AP05_1.vp


1 Introduction
If a rotating body is heated, its internal energy increases

and the concurrent temperature increase is followed by ther-
mal expansion. This results in an increase in the moment of
inertia of the body. When no external torque affects the body,
then according to the principle of conservation of angular
momentum, the angular momentum is constant [1]. Since
the angular momentum of the body is proportional to the
moment of inertia and to the angular velocity [1], the increase
in the moment of inertia is followed by a reduction in angular
velocity. The final effect is that there is a decrease in the
kinetic energy of the rotating body, which is proportional to
the moment of inertia and to the power in the angular veloc-
ity. (An example of a converse effect is the increase of the
angular velocity of a “black hole” during its gravitational
collapse).

According to the energy conservation law, the decay of
kinetic energy must convert into another form of energy.
Only thermal energy and deformation energy originat-
ing from centrifugal forces are in consideration. However,
as proved by the following mathematical calculation, defor-
mation energy also decreases with thermal expansion. This
means that the deficit of kinetic energy and of deformation
energy converts into an increase in the internal energy of the
body. In sum, if a rotating body is heated, the increase in its in-
ternal energy is higher than if the body at rest. The “addi-
tional” increase in the internal energy of a rotating body
appears to the detriment of its kinetic and deformation en-
ergy. In the following study the case of heating or cooling a
rotating body is treated mathematically, and the trend of the
quantities characterising the effect is demonstrated under
concrete parameters.

2 Assumptions
It is supposed that the body, rotating with angular velocity

�0, (subscript “0” always means the initial value of a variable) is a
cylinder with radius r0, mass m and height h0, which is small

in comparison with radius r0. The axis of rotation and the
geometric axis of the cylinder are identical. The density
of the cylinder material is �0, Young’s modulus E, Pois-
son’s number �, specific heat capacity c, and coefficient of
thermal expansion � (Fig. 1). Quantities E, c, � are con-
stant in the considered range of temperatures (approximately

300–1000 K) and the tension stress in the material does not
exceed the proportional limit. No external forces affect the
body and no uncontrolled heat transfer occurs between the
cylinder and the environment (Fig. 2). For the parameters
considered, the elastic deformation of the body originating
from centrifugal forces is very small in comparison with the
thermal expansion, and therefore its dependence on angular
velocity is neglected.

3 Mathematical treatment
It is supposed that a quantity of heat �Q is transferred

into or out of a rotating body (for instance by radiation, see
Fig. 2). A mathematical description of the consecutive behav-
iour of the body follows.

©  Czech Technical University Publishing House e http://ctn.cvut.cz/ap/ 19

Czech Technical University in Prague Acta Polytechnica Vol. 45  No. 1 /2005

Heating and Cooling Anomaly of a
Rotating Body

O. Brůha, T. Brůha

This paper deals with an effect which appears when heating or cooling a rotating body. No external forces acting on the body are supposed.
Due to thermal expansion, the moment of inertia of the body varies together with the temperature changes. In agreement with the principle of
conservation of angular momentum [1], the angular momentum is constant. This results in angular velocity changes and subsequently in
kinetic energy changes. Also the stress energy varies together with the changes in thermal dimension. To satisfy the principle of energy
conservation we have to suppose that the changes in kinetic and stress energy are compensated by the changes in internal energy, which is
correlated with temperature changes of the body. This means that the rules for the heating or cooling process of a rotating body are not the
same as those for a body at rest. This idea, applied to a cylinder rotating around its geometric axis under specific parameters, has been
mathematically treated. As a result, the difference between the final temperature of the rotating cylinder and the temperature of the cylinder
at rest has been found.

Keywords: angular momentum, energy conservation, rotating body, thermal expansion.

�

c, E, , ,� � �

r

h

Fig. 1: A rotating cylinder and its parameters



Basic relations
Moment of inertia of the cylinder

J mr�
1
2

2. (1)

Rotational kinetic energy of the cylinder

E Jk �
1
2

2� . (2)

Angular momentum
b J� �� const. (if the external torque is zero) [1], [2] (3)
The linear coefficient of thermal expansion is defined gener-
ally as:

� �
1

0x
x
t

d
d

, (4)

where x0 is initial length and dt is temperature change.
For better mathematical access another definition was

used, namely:

� �
1
x

x
t

d
d

, (4a)

where x is variable.
Relations (4) and (4a) result in practically the same values

for the parameters used.
(For � t � 700 K and � � � �16 10 6 K�1, the difference in

�x calculated according to (4a) or according to (4) is only
0.0089 �x.)

