
Describe the surface az 2 = h2 x2 + y 2
Sketch the solid G.
Sketch the projection of the solid G onto the xy-plane.
Find the volume V of the solid G.
Find the centroid of the solid G.
Find the moments of inertia of the solid G.
Find the gravitational force exerted on a point particle by the solid G if the point
particle is located at the origin (0, 0, 0). Find its limit as a → ∞ with h kept fixed,
and its limit as h → ∞ with a kept fixed.
Show the details of your work.
2. Consider the solid G bounded by the surfaces z = 0, z = h and x2 + y 2 = a2 .
(a) What is the solid?
(b) Find the gravitational force exerted on a point particle by the solid G if the point
particle is located on the z-axis at (0, 0, ζ) (ζ can be any real number)
(c) Plot its graph and the graph of ∂Fz /∂ζ for a = 1, h = 2 (use Mathematica).
(d) Find the limit of Fz as a → ∞ with h and ζ kept fixed.
(e) Find the limit of Fz as h → ∞ with a and ζ kept fixed.
p
3. Consider the solid G bounded below by the surface z = rα , α > 0, r = x2 + y 2 and
above by the plane z = 1. Note that the surface z = r is a cone, and z = r2 is a
paraboloid.
(a) Sketch the surface z = rα for α = 1/2, α = 1, α = 2, and the projection of the solid
G onto the xy-plane.
(b) Find the volume V of the solid G, and its limit as α → ∞.
(c) Find the centroid of the solid G, and its limit as α → ∞.
(d) Find the moments of inertia of the solid G, and its limit as α → ∞.
(e) Use Mathematica to find the gravitational force exerted on a point particle by the
solid G if the point particle is located at the origin (0, 0, 0).
(f) Use Mathematica to plot Fz as a function of α, and find its limit as α → ∞.
(g) Explain the limiting values obtained in (b), (c), (d), (f).
Show the details of your work.
Bonus questions (each bonus question is worth extra 25 marks)
1. Consider the solid G bounded above by the surface z = h and and below by the surface
a2 z 2 = h2 (x2 + y 2 ).
Plot the solid for a = 1, h = 2. Find the gravitational force exerted on a point particle
by the solid G if the point particle is located on the z-axis at (0, 0, ζ) (ζ can be any real
number), and plot its graph and the graph of ∂Fz /∂ζ for a = 1, h = 2 (use Mathematica).
2