Ch10 – Circular Motion
Ch11 – Rotational Equilibrium
Ch12 – Rotational Motion
Conceptual Physics
Ch10 Frames 2-47 (Asmt:38-47)
Ch11 Frames 48-111 (Asmt:98-111)
Ch11 Frames 112-158 (Asmt:147-158)
Centripetal force keeps an
object in circular motion.
10.1 Rotation and Revolution
Two types of circular motion are
rotation and revolution.
10.1 Rotation and Revolution
An axis is the straight line
around which rotation takes
place.
• When an object turns
about an internal axis—
that is, an axis located
within the body of the
object—the motion is
called rotation, or spin.
• When an object turns
about an external axis,
the motion is called
revolution.
10.1 Rotation and Revolution
The Ferris wheel turns
about an axis.
The Ferris wheel
rotates, while the riders
revolve about its axis.
10.1 Rotation and Revolution
Earth undergoes both types of rotational
motion.
• It revolves around the sun once every 365
¼ days.
• It rotates around an axis passing through
its geographical poles once every 24 hours.
10.2 Rotational Speed
Tangential speed depends on rotational
speed and the distance from the axis of
rotation.
10.2 Rotational Speed
The turntable rotates around its axis while a
ladybug sitting at its edge revolves around the
same axis. Which part of the turntable moves
faster—the outer part where the ladybug sits or a
part near the orange center?
It depends on whether
you are talking about
linear speed or
rotational speed.
10.2 Rotational Speed
Types of Speed
Linear speed is the distance traveled per unit of
time.
• A point on the outer edge of the turntable
travels a greater distance in one rotation than
a point near the center.
• The linear speed is greater on the outer edge
of a rotating object than it is closer to the axis.
• The speed of something moving along a
circular path can be called tangential speed
because the direction of motion is always
tangent to the circle.
10.2 Rotational Speed
Rotational speed (sometimes called angular speed)
is the number of rotations per unit of time.
• All parts of the rigid turntable rotate about the
axis in the same amount of time.
• All parts have the same rate of rotation, or the
same number of rotations per unit of time. It is
common to express rotational speed in
revolutions per minute (RPM).
10.2 Rotational Speed
All parts of the turntable rotate at the same
rotational speed.
a. A point farther away from the center travels a
longer path in the same time and therefore has
a greater tangential speed.
b. A ladybug sitting twice as far from the center
moves twice as fast.
10.2 Rotational Speed
In symbol form,
v ~ r
where v is tangential speed and  (pronounced
oh MAY guh) is rotational speed.
• You move faster if the rate of rotation
increases (bigger ).
• You also move faster if you are farther
from the axis (bigger r).
10.2 Rotational Speed
At the axis of the rotating platform, you have
no tangential speed, but you do have
rotational speed. You rotate in one place.
As you move away from the center, your
tangential speed increases while your
rotational speed stays the same.
Move out twice as far from the center, and you
have twice the tangential speed.
10.2 Rotational Speed
think!
At an amusement park, you and a friend sit on a large
rotating disk. You sit at the edge and have a rotational
speed of 4 RPM and a linear speed of 6 m/s. Your friend
sits halfway to the center. What is her rotational speed?
What is her linear speed?
Answer:
Her rotational speed is also 4 RPM, and her linear speed
is 3 m/s.
10.2 Rotational Speed
Railroad Train Wheels
How do the wheels of a train stay on the tracks?
The train wheels stay on the tracks because their
rims are slightly tapered.
10.2 Rotational Speed
The wheels of railroad trains are similarly tapered.
This tapered shape is essential on the curves of
railroad tracks.
•On any curve, the distance along the outer part is
longer than the distance along the inner part.
•When a vehicle follows a curve, its outer wheels
travel faster than its inner wheels. This is not a
problem because the wheels roll independent of
each other.
•For a train, however, pairs of wheels are firmly
connected like the pair of fastened cups, so they
rotate together.
10.2 Rotational Speed
When a train rounds a curve, the wheels have
different linear speeds for the same rotational
speed.
10.2 Rotational Speed
think!
Train wheels ride on a pair of tracks. For straightline motion, both tracks are the same length. But
which track is longer for a curve, the one on the
outside or the one on the inside of the curve?
Answer:
The outer track is longer—just as a circle with a
greater radius has a greater circumference.
10.3 Centripetal Force
Velocity involves both speed and direction.