Using relations (1) and (2) we can express the rotational
kinetic energy of the cylinder (see Fig. 1.)

E h rk �
1
4

4 2� � � . (5)

For r(t) and h(t) as functions of temperature we obtain from
definition (4):

r t r e t( ) � 0
�� , (6)

h t h e t( ) � 0
�� , (7)

where �t t t� � 0 .
Density �(t) and angular velocity �(t) as functions of tem-

perature can be expressed in a similar way (see Appendix,
items 1. and 2.)

� �( )t e t� �0
3�� , (8)

� �( )t e t� �0
2�� . (9)

Substituting relations (6), (7), (8) and (9) in (5) we obtain
an expression for rotational kinetic energy as a function of
temperature

E t r h e E et tk k0( ) � �
� �1

4 0
4 2

0 0
2 2� � ��

� �� � . (10)

The deformation energy should also be taken into con-
sideration. The total deformation energy corresponding to
the centrifugal forces in a rotating cylinder is (see Appendix,
item 3.)

E
r h

�
�� �

� �
�

	



�

�

 �

�

�
�

� � � �

2 4 7
2

448
2
3 2

3

3 1 3
6

E
�

�

� �
�

( )

( )( )
1
2

1
6

1 3 2
�

	



�

�

 � �

�

�
�( ) .�

(11)

Substituting relations (6), (7), (8) and (9) in (11) we obtain
for deformation energy as a function of temperature:

E t
r h e t

�
� � �

( ) ( )

(

� �
�

	



�

�

 �

�

�
�

�

�
0
2

0
4

0
7

0
6

2

448
2
3 2

3
� �

�
�

E

3 1 3
6

1
2

1
6

1 3 2

0
6

� � �
�

	



�

�

 � �

�

�
�

� �

� �
�

�

�

) ( ) ( )

.E e t�
�

(12)

The total mechanical energy is

E t E t E t� �( ) ( ) ( )� � �k k . (13)

Using relations (10) and (12), the change in total mechan-
ical energy as a function of temperature can be expressed as:

20 ©  Czech Technical University Publishing Housee http://ctn.cvut.cz/ap/

Acta Polytechnica Vol. 45  No. 1 /2005 Czech Technical University in Prague

�0 �( )t

r t( )

h0 h t( )

t0

r0

t
�0 �( )t

� �Q (radiation) �t t t� � 0

vacuum vacuum

Fig. 2: An illustration of an isolated rotating body



� �
�

� �

E t E t E t E E

E e E et t
� � ��

��

�

� �

� � � �

� � �

k k k0

k0

( ) ( ) ( )
2 61� �� �� 1 .

(14)

Now we suppose that a quantity of heat �Q is added to
(or taken away from) the rotating cylinder and we will try to
express the corresponding change in its temperature �t ac-
cording to equation (14). After the thermal field in the cyl-
inder reaches the steady state, the angular velocity of the
cylinder is again constant. There are no more reasons for
changes to it. This means that the cylinder is in thermody-
namic equilibrium both at the beginning of the process and
also at the end [4]. Consequently, these two states must satisfy
the principle of conservation of energy. Taking into account
that not only mechanical energy, but also the change in inter-
nal energy �U has to be considered, the law of conservation
of energy can be written as

� � �Q U E� � �� k , (15)
and

� � �U Q E� � �� k . (16)
For a body at rest, it is generally supposed that:

� �U Q0 � . (17)
This means that the change in internal energy, originating

from the heat transferred �Q, is not the same for a rotating
body and for the same body at rest. There exists a difference

� � � � � � � �� �� � � � � � �U U U Q E Q E0 � �k k . (18)

It is well known that a change in internal energy is corre-
lated with a temperature change in a body. [1]. This means
that a change in temperature corresponding to �Q of a rotat-
ing body also differs from that of a body at rest.

