Module MA1132 (Frolov), Advanced Calculus
Tutorial Sheet 10
To be solved during 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
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 surface x2 + y 2 + z 2 = a2 .
(a) What is the surface x2 + y 2 + z 2 = a2 ?
1
(b) Find the volume V of the solid G.
(c) Find the moments of inertia of the solid G.
(d) Find the gravitational force F(ξ, η, ζ) exerted on a point particle located at (ξ, η, ζ)
by the solid G.
(e) Find the gravitational potential field U (ξ, η, ζ) of the solid G.
(f) Plot Fz (0, 0, ζ) and U (0, 0, ζ) for a = 1.
2. Consider the spherical shell, i.e. the solid G: a21 ≤ x2 + y 2 + z 2 ≤ a22 . Use the results
obtained for a sphere, and express your answers in terms of the shell’s mass.
(a) Find the volume V of the solid G.
(b) Find the moments of inertia of the solid G.
(c) Find the gravitational force F(ξ, η, ζ) exerted on a point particle located at (ξ, η, ζ)
by the solid G.
(d) Find the gravitational potential field U (ξ, η, ζ) of the solid G.
(e) Plot Fz (0, 0, ζ) and U (0, 0, ζ) for a1 = 1 and a2 = 2.
3. Consider the cylinder G bounded by the planes z = 0, z = h and by the surface
x 2/3
a
+
y 2/3
b
= 1.
(a) Sketch the solid G and its projection onto the xy-plane.
(b) G can be expressed parametrically by a generalisation of cylindrical coordinates
x = a r cos3 θ , y = b r sin3 θ , z = z ,
0 ≤ r ≤ 1 , 0 ≤ θ ≤ 2π , 0 ≤ u ≤ h .
Think about (4) as a change go variable. Find the Jacobian of the change.
(c) Find the volume V of the solid G.
(d) Find the moments of inertia of the solid G (you may use Mathematica).
Show the details of your work.
2
(4)