• When an object moves in a circle, even at
constant speed, the object still undergoes
acceleration because its direction is
changing.
• This change in direction is due to a net
force (otherwise the object would
continue to go in a straight line).
• Any object moving in a circle undergoes an
acceleration that is directed to the center
of the circle—a centripetal acceleration.
10.3 Centripetal Force
Centripetal means “toward the center.”
The force directed toward a fixed center that
causes an object to follow a circular path is
called a
centripetal force.
10.3 Centripetal Force
Examples of Centripetal Forces
If you whirl a tin can on the end of a string, you
must keep pulling on the string—exerting a
centripetal force.
The string transmits the centripetal force, pulling
the can from a straight-line path into a circular
path.
The force exerted
on a whirling can is
toward the center.
No outward force
acts on the can.
10.3 Centripetal Force
Centripetal force holds a car in a curved path.
a. For the car to go around a curve, there must be
sufficient friction to provide the required
centripetal force.
b. If the force of friction is not great enough,
skidding occurs.
10.3 Centripetal Force
The clothes in a washing machine are forced into a
circular path, but the water is not, and it flies off
tangentially.
Calculating Centripetal Forces
Greater speed and greater mass require greater
centripetal force.
Traveling in a circular path with a smaller radius of
curvature requires a greater centripetal force.
Centripetal force, Fc, is measured in newtons when
m is expressed in kilograms, v in meters/second,
and r in meters.
The string of a conical pendulum sweeps out a
cone. Only two forces act on the bob: mg, the force due
to gravity, and T, tension in the string.
• Both are vectors.
The vector T can be resolved
into two perpendicular
components, Tx (horizontal),
and Ty (vertical).
If vector T were replaced with
forces represented by these
component vectors, the bob
would behave just as it does
when it is supported only by T.
The vector T can be resolved into a horizontal
(Tx) component and a vertical (Ty) component.
The vector T can be resolved
into two perpendicular
components, Tx (horizontal),
and Ty (vertical).
If vector T were replaced with
forces represented by these
component vectors, the bob
would behave just as it does
when it is supported only by
T.
10.3 Centripetal Force
Since the bob doesn’t accelerate vertically, the
net force in the vertical direction is zero.
Therefore Ty must be equal and opposite to
mg.
Tx is the net force on the bob–the
centripetal force. Its magnitude
is mv/r2, where r is the radius
of the circular path.
10.3 Centripetal Force
Centripetal force keeps the vehicle in a circular
path as it rounds a banked curve.
10.3 Centripetal Force
Suppose the speed of the vehicle is such that
the vehicle has no tendency to slide down the
curve or up the curve.
At that speed, friction plays no role in keeping
the vehicle on the track.
Only two forces act on the vehicle, one mg,
and the other the normal force n (the support
force of the surface). Note that n is resolved
into nx and ny components.
10.4 Centripetal and Centrifugal Forces
Sometimes an outward force is also
attributed to circular motion.
This apparent outward force on a rotating or
revolving body is called centrifugal force.
Centrifugal means “center-fleeing,” or “away
from the center.”
10.4 Centripetal and Centrifugal Forces
When the string breaks, the whirling can
moves in a straight line, tangent to—not
outward from the center of—its circular
path.
10.4 Centripetal and Centrifugal Forces
In the case of the whirling can, it is a
common misconception to state that a
centrifugal force pulls outward on the can.
In fact, when the string breaks the can goes
off in a tangential straight-line path because
no force acts on it.
So when you swing a tin can in a circular
path, there is no force pulling the can
outward.
Only the force from the string acts on the can
to pull the can inward. The outward force is
on the string, not on the can.
10.5 Centrifugal Force in a Rotating Reference Frame
Centrifugal force is an effect of rotation. It is
not part of an interaction and therefore it
cannot be a true force.
10.5 Centrifugal Force in a Rotating Reference Frame
From the reference frame of the ladybug
inside the whirling can, the ladybug is being
held to the bottom of the can by a force that is
directed away from the center of circular
motion.
10.5 Centrifugal Force in a Rotating Reference Frame
•From a stationary frame of reference outside the
whirling can, we see there is no centrifugal force
acting on the ladybug inside the whirling can.
•However, we do see centripetal force acting on the
can, producing circular motion.
• Nature seen from the reference frame of the
rotating system is different.