Let us accept a simplified presumption, that the internal
energy of a rigid body is entirely a function of the kinetic
energy of chaotic motion of all particles and that an increase
in it corresponds to an increase in the body temperature.
Then we obtain a simplified expression for the difference
	� * )t :

� � �
�

* t t t
U

m
� � �

�
s c

[1], (19)

where �t is a change in temperature of the rotating body,
and �ts is a change in temperature of the still body, m is mass
and c is specific heat capacity of the body.

Equation (19) combined with (14) and (18) yields:

�
� �

*
( ) ( )

t
E e E e

m

t t
�

� � �� �k0
c

1 12 0
6� �

� . (20)

According to (19)
� � �t t t� �s *

and then (20) reads:

�
� � � �

*
( ) ( )( * ) ( * )

t
E e E e

m

t t t t
�

� � �� � � �k0 s s

c
1 12 0

6� �
� , (21)

where � ts is:

�
�

t
Q

m
s c

� [1] (22)

Expression (21) is a transcendent equation for unknown
quantity � * t and given quantity � ts . This equation was
solved with the help of the MAPLE mathematical program,
and the following real solutions for � * t as a function of � ts
were found:

� �* ( )

(

t f t

E e Zm E e Z E e Z E

� �

� �
� � � � �

s

RootOf c k
�

� �
0

6 0
2

0
2� � � 0 0

0
6

0
6

0
2

� �

� � �
�

�

� �

� �

E t m

Zm E e Z E e Z

m

E e

k s

k

c

k
RootOf c

c

� )

( � � � �

�
�

� � �E E t m

m
E E

m

�

�

0 0

0 0 23

k s c

k

c

c

� )

( )

(The symbol Z in this equation represents an auxiliary vari-
able for the solution of partial expressions for given � ts .)

The solution (23) is illustrated (Fig. 3) for the following pa-
rameters of the rotating cylinder:

©  Czech Technical University Publishing Housee http://ctn.cvut.cz/ap/ 21

Czech Technical University in Prague Acta Polytechnica Vol. 45  No. 1 /2005

Cooling of body

Heating of body

| | [K]�ts

| [K]��t|

Fig. 3: Curves � * t versus �ts as a solution of the equation (23)



material – steel
r0 1� m
� �0 2 100� rad×s

�1

h0 01� . m
�0 7800� kg×m

�3 [2]
E � �2 1011 Pa
a � � �16 10 6 K�1 [2]
c � 483 J×kg�1×K�1 [2]
� � 0 29. [3]

Conclusions

Fig. 3 shows the curves of � * t versus � ts (in absolute val-
ues) as a solution of equation (23), both for heating and for
cooling the cylinder. For � ts max � 700 K, and for the parame-

ters supposed, the value of � * t is:

� * ( ) .t 700 4 62K K� for heating and
� * ( ) .t 700 4 72K K� for cooling.

The results show that the effect is more readable for cool-
ing than for the heating process.

This corresponds to the fact that for the heating process
the temperature difference � * t has the finite asymptotic
value:

� �* ( )t
E E

m
t �� �

�k
c

0 0� (follows from (21)).

(This relation describes a theoretical extreme case when the
whole initial kinetic and deformation energy converts into
inner energy and the cylinder “comes to a standstill”).

In the case of cooling, the mathematical expression for
� * t has no limit.

In the study it is supposed that the coefficient of thermal
expansion of the cylinder material is positive. If the coef-
ficient of thermal expansion is negative, the heating of a ro-
tating cylinder would result in an increase in kinetic energy on
account of the heat transferred. As materials with a negative
coefficient of thermal expansion are very rare (for instance
quenched irreversible Fe-Ni invar alloys), this case is not
considered in detail in the study.

Appendix
1. Relation (8):

From its definition, the density is given by:

� �
m
V

(A1)

and according to Fig. 1:

�
�

�
m
r h2

, (A2)

or:

�
�

�
�

0
0
2

0
2

� �
m
r h

t
m

r t h t
, ( )

( ) ( )
, (A3)

and from relation (6, 7) follows:

�
� �

�( )t
m

r e h e
m

r h e
e

t t t
t� � � �

0
2 2

0 0
2

0
3 0

3
� � �

�
� � �

� . (A4)

2. Relation (9):
According to relation (3):

J const� � . , (A5)

or:

J J t t0 0� �� ( ) ( ) , (A6)

� �� �( )
( )

t
J
J t

0
0 (A7)

and from relation (1, 6) follows:

� � �( )t
mr

mr e
e

t
t� � �

1
2

1
2

0
2

0
2 2

0 0
2

�

�

�

� . (A8)

3. Relation (11):
Note: since the height of the cylinder is small in comparison
with its radius, the axial stress is neglected.

