Module MA1132 (Frolov), Advanced Calculus
Homework Sheet 10
Due: at the beginning of the tutorial session Thursday/Friday, 7/8 April 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
Ix =
(y 2 + z 2 )δ(x, y, z)dV ,
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
G
ZZZ
y−η
δ(x, y, z)dV ,
Fy (ξ, η, ζ) = Gm
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
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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 spherical coordinates x = r cos θ sin φ, y = r sin θ sin φ, z = r cos φ. Consider
the solid G made of points with the coordinates r, θ, φ satisfying 0 ≤ r ≤ a, 0 ≤ θ ≤ γ,
0 ≤ φ ≤ π.
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(a) What is the surface r = a?
(b) What are the surfaces θ = 0 and θ = γ?
(c) What are the sets of points φ = 0 and φ = π?
(d) Sketch and describe the solid G and its projection onto the xy-plane.
(e) Find the volume V of the solid G. Specify your answer for γ = π and γ = 2π and
explain the result.
(f) Find the centroid of the solid G. Specify your answer for γ = π and γ = 2π.
(g) Find the moments of inertia of the solid G. Specify your answer for γ = π and
γ = 2π.
(h) Set γ = π, and find the gravitational force exerted on a point particle by the solid
G if the point particle is located at (0, ζ, 0). To simplify the computation introduce
different spherical coordinates by exchanging y and z. Why does it simplify the
computation?
(i) Set γ = π, and find the gravitational potential field U (0, ζ, 0) of the solid G.
(j) Plot Fz (0, ζ, 0) and U (0, ζ, 0)) for γ = π and a = 1.
Show the details of your work.
2. Consider the solid G that is generated by revolving the disc
z 2 + (x − b)2 ≤ a2 ,
0<a<b
in the xz-plane about the z-axis.
(a) What is the solid?
(b) Show that this solid can be expressed parametrically as
x = (b + ρ cos φ) cos θ , y = (b + ρ cos φ) sin θ ,
0 ≤ ρ ≤ a , 0 ≤ θ ≤ 2π , 0 ≤ φ ≤ 2π .
z = ρ sin φ ,
(c) Write the parametrisation of the boundary of this solid induced by (4).
(d) Find the area of the boundary.
(e) Think about (4) as a change go variable. Find the Jacobian of the change.
(f) Find the volume V of the solid G.
(g) Find the moments of inertia of the solid G.
Show the details of your work.
3. Consider the solid G bounded by the “astroidal” hyperboloid of one sheet
x 2/3
a
+
y 2/3
b
and by the planes z = −c and z = c.
2
−
z 2/3
c
= 1,
(4)
(a) Plot the astroidal hyperboloid for a = b = c = 1.
(b) G can be expressed parametrically by a generalisation of hyperbolic coordinates
x = a r(cosh p cos θ)3 ,
0 ≤ r ≤ 1,
y = b r(cosh p sin θ)3 ,
z = c r sinh3 p ,
− sinh−1 1 ≤ p ≤ sinh−1 1 ,
0 ≤ θ ≤ 2π ,
(5)
where sinh−1 is the function inverse to sinh: sinh−1 (sinh x) = sinh(sinh−1 x) = x.
Think about (5) as a change of variables. Find the Jacobian of the change.
(c) Find the volume V of the solid G.
(d) Use Mathematica to find the moments of inertia of the solid G.
Show the details of your work.
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