•In the rotating frame of reference of the whirling
can, both centripetal force (supplied by the can) and
centrifugal force act on the ladybug.
10.5 Centrifugal Force in a Rotating Reference Frame
In a rotating reference frame the centrifugal
force has no agent such as mass—there is no
interaction counterpart.
For this reason, physicists refer to centrifugal
force as a fictitious force, unlike gravitational,
electromagnetic, and nuclear forces.
Nevertheless, to observers who are in a rotating
system, centrifugal force is very real. Just as
gravity is ever present at Earth’s surface,
centrifugal force is ever present within a rotating
system.
Assessment Questions
1. Whereas a rotation takes place about an axis
that is internal, a revolution takes place about
an axis that is
a. external.
b. at the center of gravity.
c. at the center of mass.
d. either internal or external.
Assessment Questions
1. Whereas a rotation takes place about an axis
that is internal, a revolution takes place about
an axis that is
a. external.
b. at the center of gravity.
c. at the center of mass.
d. either internal or external.
Answer: A
Assessment Questions
2. When you roll a tapered cup across a table, the
path of the cup curves because the wider end
rolls
a. slower.
b. at the same speed as the narrow part.
c. faster.
d. in an unexplained way.
Assessment Questions
2. When you roll a tapered cup across a table, the
path of the cup curves because the wider end
rolls
a. slower.
b. at the same speed as the narrow part.
c. faster.
d. in an unexplained way.
Answer: C
Assessment Questions
3. When you whirl a tin can in a horizontal circle
overhead, the force that holds the can in the
path acts
a. in an inward direction.
b. in an outward direction.
c. in either an inward or outward direction.
d. parallel to the force of gravity.
Assessment Questions
3. When you whirl a tin can in a horizontal circle
overhead, the force that holds the can in the
path acts
a. in an inward direction.
b. in an outward direction.
c. in either an inward or outward direction.
d. parallel to the force of gravity.
Answer: A
Assessment Questions
4. When you whirl a tin can in a horizontal circle
overhead, the force that the can exerts on the
string acts
a. in an inward direction.
b. in an outward direction.
c. in either an inward or outward direction.
d. parallel to the force of gravity.
Assessment Questions
4. When you whirl a tin can in a horizontal circle
overhead, the force that the can exerts on the
string acts
a. in an inward direction.
b. in an outward direction.
c. in either an inward or outward direction.
d. parallel to the force of gravity.
Answer: B
Assessment Questions
5. A bug inside a can whirled in a circle feels a
force of the can on its feet. This force acts
a. in an inward direction.
b. in an outward direction.
c. in either an inward or outward direction.
d. parallel to the force of gravity.
Assessment Questions
5. A bug inside a can whirled in a circle feels a
force of the can on its feet. This force acts
a. in an inward direction.
b. in an outward direction.
c. in either an inward or outward direction.
d. parallel to the force of gravity.
Answer: A
Chapter 11
An object will remain in
rotational equilibrium if its
center of mass is above the
area of support.
11.1 Torque
To make an object turn or
rotate, apply a torque.
11.1 Torque
A torque produces rotation.
11.1 Torque
When a perpendicular force is applied, the
lever arm is the distance between the
doorknob and the edge with the hinges.
11.1 Torque
When the force is perpendicular, the distance
from the turning axis to the point of contact is
called the lever arm.
If the force is not at right angle to the lever
arm, then only the perpendicular component of
the force will contribute to the torque.
T = Fd
units: nm
11.1 Torque
Although the magnitudes of the applied forces
are the same in each case, the torques are
different.
11.2 Balanced Torques
When balanced torques act on an object,
there is no change in rotation.
11.2 Balanced Torques
A pair of torques can balance each other. Balance
is achieved if the torque that tends to produce
clockwise rotation by the boy equals the torque
that tends to produce counterclockwise rotation
by the girl.
11.2 Balanced Torques
do the math!
What is the weight of the block hung at the 10-cm mark?
11.2 Balanced Torques
do the math!
The block of unknown weight tends to rotate the system of blocks
and stick counterclockwise, and the 20-N block tends to rotate the
system clockwise. The system is in balance when the two torques
are equal:
counterclockwise torque = clockwise torque
11.2 Balanced Torques
do the math!
Rearrange the equation to solve for the unknown weight:
The lever arm for the unknown weight is 40 cm.
