On A Roll!
The Theoretical Background
A. D. Andrew
February 2005
Introduction
In the first project this semester, you calculated the time required for a bead to
slide down a wire connecting two points, and then tried to find a fast path between the
points. The calculation of descent time began with the statement of conservation of
energy —the potential energy lost as the bead slides is converted to kinetic energy.
In this project you'll consider bodies rolling down an inclined plane. The
calculations involve the use of triple integration to find the moment of inertia of a solid
body.
The project is comprised of three documents. The theoretical background is
explained in this document. The Maple illustrations and exercises are in the Maple
worksheet, On A Roll! Examples and Exercises, which is available from my web pages
for this course. The Cover Sheet, which you should attach to your project when you
submit it is also posted on my web pages.
Part I. Kinetic Energy of a Rotating Body
In these problems, the conservation of energy statement is slightly more
complicated than in the case of the sliding beads -- part of the kinetic energy comes from
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the translation of the body ( m v 2 , just as in the case of the sliding bead), and part of it
2
comes from the rotation of the body. Now this is especially interesting, since we'll see
that the fraction of kinetic energy due to rotation depends on the geometry of the body.
To begin, assume there are several objects at the top of a ramp, with the ramp
making an angle  with horizontal. The objects are a solid sphere, a hollow sphere (or a
spherical shell), a solid cylinder, and a hollow cylinder. If they are released at the same
time, in what order do they reach the bottom of the ramp? Here are two diagrams of this
situation.
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Figure 1
Objects on a Ramp
To calculate the time required to roll down the ramp, we again equate potential
energy (mgH) at the top of the ramp with kinetic energy at the bottom of the ramp. What
we need to determine is the kinetic energy of a body rotating with angular velocity 
about an axis L.
Suppose that a solid S, having mass density (x,y,z) is rotating about an axis L
with angular velocity  radians per second. Such a solid is pictured in Figure 2. To
calculate its kinetic energy, note that we may approximate the body by a bunch of little
1
m v 2 , and we can
volumes as shown. Each of these little volumes has kinetic energy
2
express this in terms of . The volume element dV shown below has mass  dV, and it's
moving in a circle of radius d , so it has velocity v = d. Thus the kinetic energy of the
volume element dV is
dE 
1
 ( d) 2 dV ,
2
and the total kinetic energy of the rotating body is
E 
1
2
  d
2
 2 dV 
S
1
I 2 ,
2
where I is the moment of inertia
I 
  d dV
2
.
S
Notice that d is the distance to the axis of rotation L. Thus if the axis of rotation is the
2
2
2
2
2
2
z-axis, then d  x  y and if the axis of rotation is the x -axis, d  y  z .
The exact form of d will depend on the coordinate system you are using and the location
of the axis.
2
Figure 2
A rotating body
Note the similarities between the expressions
1
1
m v 2 and
I 2.
2
2
Part II. Calculation of the Moment of Inertia
Here is an example of the calculation of a moment of inertia. We consider a
cylinder of uniform density with base radius a and height h rotating about its axis, as
shown below.
3
We'll calculate the moment of inertia using cylindrical coordinates ( r, ,z) so the
distance d to the axis is r and dV = r dr d dz. Thus
2
I  
h
a
   r dr dz d
3
 0 z 0 r 0
m
(m is the total mass of the cylinder and V is its volume). This
V
is worked out easily by hand, and we illustrate the calculation with Maple on the
worksheet for this project.
where  is the density

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Part III. Calculation of the Descent Time
Suppose the height of the ramp is H, and that the angle between the ramp and the
horizontal is . Suppose further that a body with radius a, after starting at rest from the
top of the ramp, has rolled part way down the ramp, achieving a velocity v after having
descended a vertical distance y.
Figure 3
It has lost potential energy mgy , and it has total kinetic energy
1
1
mv 2 
I 2.
2
2
If the object is rolling without slipping, then v and  are related by v  a  .
v
We may then replace  by
, and equate the potential energy lost with the kinetic
a
energy. That is,
2
1
1 v
m g y  m v 2  I 
2
2 a 
I 
2 
2 g y  v 1
.
 m a 2 
I
is called the normalized moment of inertia, and it's the parameter
m a2
I
to keep track of. Let's denote it by J =
, and see how it determines the descent time.
m a2
We now see that
The ratio
v2 
2gy
,
1 J
and since the vertical component of velocity is v sin(), we obtain
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dy
 v sin( ) 
dt
2g y
sin( ) .
1 J
Separating variables yields
y
 12
2g
sin( ) dt .
1  J
dy 
Finally, denoting the descent time by T, we integrate this expression to obtain
H
y
 12
T
dy 
0

0
2g
sin( )dt .
1  J
Thus
2 H
T 

2g
sin( )T , and solving for T:
1  J
2H (1  J)
.
2
gsin ( )
Let's meditate on this formula for descent time. First, if the ramp height H is
increased, T is increased. Second, if the angle of elevation  is small, the descent time is
large. Third, if the acceleration of gravity g is increased (say by rolling objects on
Jupiter) the descent time T is decreased. All of these features of the descent time agree
with our intuition. Finally, note the parameter J is the only parameter controlled by the
geometry of the rolling body. The smaller J is, the shorter the descent time.
Part IV. Some figures
You may find these figures useful when you do the exercises.
A hollow sphere having mass m, inner radius ai , and outer radius ao
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A hollow cylinder of mass m, length l, inner radius ai , and outer radius ao .
References
Andrew, Cain, Crum, Morley, Calculus Projects Using Mathematica, McGraw-Hill,
1996.
Drucker, A Mathematical Roller Derby, The College Mathematics Journal, Vol 23, No.
5, 1992, 396-401
Salas, Hille, Etgen, Calculus: one and several variables, ninth edition, John Wiley and
Sons, 2003.
Strang, Calculus, Wellesley-Cambridge Press, Wellesley, MA, 1991.
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