Chapter 8
Rotational Motion
Objectives
Distinguish between inertia and moment of
inertia.
Calculate the moment of inertia of various
objects.
Explain the meaning of the radius of
gyration. Use the radius of gyration to solve
for an object's moment of inertia.
Engagement
Discussion of Launch Activity
1. List objects from greatest to least
acceleration.
2. Which of the object’s properties may
have contributed to their behavior?
3. List properties that were the same and
different for each object.
4. Demonstration
5. Demonstration 2
Rotational Inertia and Rolling
Which will roll down an incline with greater
acceleration, a hollow cylinder or a solid
cylinder of the same mass and radius?
The answer is the cylinder with the smaller
rotational inertia because the cylinder with the
greater rotational inertia requires more time to
get rolling.
Rotational Inertia and Rolling
Inertia of any kind is a measure of “laziness.”
The cylinder with its mass concentrated
farthest from the axis of rotation—the hollow
cylinder—has the greater rotational inertia.
The solid cylinder will roll with greater
acceleration.
Rotational Inertia and Rolling
Any solid cylinder will
roll down an incline
with more acceleration
than any hollow
cylinder, regardless of
mass or radius.
A hollow cylinder has
more “laziness per
mass” than a solid
cylinder.
Rotational Inertia and Rolling
A solid cylinder rolls down an incline
faster than a hollow one, whether or not
they have the same mass or diameter.
Rotational Inertia and Rolling
think!
A heavy iron cylinder and a light wooden cylinder, similar in
shape, roll down an incline. Which will have more
acceleration?
Rotational Inertia and Rolling
think!
A heavy iron cylinder and a light wooden cylinder, similar in
shape, roll down an incline. Which will have more
acceleration?
Answer:
The cylinders have different masses, but the same rotational
inertia per mass, so both will accelerate equally down the
incline. Their different masses make no difference, just as the
acceleration of free fall is not affected by different masses. All
objects of the same shape have the same “laziness per
mass” ratio.
Rotational Inertia and Rolling
think!
Would you expect the rotational inertia of a hollow sphere
about its center to be greater or less than the rotational inertia
of a solid sphere? Defend your answer.
Rotational Inertia and Rolling
think!
Would you expect the rotational inertia of a hollow sphere
about its center to be greater or less than the rotational inertia
of a solid sphere? Defend your answer.
Answer:
Greater. Just as the value for a hoop’s rotational inertia is
greater than a solid cylinder’s, the rotational inertia of a
hollow sphere would be greater than that of a same-mass
solid sphere for the same reason: the mass of the hollow
sphere is farther from the center.
The shape of an object determines
how easy or hard it is to spin
Hinge
For objects of the same mass, the longer
one is tougher to spin  takes more torque
It matters where the hinge is
The stick with the hinge at the end takes 4 times
more torque to get it spinning than the stick with
the hinge in the center.
Rotational Inertia
(moment of inertia)
• Rotational inertia is a parameter that is
used to quantify how much torque it takes
to get a particular object rotating
• it depends not only on the mass of the
object, but where the mass is relative to
the hinge or axis of rotation
• the rotational inertia is bigger, if more
mass is located farther from the axis.
How fast does it spin?
• For spinning or rotational motion, the
rotational inertia of an object plays the
same role as ordinary mass for simple
motion
• For a given amount of torque applied to an
object, its rotational inertia determines its
rotational acceleration  the smaller the
rotational inertia, the bigger the rotational
acceleration
Rotational Inertia
Newton’s first law, the law of inertia, applies
to rotating objects.
• An object rotating about an internal axis
tends to keep rotating about that axis.
• Rotating objects tend to keep rotating,
while non-rotating objects tend to remain
non-rotating.
• The resistance of an object to changes in
its rotational motion is called rotational
inertia (sometimes moment of inertia).
Rotational Inertia
Just as it takes a force to change the linear
state of motion of an object, a torque is
required to change the rotational state of
motion of an object.
In the absence of a net torque, a rotating
object keeps rotating, while a non-rotating
object stays non-rotating.
Same torque,
different
rotational inertia
Big rotational
inertia
Small rotational
inertia
spins
slow
spins
fast
Rotational Inertia
Rotational Inertia and Mass
Like inertia in the linear
sense, rotational inertia
depends on mass, but
unlike inertia, rotational
inertia depends on the
distribution of the mass.
The greater the distance
between an object’s
mass concentration and
the axis of rotation, the
greater the rotational
inertia.
8-5 Rotational Dynamics; Torque and
Rotational Inertia
Knowing that
, we see that
(8-11)
This is for a single point
mass; what about an
extended object?
As the angular
acceleration is the same
for the whole object, we
can write:
(8-12)
8-5 Rotational Dynamics; Torque and
Rotational Inertia
The quantity
is called the
rotational inertia of an object.