The deformation energy E� per volume unit correspond-
ing to the tension stress � is:

E
V

� ��
2

2E
[2] (A9)

and

E
V

�
�

�
2

2E
, (A10)

and in elementary form:

d
d
E

E
V

�
�

�
2

2
. (A11)

In the case that the height of the rotating cylinder is small
in comparison with its radius, radial and tangential stress �r
and �t, originating from centrifugal force, can be expressed as
(see Fig. 4)

� ��r( ) ( )r r rx x�
�

�
3

8
2 2 2� (A12)

and

�
��


 
t( ) ( ( ( )r r rx x� � � �
2

2 2

8
3 1 3� � , [3] (A13)

where rx is the radius of stress action.

The elementary deformation energy corresponding to the
radial and tangential stress is

d
d

E r t r t
E r

V
r r r rx x x x x� � � � �( ) ( ( ) ( ) ( ) ( ))� � �2

22 2 � . (A14)

As shown in Fig. 4.:

d d d dV l S r h rx� � � . (A15)

Substituting relation (A15) in (A14) we obtain for elemen-
tary deformation energy:

d
E

d d

r t

r t

E r
h

r r r

r r r

x x x x

x x

� � �

� � �

( ) ( ( ) ( )

( ) ( ))

� � �

�
2

2

2 2

� �

(A16)

Substituting relations (A12), (A13) in (A16) and integrat-
ing we obtain the integral deformation energy accumulated
in the rotating cylinder:

22 ©  Czech Technical University Publishing Housee http://ctn.cvut.cz/ap/

Acta Polytechnica Vol. 45  No. 1 /2005 Czech Technical University in Prague



E E r r

r h

x

r

� �

�

�

�� �

�

� �
�

	



�

�

 �

�� d d d

E

( )

( )

0

2

0
2 4 7

448
2
3 2

3
�

� 2

23 1 3
6

1
2

1
6

1 3

�

	




� � � �
�

	



�

�

 � �

�

�


( ) ( ) ( ) .� �

�
�

(A17)

References
[1] Tipler, P. A.: “Physics.” New York, Worth Publisher Inc.,

1982, p. 80, 295–297, 495–496, 498–500.
[2] Horák, Z., Krupka, F., Šindelář, V.: „Technická fyzika.“

Praha, SNTL, 1960, p. 187–188, 310, 1411.
[3] Michalec, J. et al.: „Pružnost a pevnost II.“ Praha, Vyda-

vatelství ČVUT, 2001, p. 78–82.
[4] Landau, L. D., Lifschitz, E. M.: “Lehrbuch der

Theoretischen Physik V: Statistische Physik.” Berlin,
Akademie-Verlag, 1966, Kap. 2, § 10, p. 38–40.

Doc. Ing. Oldřich Brůha, CSc.
phone: +420 602 458 314
fax: +420 251 561 115
e-mail: oldrich.bruha@volny.cz

Department of Physics

Czech Technical University in Prague
Faculty of Mechanical Engineering
Technická 4
166 07 Praha 6, Czech Republic

Ing. Tomáš Brůha
phone: +420 777 895 762
e-mail: enex@volny.cz

ENEX s.r.o.
Na Míčánce 13
160 00 Praha 6, Czech Republic

©  Czech Technical University Publishing Housee http://ctn.cvut.cz/ap/ 23

Czech Technical University in Prague Acta Polytechnica Vol. 45  No. 1 /2005

dm r
 
�
2

x

d /2�

d�

d /2�

� 
 
t h rd
� 
 
t h rd

( d ) d� � � 
 �
r r hrx

� 
 �
r d h
rx

h

dl �

drrx

Fig. 4: Centrifugal and stress forces in a rotating cylinder