The lever arm for the 20-N block is 30 cm.
The unknown weight is thus 15 N.
11.3 Center of Mass
The center of mass of an object is the point
located at the object’s average position of
mass.
11.3 Center of Mass
A baseball thrown into the air follows a smooth
parabolic path. A baseball bat thrown into the air does
not follow a smooth path.
The bat wobbles about a special point. This point stays
on a parabolic path, even though the rest of the bat
does not.
The motion of the bat is the sum of two motions:
• a spin around this point, and
• a movement through the air as if all the mass
were concentrated at this point.
This point, called the center of mass, is where all the
mass of an object can be considered to be
concentrated.
11.3 Center of Mass
The centers of mass of the baseball and of the
spinning baseball bat each follow parabolic
paths.
11.3 Center of Mass
Location of the Center of Mass
For a symmetrical object, such as a baseball, the
center of mass is at the geometric center of the
object.
For an irregularly shaped object, such as a baseball
bat, the center of mass is toward the heavier end.
11.3 Center of Mass
The center of mass for each object is shown by the
red dot.
11.3 Center of Mass
The center of mass of the toy is below its
geometric center.
11.3 Center of Mass
The center of mass of the rotating wrench
follows a straight-line path as it slides across a
smooth surface.
If the wrench were tossed into the air, its center
of mass would follow a smooth parabola.
11.3 Center of Mass
Applying Spin to an Object
When you throw a ball and apply spin to it, or
when you launch a plastic flying disk, a force must
be applied to the edge of the object.
This produces a torque that adds rotation to the
projectile.
A skilled pool player strikes the cue ball below its
center to put backspin on the ball.
11.3 Center of Mass
A force must be applied to the edge of an
object for it to spin.
a. If the football is kicked in line with its
center, it will move without rotating.
b. If it is kicked above or below its center, it
will rotate.
11.4 Center of Gravity
For everyday objects, the center of gravity is
the same as the center of mass.
11.4 Center of Gravity
Center of mass is often called center of gravity, the
average position of all the particles of weight that
make up an object. For almost all objects on and
near Earth, these terms are interchangeable.
There can be a small difference between center of
gravity and center of mass when an object is large
enough for gravity to vary from one part to
another.
The center of gravity of the Sears Tower in Chicago
is about 1 mm below its center of mass because
the lower stories are pulled a little more strongly
by Earth’s gravity than the upper stories.
11.4 Center of Gravity
If all the planets were lined up on one side of the
sun, the center of gravity of the solar system
would lie outside the sun.
11.4 Center of Gravity
Locating the Center of Gravity
The center of gravity (COG) of a uniform object is
at the midpoint, its geometric center.
• The COG is the balance point.
• Supporting that single point supports the
whole object.
If you suspend any object at a single point, the
COG of the object will hang directly below (or at)
the point of suspension.
To locate an object’s CG:
•Construct a vertical line beneath the point of
suspension.
•The COG lies somewhere along
that line.
•Suspend the object from some
other point and construct a
second vertical line.
•The COG is where the two lines intersect.
•You can use a plumb bob to find the
COG for an irregularly shaped object.
11.4 Center of Gravity
There is no material at the COG of these
objects.
11.5 Torque and Center of Gravity
If the center of gravity of an object is above
the area of support, the object will remain
upright.
11.5 Torque and Center of Gravity
The block topples when the COG extends
beyond its support base.
•The Leaning Tower of Pisa does not topple because
its COG does not extend beyond its base.
•A vertical line below the COG falls inside the base,
and so the Leaning Tower has stood for centuries.
•If the tower leaned far enough
that the COG extended beyond
the base, an unbalanced torque
would topple the tower.
•The Leaning Tower of Pisa
does not topple over because
its COG lies above its base.
11.5 Torque and Center of Gravity
The Leaning Tower
of Pisa does not
topple over
because its COG lies
above its base.
The Moon’s COG
Only one side of the moon continually faces Earth.
Because the side of the moon nearest Earth is
gravitationally tugged toward Earth a bit more than
farther parts, the moon’s COG is closer to Earth
than its center of mass.
•While the moon rotates about its center of mass,
Earth pulls on its COG.
•This produces a torque when the moon’s COG is
not on the line between the moon’s and Earth’s
centers.
•This torque keeps one hemisphere of the moon
facing Earth.