The distribution of mass matters here – these
two objects have the same mass, but the one on
the left has a greater rotational inertia, as so
much of its mass is far from the axis of rotation.
Demo with inertial rods
Rotational Inertia
Rotational inertia depends on the distance of
mass from the axis of rotation.
Rotational Inertia
By holding a long pole, the tightrope walker
increases his rotational inertia.
Rotational Inertia
A long baseball bat held near its thinner end
has more rotational inertia than a short bat of
the same mass.
• Once moving, it has a greater tendency to
keep moving, but it is harder to bring it up
to speed.
• Baseball players sometimes “choke up”
on a bat to reduce its rotational inertia,
which makes it easier to bring up to
speed.
A bat held at its end, or a long bat, doesn’t
swing as readily.
Rotational Inertia
The short pendulum will swing back and forth
more frequently than the long pendulum.
Rotational Inertia
For similar mass
distributions, short
legs have less
rotational inertia
than long legs.
Rotational Inertia
The rotational inertia of an object is not
necessarily a fixed quantity.
It is greater when the mass within the
object is extended from the axis of
rotation.
Rotational Inertia
You bend your legs when you run to reduce their
rotational inertia. Bent legs are easier to swing
back and forth.
Rotational Inertia
think!
When swinging your leg from your hip, why is the rotational
inertia of the leg less when it is bent?
Rotational Inertia
think!
When swinging your leg from your hip, why is the rotational
inertia of the leg less when it is bent?
Answer:
The rotational inertia of any object is less when its mass is
concentrated closer to the axis of rotation. Can you see that a
bent leg satisfies this requirement?
8-5 Rotational
Dynamics; Torque
and Rotational
Inertia
The rotational inertia of
an object depends not
only on its mass
distribution but also the
location of the axis of
rotation – compare (f)
and (g), for example.
Rotational Inertia
Formulas for Rotational Inertia
When all the mass m of an object is
concentrated at the same distance r from a
rotational axis, then the rotational inertia is I =
mr2.
When the mass is more spread out, the
rotational inertia is less and the formula is
different.
Rotational Inertia and Gymnastics
The human body can rotate freely about three
principal axes of rotation.
Each of these axes is at right angles to the
others and passes through the center of
gravity.
The rotational inertia of the body differs about
each axis.
Rotational Inertia and Gymnastics
The human body has three principal axes of
rotation.
Rotational Inertia and Gymnastics
Longitudinal Axis
Rotational inertia is least about the longitudinal
axis, which is the vertical head-to-toe axis,
because most of the mass is concentrated
along this axis.
• A rotation of your body about your
longitudinal axis is the easiest rotation to
perform.
• Rotational inertia is increased by simply
extending a leg or the arms.
• Period 1 stopped here.
Rotational Inertia and Gymnastics
An ice skater rotates around her longitudinal
axis when going into a spin.
a.The skater has the least amount of
rotational inertia when her arms are tucked
in.
Rotational Inertia and Gymnastics
An ice skater rotates around her longitudinal axis
when going into a spin.
a.The skater has the least amount of rotational
inertia when her arms are tucked in.
b.The rotational inertia when both arms are
extended is about three times more than in the
tucked position.
Rotational Inertia and Gymnastics
c and d. With your leg and arms extended, you
can vary your spin rate by as much as
six times.
Rotational Inertia and Gymnastics
Transverse Axis
You rotate about your transverse axis when you
perform a somersault or a flip.
12.2 Rotational Inertia and Gymnastics
A flip involves rotation about the transverse axis.
a. Rotational inertia is least in the tuck position.
12.2 Rotational Inertia and Gymnastics
A flip involves rotation about the transverse axis.
a. Rotational inertia is least in the tuck position.
b. Rotational inertia is 1.5 times greater.
Rotational Inertia and Gymnastics
A flip involves rotation about the transverse axis.
a. Rotational inertia is least in the tuck position.
b. Rotational inertia is 1.5 times greater.
c. Rotational inertia is 3 times greater.
Rotational Inertia and Gymnastics
A flip involves rotation about the transverse axis.
a. Rotational inertia is least in the tuck position.
b. Rotational inertia is 1.5 times greater.
c. Rotational inertia is 3 times greater.
d. Rotational inertia is 5 times greater than in the tuck position.
Rotational Inertia and Gymnastics
Rotational inertia is greater when the axis
is through the hands, such as when doing a
somersault on the floor or swinging from a
horizontal bar with your body fully
extended.
Rotational Inertia and Gymnastics
The rotational inertia of a
body is with respect to the
rotational axis.
a.The gymnast has the
greatest rotational
inertia when she pivots
about the bar.
Rotational Inertia and Gymnastics
The rotational inertia of a
body is with respect to the
rotational axis.
a.The gymnast has the
greatest rotational
inertia when she pivots
about the bar.
b.The axis of rotation
changes from the bar
to a line through her
center of gravity when
she somersaults in the
tuck position.
