Lecture 9
Chapter 8
Rotational Motion
Quiz -- Monday Sept. 20 (Chaps. 4-7)
14-Sep-10
Physics 101
Announcements
• My office hour today was 1010-1100
instead of 1310-1400.
• My help session today is 1510-1600 in
TH118.
• Quiz Monday
Physics 101
Rotational Motion
In physics we distinguish two types of motion
for objects:
• Translational Motion (change of location).
Whole object moves through space.
• Rotational Motion - object turns around an
axis (axle); axis does not move. (Wheels)
We’ve mostly discussed translational motion;
today we consider rotation.
Physics 101
Rotational Motion
• Study rotational motion of solid object with
fixed axis of rotation (axle)
• Have “angular” versions of the quantities
we studied in translational motion position, distance, velocity, acceleration
• Later look at angular versions of force,
mass, momentum, and kinetic energy.
• Use Greek letters for most angular
quantities
Physics 101
Angular Position
We will measure
angular position
in revolutions
Counterclockwise
(CCW): positive
rotation
Clockwise (CW):
Negative rotation
Physics 101
Circles
Circumference is
distance around the
perimeter of a circle.
Radius
Diameter
Axis
For circle of radius R,
circumference is
2πR.
If point on circle makes one revolution, it
travels a distance 2πR.
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Physics 101
•
Translational (Tangential) Distance d
vs. Angular Distance ∆θ
For a point at
radius R on the
wheel,
d = 2πR∆θ
R
∆θ
for ∆θ in
revolutions
Physics 101
(Average) Angular Velocity ω
Avg. Angular Velocity = # Revolutions/(Time Taken)
ω = ∆θ / t
Unit: Revolutions/s or Revolutions/min (RPM)
Sign convention : ω is positive for counterclockwise
rotation, negative for clockwise rotation
Example: Wheel turns CCW through 60 revolutions
in 2 minutes. What is the rotational speed?
ω = +60 Rev/(2 min) = 30 RPM or 0.5 rev/s
Physics 101
Translational vs. Rotational Speed
Every point on a rotating body has an angular
velocity ω and a tangential velocity vt.
vt = d/∆t = 2πR(∆θ)/∆t = 2πRω
vt
d
R
Physics 101
Speed in Circular Motion
Rotational Speed ω: Revolutions/s or RPM
Tangential Speed vt: Total distance per second
(Depends on ω and radius)
Same Rotational Speed
Different Tangential Speeds
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Physics 101
Check Yourself
A pair of wheels are connected by a chain.
Which wheel has higher
rotational speed?
By how much?
1
Faster tangential speed on rim?
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2
Physics 101
Bicycles
Notice how gears work on a modern bicycle
FAST
SLOW
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Physics 101
Acceleration in Rotational Motion
If the angular velocity of a
rotating object changes, it
has an angular
acceleration. Each point
on the object has a
corresponding tangential
acceleration.
Even if the angular
velocity is constant, each
point on the object has a
centripetal acceleration
because the velocity
direction
is changing.
Physics 101
Simple & Complex Objects
Motion of simple objects: Position
Motion of complex objects: Position & Rotation
Axis of
Rotation
COMPLEX
SIMPLE
14-Sep-10
Physics 101
Rotational Inertia
Mass is a measure of inertia for linear motion.
Gold brick
Normal brick
M
m
Easy to accelerate
Difficult to start moving
Rotational inertia is similar concept for rotation.
Wood Bat
Plastic Pee-wee Bat
x
x
Difficult to Start Rotating
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Easy to Start Rotating
Physics 101
Rotational Inertia
• An object rotating about an axis tends to
remain rotating about the same axis at the
same rotational speed unless interfered
with by some external influence.
• The property of an object to resist changes
in its rotational state of motion is called
rotational inertia.
Physics 101
Rotational Inertia
Rotational inertia depends on
• Total mass of the object
• Distribution of the mass relative to axis
Farther the mass is from the axis of rotation, the
larger the rotational inertia.
