Chapter 3: Many Variables; Section 2: Integration; page 4
Multidimensional Transformation, Exercise 1
Question
A uniform solid body ( density ρ ) consists of two inseparable parts, one on top of the other according to the Fig. Sketch for exercise 1:
The bottom half is a cylinder with radius a and height
h1 ,
The top half is a cylinder with the same radius, but height -
h2 . This cylinder has a cavity of an
inverted cone, with its vertex on the bottom of the cylinder, with the
same radius a of the base, and height -
h2 .
The centre of mass of the whole is at the vertex of the conic cavity. Use
cylindrical coordinates in order to answer numerically the following
questions:
What is the ratio
?
What is the ratio
, where M is the mass of the corresponding
part?
What is the ratio
, where is the moment of inertia about the axis of symmetry of the
corresponding part?
You don't have to repeat the integrations that were done in the relevant studied examples.
Reminder
(3.2.2.16)
is applicable for the volume V of any pyramid or conical shape, where A is the area of the base, and h is the height representing the distance between the head (the vertex) and the plane of the base. .....
................
The cylindrical coordinates were defined... as
(3.2.4.5)
.... The Jacobian is
(3.2.4.6)
................
The integration of a function f of the variables over a region of a straight cylinder with radius a and length h is done with cylindrical coordinates by taking the z axis as the axis of the cylinder:
(3.2.4.8)
As an example we'll obtain the mass M of a uniform cylinder with density ρ. From (3.2.4.8) after substituting f with
ρ we have
(3.2.4.9)
As a continuation of this example we'll obtain the moment of inertia of the same cylinder about the axis of symmetry (z), by substituting f from (3.2.4.8) with
:
(3.2.4.10)
.... We already know, from integration in the Cartesian coordinate system, how to code the limits of integration of any given boundary surface. The same procedure can be applied for the spherical and the cylindrical coordinate systems.....
Parts 1-7
Solution of question 1.
By definition
where dV means differential of volume, and (1) or (2) mean integration limits corresponding to part 1 or 2 of the body.
Since the zcm is fixed, it is sensible to choose
in order not to calculate the masses. One should take this in account for the integration limits over z .
where
yielding
From parts 1, 2, 3, 6, one obtains
yielding
Part 8
Solution of question 2.
From (3.2.4.9)
and by use of (3.2.2.16)
yielding
Parts 9-10
Solution of question 3.
With help of parts 4, 5
From (3.2.4.10)
which together with part 9 yields
Score
Note. Fixing the limits for the integration over the second body are worth 2 points (parts 4 and 5 of question 1). If question 1 is not the first to be answered, care should be taken for these 2 points to be assigned elsewhere.
By parts.
All the 10 parts are worth 1 point each.
By questions.
Question 1 is worth 7 points.
Question 2 is worth 1 point .
Question 3 is worth 2 points.