Rotational Inertia and Gymnastics
The rotational inertia of a gymnast is up to 20
times greater when she is swinging in a fully
extended position from a horizontal bar than
after dismount when she somersaults in the
tuck position.
Rotation transfers from one axis to another,
from the bar to a line through her center of
gravity, and she automatically increases her
rate of rotation by up to 20 times.
This is how she is able to complete two or
three somersaults before contact with the
ground.
Rotational Inertia and Gymnastics
Medial Axis
The third axis of rotation for the human body is
the front-to-back axis, or medial axis.
This is a less common axis of rotation and is
used in executing a cartwheel.
8-6 Solving Problems in Rotational
Dynamics
1. Draw a diagram.
2. Decide what the system comprises.
3. Draw a free-body diagram for each object
under consideration, including all the forces
acting on it and where they act.
4. Find the axis of rotation; calculate the torques
around it.
8-6 Solving Problems in Rotational
Dynamics
5. Apply Newton’s second law for rotation. If
the rotational inertia is not provided, you
need to find it before proceeding with this
step.
6. Apply Newton’s second law for translation
and other laws and principles as needed.
7. Solve.
8. Check your answer for units and correct
order of magnitude.
Elaboration
Moments of Inertia hand-out
Practice Problem 1 (#28 in text)
Calculate the moment of inertia of a
66.7cm-diameter bicycle wheel. The rim
and tire have a combined mass of 1.25kg.
The mass of the hub can be ignored
(why?).
Practice Problem 1 (#28 in text)
Calculate the moment of inertia of a 66.7cm-diameter bicycle wheel. The
rim and tire have a combined mass of 1.25kg. The mass of the hub can be
ignored (why?).
The moments of inertia are listed on p. 223, and a
thin hoop through its center is:
I = mr2
m = 1.25 kg
r = (.667 m)/2 = .3335 m (They gave you the
diameter)
so
I = (1.25 kg)(.3335 m)2 = 0.139 kgm2
Practice Problem 2 (similar to #29)
A small 1.05kg ball on the end of the light
rod is rotated in a horizontal circle of a
radius 0.900 m. Calculate (a) the moment
of inertia of the system about the axis of
rotation, and (b) the torque needed to
keep the ball rotating at a constant
angular velocity if air resistance exerts a
force of 0.0800N on the ball.
Practice Problem 2 (similar to #29)
A small 1.05kg ball on the end of the light rod is rotated in a horizontal circle of a radius
0.900 m. Calculate (a) the moment of inertia of the system about the axis of rotation, and (b)
the torque needed to keep the ball rotating at a constant angular velocity if air resistance
exerts a force of 0.0800N on the ball.
The small ball can be treated as a particle for calculating the
moment of inertia.
I = mr2
I = (1.05 kg)(.900)2 = 0.8505 kgm2
If the mass is moving at a constant angular velocity, then it is not
accelerating (net torque is zero), and the only necessary torque
needed is the same magnitude as the torque caused by the
frictional force of .0800 N acting at a radius of .900 m on the ball:
Since force is applied at a 90o angle to the radius, so the factor
sinbecomes 1, and really the torque is:
(.0800 N)(.900 m) = .072 Nm
Practice Problem 3 (#40)
A helicopter rotor blade can be considered a
long thin rod, as shown in Fig. 8-42. If each of
the three rotor helicopter blades is 3.75m long
and has a mass of 160kg, calculate the moment
of inertia of the three rotor blades about the
axis of rotation. How much torque must the
motor apply to bring the blades up to a speed
of 5.0rev/s in 8.0s?
Practice Problem 3 (#40)
A helicopter rotor blade can be considered a long thin rod, as shown in Fig. 8-42. If each
of the three rotor helicopter blades is 3.75m long and has a mass of 160kg, calculate the
moment of inertia of the three rotor blades about the axis of rotation. How much torque
must the motor apply to bring the blades up to a speed of 5.0rev/s in 8.0s?
The moments of inertia are listed on p. 208, and a long thin rod through its end is:
I = 1/3ML2
so each rotor has a moment of inertia of:
I = 1/3(160 kg)(3.75 m)2 = 2500 kgm2 = 750 kgm2
Three rotors would have three times this moment:
I = (750 kgm2)3 = 2250 kgm2
Now we need to solve a kinematics question:
o = 0
 = (5 Revolutions/s)(2radians/revolution) = 31.416 rad/s
t = 8.0 s
Now apply:
o + t
 = 3.927 rad/s/s
And now find the torque using F = ma:

= (2250 kgm2)(3.927 rad/s/s) = 8836 Nm = 8800 Nm of torque
Homework
• Chapter 8 Problems #27, 31, 33
Closure
• Kahoot 8-5 and 8-6