11.6 Center of Gravity of People
The center of gravity of a person is not
located in a fixed place, but depends
on body orientation.
11.6 Center of Gravity of People
When you stand erect with your arms hanging at
your sides, your COG is within your body, typically
2 to 3 cm below your navel, and midway between
your front and back.
Raise your arms vertically overhead. Your COG
rises 5 to 8 cm.
Bend your body into a U or C shape and your COG
may be located outside your body altogether.
11.6 Center of Gravity of People
A high jumper executes a “Fosbury flop” to clear
the bar while his CG nearly passes beneath the bar.
11.6 Center of Gravity of People
When you stand, your COG is somewhere above
your support base, the area bounded by your feet.
• In unstable situations, as in standing in the
aisle of a bumpy-riding bus, you place your
feet farther apart to increase this area.
• Standing on one foot greatly decreases this
area.
• In learning to walk, a baby must learn to
coordinate and position the COG above a
supporting foot.
11.6 Center of Gravity of People
You can probably bend over and touch your toes
without bending your knees.
In doing so, you unconsciously extend the lower
part of your body so that your COG, which is now
outside your body, is still above your supporting
feet.
Try it while standing with your heels to a wall. You
are unable to adjust your body, and your COG
protrudes beyond your feet. You are off balance
and torque topples you over.
11.6 Center of Gravity of People
You can lean over and touch your toes without
toppling only if your COG is above the area
bounded by your feet.
11.6 Center of Gravity of People
think!
When you carry a heavy load—such as a pail of water—
with one arm, why do you tend to hold your free arm out
horizontally?
Answer:
You tend to hold your free arm outstretched to shift the
COG of your body away from the load so your combined
COG will more easily be above the base of support. To
really help matters, divide the load in two if possible, and
carry half in each hand. Or, carry the load on your head!
11.7 Stability
It is nearly impossible to balance a pen upright on
its point, while it is rather easy to stand it upright
on its flat end.
• The base of support is inadequate for the
point and adequate for the flat end.
• Also, even if you position the pen so that its
COG is exactly above its tip, the slightest
vibration or air current can cause it to
topple.
11.7 Stability
Change in the Location of the COG Upon Toppling
What happens to the COG of a cone standing on its
point when it topples?
The COG is lowered by any movement.
We say that an object balanced so that any
displacement lowers its center of mass is in
unstable equilibrium.
11.7 Stability
A cone balances easily on its base.
To make it topple, its COG must be raised.
This means the cone’s potential energy must be
increased, which requires work.
We say an object that is balanced so that any
displacement raises its center of mass is in stable
equilibrium.
11.7 Stability
A cone on lying on its side is balanced so that any
small movement neither raises nor lowers its
center of gravity.
The cone is in neutral equilibrium.
11.7 Stability
a. Equilibrium is unstable when the COG is
lowered with displacement.
b. Equilibrium is stable when work must be
done to raise the COG.
c. Equilibrium is neutral when displacement
neither raises nor lowers the COG.
11.7 Stability
Objects in Stable Equilibrium
The horizontally balanced pencil is in unstable
equilibrium. Its COG is lowered when it tilts.
But suspend a potato from each end and the pencil
becomes stable because the COG is below the point
of support, and is raised when the pencil is tilted.
11.7 Stability
A pencil balanced on the edge of a hand is in
unstable equilibrium.
a. The COG of the pencil is lowered when it tilts.
b. When the ends of the pencil are stuck into
long potatoes that hang below, it is stable
because its COG rises when it is tipped.
11.7 Stability
The toy is in stable equilibrium because the CG
rises when the toy tilts.
11.7 Stability
The COG of a building is lowered if much of the
structure is below ground level.
This is important for tall, narrow structures.
11.7 Stability
The COG of an object has a tendency to take the
lowest position available.
a. A table tennis ball is placed at the bottom
of a container of dried beans.
b. When the container is shaken from side
to side, the ball is nudged to the top.
11.7 Stability
The COG of the glass of water is affected by the
position of the table tennis ball.
a. The COG is higher when the ball is
anchored to the bottom.
b. The COG is lower when the ball floats.
Assessment Questions
1. Applying a longer lever arm to an object so it
will rotate produces
a. less torque.
b. more torque.
c. less acceleration.
d. more acceleration.
Assessment Questions
1. Applying a longer lever arm to an object so it
will rotate produces
a. less torque.
b. more torque.
c. less acceleration.
d. more acceleration.
