The Kinetic Energy of Rolling
Let us now calculate the kinetic energy of the rolling wheel as measured by the
stationary observer. If we view the rolling as pure rotation about an axis through
P in Fig. 11-6, then from Eq. 10-34 we have
(11-3)
in which v is the angular speed of the wheel and I P is the rotational inertia of the
wheel about the axis through P. From the parallel-axis theorem of Eq. 10-36
(I ϭ I com ϩ Mh
2
), we have
I P ϭ I com ϩ MR
2
,
(11-4)
in which M is the mass of the wheel, I com is its rotational inertia about an axis
through its center of mass, and R (the wheel’s radius) is the perpendicular
distance h. Substituting Eq. 11-4 into Eq. 11-3, we obtain
and using the relation v com ϭ vR (Eq. 11-2) yields
(11-5)
We can interpret the term
as the kinetic energy associated with the
rotation of the wheel about an axis through its center of mass (Fig. 11-4a), and the
term
as the kinetic energy associated with the translational motion of the
wheel’s center of mass (Fig. 11-4b). Thus, we have the following rule:
1
2 Mv com
2
1
2 I com v
2
K ϭ
1
2 I com v
2 ϩ
1
2 ⌴v
2
com .
K ϭ
1
2 I com v
2 ϩ
1
2 ⌴R
2 v
2
,
K ϭ
1
2 I P v
2
,
298
CHAPTE R 11 ROLLI NG, TORQU E, AN D ANG U L AR M OM E NTU M
● A smoothly rolling wheel has kinetic energy
where I com is the rotational inertia of the wheel about its center of mass and M is the mass of the wheel.
● If the wheel is being accelerated but is still rolling smoothly,
the acceleration of the center of mass
is related to the
a
:
com
K ϭ
1
2 I com v
2 ϩ
1
2 ⌴v
2
com ,
angular acceleration a about the center with
a com ϭ aR.
● If the wheel rolls smoothly down a ramp of angle u, its
acceleration along an x axis extending up the ramp is
a com, x ϭ Ϫ
g sin u
1 ϩ I com /MR
2
.
Key Ideas
A rolling object has two types of kinetic energy: a rotational kinetic energy
due to its rotation about its center of mass and a translational kinetic
(
1
2 I com v
2
)
energy
due to translation of its center of mass.
(
1
2 Mv com
2
)
11-2 FORCES AND KINETIC ENERGY OF ROLLING
After reading this module, you should be able to . . .
11.03 Calculate the kinetic energy of a body in smooth rolling as
the sum of the translational kinetic energy of the center of mass
and the rotational kinetic energy around the center of mass.
11.04 Apply the relationship between the work done on a
smoothly rolling object and the change in its kinetic energy.
11.05 For smooth rolling (and thus no sliding), conserve mechanical energy to relate initial energy values to the values
at a later point.
11.06 Draw a free-body diagram of an accelerating body that is
smoothly rolling on a horizontal surface or up or down a ramp.
11.07 Apply the relationship between the center-of-mass
acceleration and the angular acceleration.
11.08 For smooth rolling of an object up or down a
ramp, apply the relationship between the object’s
acceleration, its rotational inertia, and the angle of
the ramp.
Learning Objectives
Let us now calculate the kinetic energy of the rolling wheel as measured by the
stationary observer. If we view the rolling as pure rotation about an axis through
P in Fig. 11-6, then from Eq. 10-34 we have
(11-3)
in which v is the angular speed of the wheel and I P is the rotational inertia of the
wheel about the axis through P. From the parallel-axis theorem of Eq. 10-36
(I ϭ I com ϩ Mh
2
), we have
I P ϭ I com ϩ MR
2
,
(11-4)
in which M is the mass of the wheel, I com is its rotational inertia about an axis
through its center of mass, and R (the wheel’s radius) is the perpendicular
distance h. Substituting Eq. 11-4 into Eq. 11-3, we obtain
and using the relation v com ϭ vR (Eq. 11-2) yields
(11-5)
We can interpret the term
as the kinetic energy associated with the
rotation of the wheel about an axis through its center of mass (Fig. 11-4a), and the
term
as the kinetic energy associated with the translational motion of the
wheel’s center of mass (Fig. 11-4b). Thus, we have the following rule:
1
2 Mv com
2
1
2 I com v
2
K ϭ
1
2 I com v
2 ϩ
1
2 ⌴v
2
com .
K ϭ
1
2 I com v
2 ϩ
1
2 ⌴R
2 v
2
,
K ϭ
1
2 I P v
2
,
298
CHAPTE R 11 ROLLI NG, TORQU E, AN D ANG U L AR M OM E NTU M
● A smoothly rolling wheel has kinetic energy
where I com is the rotational inertia of the wheel about its center of mass and M is the mass of the wheel.
● If the wheel is being accelerated but is still rolling smoothly,
the acceleration of the center of mass
is related to the
a
:
com
K ϭ
1
2 I com v
2 ϩ
1
2 ⌴v
2
com ,
angular acceleration a about the center with
a com ϭ aR.
● If the wheel rolls smoothly down a ramp of angle u, its
acceleration along an x axis extending up the ramp is
a com, x ϭ Ϫ
g sin u
1 ϩ I com /MR
2
.
Key Ideas
A rolling object has two types of kinetic energy: a rotational kinetic energy
due to its rotation about its center of mass and a translational kinetic
(
1
2 I com v
2
)
energy
due to translation of its center of mass.
(
1
2 Mv com
2
)
11-2 FORCES AND KINETIC ENERGY OF ROLLING
After reading this module, you should be able to . . .
11.03 Calculate the kinetic energy of a body in smooth rolling as
the sum of the translational kinetic energy of the center of mass
and the rotational kinetic energy around the center of mass.
11.04 Apply the relationship between the work done on a
smoothly rolling object and the change in its kinetic energy.
11.05 For smooth rolling (and thus no sliding), conserve mechanical energy to relate initial energy values to the values
at a later point.
11.06 Draw a free-body diagram of an accelerating body that is
smoothly rolling on a horizontal surface or up or down a ramp.
11.07 Apply the relationship between the center-of-mass
acceleration and the angular acceleration.
11.08 For smooth rolling of an object up or down a
ramp, apply the relationship between the object’s
acceleration, its rotational inertia, and the angle of
the ramp.
Learning Objectives