Rotational inertia goes as (mass) x (axis_distance)2
14-Sep-10
Physics 101
Rotational Inertia
Depends upon
• mass of object.
• distribution of mass
around axis of rotation.
– The greater the distance
between an object’s mass
concentration and the axis,
the greater the rotational
inertia.
Physics 101
Rotational Inertia
• The greater the rotational inertia of an
object, the harder to change object’s
rotational state.
– A tightrope walker carries a long pole that has a high
rotational inertia, so it does not easily rotate.
– Keeps the tightrope walker stable.
Physics 101
Rotational Inertia
Depends upon the axis
around which it rotates
• Easier to rotate pencil
around an axis passing
through it.
• Harder to rotate it around
vertical axis passing through
center.
• Hardest to rotate it around
vertical axis passing through
the end.
Physics 101
Check Yourself
Which dancer has greater rotational inertia?
Axis of Rotation
A
Physics 101
Demo: Long Legs
Long legs have greater
rotational inertia than
short legs so long
legged animals have
a slow walking stride.
14-Sep-10
Physics 101
B
Demo: Ramp Racing
The two disks and rods have the same mass but
one has connecting rods near the center while
the other has rods near the rim.
When rolled down a ramp, which wins the race?
Winner
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Physics 101
Demo: Hoop & Disk Racing
Roll a hoop and a disk down a ramp;
which wins the race?
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Physics 101
Torque
When a force tends to cause a rotation, we say the
force produces a torque.
Torque depends on
• Magnitude of Force
• Direction of Force
• Lever Arm
(Torque) = (Force) x (Lever Arm)
Positive torque gives counterclockwise rotation
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Physics 101
Lever Arm
Lever arm is perpendicular distance
from axis of rotation to the direction of
the force.
14-Sep-10
Physics
Physics
1 (Garcia)
101 SJSU
Check Yourself
Identify the lever arm for each wrench.
Force
Force
Which wrench design gives more torque for the
same force?
Physics 101
14-Sep-10
Torque for a Pirouette
The farther the distance between the feet,
the greater the lever arm, so the greater
the torque for creating the rotation.
Reaction
Force
Push on
Floor
Lever
Arm
Feet apart
Feet
together
Physics
101
Check Yourself
In which case are you exerting more torque?
A
Lever Arm
14-Sep-10
B
Lever Arm
almost zero
Physics 101
Balance of Torques
Torques clockwise and counter-clockwise
balance in both cases since
(250 N) x (3 m) = (500 N) x (1.5 m)
14-Sep-10
Physics 101
Her lever arm
is still 3 meters
Center of Mass/Gravity
Average position of all the mass in an object
is called the center of mass (CM) of object.
Average position of the weight distribution is
called the center of gravity (CG).
When gravity is constant (usually the case),
these two locations are the same.
14-Sep-10
Physics 101
Locating Center of Gravity
Balance an object to find center of gravity
Center
of
Gravity
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Physics 101
Key Points of Lecture 9
• Rotational Motion
• Angular position
• Angular (rotational) velocity
• Angular acceleration
• Rotational Inertia
• Torque
• Torque Balance
• Center of Mass
z Before next lecture, read Hewitt through 1st half Chap.8
z Homework Assignment #5 is due before 11:00 PM on
Thursday, Sept. 16.
z Homework Assignment #6 is due before 11:00 PM on
Sunday, Sept. 19.
Physics 101
Stability
Object is stable if CG is above the base.
STABLE
CG
BASE
14-Sep-10
Weight
Weight
CG
Axis
BASE
Physics 101
Axis
UNSTABLE
Demo: Balance the Can
PE
PS
I
If a small amount of water is added to an
empty soda can, then the can may be
balanced on its bottom edge.
x CG
Physics 101
14-Sep-10
Check Yourself
Three trucks
are parked on
a slope. Which
truck(s) tip
over?
CG
CG
CG
BASE
14-Sep-10
Physics 101