Answer: B
Assessment Questions
2. When two children of different weights balance
on a seesaw, they each produce
a. equal torques in the same direction.
b. unequal torques.
c. equal torques in opposite directions.
d. equal forces.
Assessment Questions
2. When two children of different weights balance
on a seesaw, they each produce
a. equal torques in the same direction.
b. unequal torques.
c. equal torques in opposite directions.
d. equal forces.
Answer: C
Assessment Questions
3. The center of mass of a donut is located
a. in the hole.
b. in material making up the donut.
c. near the center of gravity.
d. over a point of support.
Assessment Questions
3. The center of mass of a donut is located
a. in the hole.
b. in material making up the donut.
c. near the center of gravity.
d. over a point of support.
Answer: A
Assessment Questions
4. The center of gravity of an object
a. lies inside the object.
b. lies outside the object.
c. may or may not lie inside the object.
d. is near the center of mass.
Assessment Questions
4. The center of gravity of an object
a. lies inside the object.
b. lies outside the object.
c. may or may not lie inside the object.
d. is near the center of mass.
Answer: C
Assessment Questions
5. An unsupported object will topple over when
its center of gravity
a. lies outside the object.
b. extends beyond the support base.
c. is displaced from its center of mass.
d. lowers at the point of tipping.
Assessment Questions
5. An unsupported object will topple over when
its center of gravity
a. lies outside the object.
b. extends beyond the support base.
c. is displaced from its center of mass.
d. lowers at the point of tipping.
Answer: B
Assessment Questions
6. The center of gravity of your best friend is
located
a. near the belly button.
b. at different places depending on body
orientation.
c. near the center of mass.
d. at a fulcrum when rotation occurs.
Assessment Questions
6. The center of gravity of your best friend is
located
a. near the belly button.
b. at different places depending on body
orientation.
c. near the center of mass.
d. at a fulcrum when rotation occurs.
Answer: B
Assessment Questions
7. When a stable object is made to topple over, its
center of gravity
a. is at first raised.
b. is lowered.
c. plays a minor role.
d. plays no role.
Assessment Questions
7. When a stable object is made to topple over, its
center of gravity
a. is at first raised.
b. is lowered.
c. plays a minor role.
d. plays no role.
Answer: A
Chapter 12
Rotating objects tend to
keep rotating while nonrotating objects tend to
remain non-rotating.
12.1 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).
12.1 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.
12.1 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.
12.1 Rotational Inertia
By holding a long pole, the tightrope walker
increases his rotational inertia.
12.1 Rotational Inertia
The short pendulum will swing back and forth
more frequently than the long pendulum.
12.1 Rotational Inertia
You bend your legs when you run to reduce their
rotational inertia. Bent legs are easier to swing
back and forth.
12.1 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.
12.1 Rotational Inertia
Rotational
inertias of
various objects
are different. (It
is not important
for you to learn
these values, but
you can see how
they vary with
the shape and
axis.)
12.1 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?
12.2 Rotational Inertia and Gymnastics
The human body has three principal axes of
rotation.
12.2 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.
12.2 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.
12.2 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.
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.
c. Rotational inertia is 3 times greater.
d. Rotational inertia is 5 times greater than in the
tuck position.
12.2 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.
12.2 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.
•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.
•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.
12.3 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.
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.
12.3 Rotational Inertia and Rolling
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.
12.4 Angular Momentum
Newton’s first law of inertia for rotating
systems states that an object or system of
objects will maintain its angular momentum
unless acted upon by an unbalanced external
torque.
12.4 Angular Momentum, L
Anything that rotates keeps on rotating until
something stops it.
Angular momentum is defined as the product
of rotational inertia, I, and rotational velocity,
.
angular momentum = rotational inertia ×
rotational velocity ()
L = I ×
12.4 Angular Momentum, L
Like linear momentum, angular momentum is a
vector quantity and has direction as well as
magnitude.
• When a direction is assigned to rotational
speed, we call it rotational velocity.
• Rotational velocity is a vector whose
magnitude is the rotational speed.
12.4 Angular Momentum,L
For the case of an object that is small compared
with the radial distance to its axis of rotation,
the angular momentum is simply equal to the
magnitude of its linear momentum, mv,
multiplied by the radial distance, r.
