Module MA1132 (Frolov), Advanced Calculus
Tutorial Sheet 9
To be solved during the tutorial session Thursday/Friday, 24/25 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
Fx (ξ, η, ζ) = Gm
δ(x, y, z)dV ,
r3
Z Z ZG
y−η
δ(x, y, z)dV ,
Fy (ξ, η, ζ) = Gm
r3
(2)
Z Z ZG
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. Consider the solid G bounded by the surfaces z = 0, z = h and and x2 + y 2 = a2 .
(a) What are the surfaces z = 0 and z = h?
1
(b) What is the surface x2 + y 2 = a2 ?
(c) Sketch the solid G.
(d) Sketch the projection of the solid G onto the xy-plane.
(e) Find the volume V of the solid G.
(f) Find the centroid of the solid G.
(g) Find the moments of inertia of the solid G.
(h) 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 above by the surface
1
15 p 2
− x + y 2 and below by the surface z = (x2 + y 2 ) .
2
2
p
(a) What is the surface z = 15
− x2 + y 2 ?
2
z=
(b) What is the surface z = 21 (x2 + y 2 )?
(c) Sketch the solid G.
(d) Sketch the projection of the solid G onto the xy-plane.
(e) Find the volume V of the solid G.
(f) Find the mass M of the solid G if its density is
√
2
2
e x +y − 1
p
.
δ(x, y, z) =
x2 + y 2 + 5 x2 + y 2
(g) What is the density of the solid G at the origin: δ(0, 0, 0) =?
Show the details of your work.
2