L= mvr
This applies to a tin can swinging from a long
string or a planet orbiting in a circle around the
sun.
12.4 Angular Momentum
An object of
concentrated mass m
whirling in a circular
path of radius r with a
speed v has angular
momentum mvr.
It is easier to balance on a moving bicycle than
on one at rest.
• The spinning wheels have angular
momentum.
• When our center of gravity is not above a
point of support, a slight torque is
produced.
• When the wheels are at rest, we fall over.
• When the bicycle is moving, the wheels
have angular momentum, and a greater
torque is required to change the direction
of the angular momentum.
12.4 Angular Momentum
The lightweight wheels on racing bikes have
less angular momentum than those on
recreational bikes, so it takes less effort to get
them turning.
12.5 Conservation of Angular Momentum
Angular momentum is conserved for systems in
rotation.
The law of conservation of angular
momentum states that if no unbalanced
external torque acts on a rotating system, the
angular momentum of that system is constant.
With no external torque, the product of
rotational inertia and rotational velocity at one
time will be the same as at any other time.
12.5 Conservation of Angular Momentum
When the man pulls his arms and the whirling
weights inward, he decreases his rotational
inertia, and his rotational speed
correspondingly increases.
12.5 Conservation of Angular Momentum
Although the cat is dropped
upside down, it is able to
rotate so it can land on its
feet.
12.6 Simulated Gravity
From within a rotating frame of reference,
there seems to be an outwardly directed
centrifugal force, which can simulate gravity.
12.6 Simulated Gravity
Challenges of Simulated Gravity
The comfortable 1 g we experience at Earth’s
surface is due to gravity.
Inside a rotating spaceship the acceleration
experienced is the centripetal/centrifugal
acceleration due to rotation.
The magnitude of this acceleration is directly
proportional to the radial distance and the
square of the rotational speed.
At the axis where radial distance is zero, there is
no acceleration due to rotation.
12.6 Simulated Gravity
If the structure rotates so that
inhabitants on the inside of
the outer edge experience 1 g,
then halfway between the axis
and the outer edge they would
experience only 0.5 g.
Assessment Questions
1. The rotational inertia of an object is greater
when most of the mass is located
a. near the center.
b. off center.
c. on the rotational axis.
d. away from the rotational axis.
Assessment Questions
1. The rotational inertia of an object is greater
when most of the mass is located
a. near the center.
b. off center.
c. on the rotational axis.
d. away from the rotational axis.
Answer: D
Assessment Questions
2. How many principal axes of rotation are found
in the human body?
a. one
b. two
c. three
d. four
Assessment Questions
2. How many principal axes of rotation are found
in the human body?
a. one
b. two
c. three
d. four
Answer: C
Assessment Questions
3. For round objects rolling on an incline, the
faster objects are generally those with the
a. greatest rotational inertia compared with
mass.
b. lowest rotational inertia compared with
mass.
c. most streamlining.
d. highest center of gravity.
Assessment Questions
3. For round objects rolling on an incline, the
faster objects are generally those with the
a. greatest rotational inertia compared with
mass.
b. lowest rotational inertia compared with
mass.
c. most streamlining.
d. highest center of gravity.
Answer: B
Assessment Questions
4. For an object traveling in a circular path, its
angular momentum doubles when its linear
speed
a. doubles and its radius remains the same.
b. remains the same and its radius doubles.
c. and its radius remain the same and its mass
doubles.
d. all of the above
Assessment Questions
4. For an object traveling in a circular path, its
angular momentum doubles when its linear
speed
a. doubles and its radius remains the same.
b. remains the same and its radius doubles.
c. and its radius remain the same and its mass
doubles.
d. all of the above
Answer: D
Assessment Questions
5. The angular momentum of a system is
conserved
a. never.
b. at some times.
c. at all times.
d. when angular velocity remains unchanged.
Assessment Questions
5. The angular momentum of a system is
conserved
a. never.
b. at some times.
c. at all times.
d. when angular velocity remains unchanged.
Answer: B
Assessment Questions
6. Gravity can be simulated for astronauts in
outer space if their habitat
a. is very close to Earth.
b. is in free fall about Earth.
c. rotates.
d. revolves about Earth.
Assessment Questions
6. Gravity can be simulated for astronauts in
outer space if their habitat
a. is very close to Earth.
b. is in free fall about Earth.
c. rotates.
d. revolves about Earth.
Answer: C