Physics 10114-Sep-10 Physics 101
Chapter 8Rotational Motion
Lecture 9
Quiz -- Monday Sept. 20 (Chaps. 4-7)
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 MotionIn 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 PositionAngular Position
We will measure angular position in revolutions
Counterclockwise (CCW): positive rotation
Clockwise (CW): Negative rotation
Physics 10114-Sep-10
Circles
Circumference is distance around the perimeter of a circle.
For circle of radius R, circumference is 2πR.
Axis
Radius
Diameter
If point on circle makes one revolution, it travels a distance 2πR.
•
Physics 101
Translational (Tangential) Distance d vs. Angular Distance ∆θ
Translational (Tangential) Distance d vs. Angular Distance ∆θ
∆θ
For a point at radius R on the wheel, d = 2πR∆θ R
for ∆θ in revolutions
Physics 101
(Average) Angular Velocity ω(Average) Angular Velocity ωAvg. 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 SpeedEvery point on a rotating body has an angular velocity ω and a tangential velocity vt. vt = d/∆t = 2πR(∆θ)/∆t = 2πRω
vt
R
d
Physics 10114-Sep-10
Speed in Circular MotionRotational Speed ω: Revolutions/s or RPMTangential Speed vt: Total distance per second
(Depends on ω and radius)Same Rotational Speed
Different Tangential Speeds
Physics 10114-Sep-10
Check Yourself
A pair of wheels are connected by a chain.Which wheel has higher rotational speed?
By how much?
Faster tangential speed on rim?1
2
Physics 10114-Sep-10
Bicycles
FASTSLOW
Notice how gears work on a modern bicycle
Physics 101
Acceleration in Rotational MotionIf the angular velocity of a rotating object changes, it has an angular acceleration. Each point on the object has a corresponding tangentialacceleration.
Even if the angular velocity is constant, each point on the object has a centripetal acceleration because the velocity direction is changing.
Physics 10114-Sep-10
Simple & Complex Objects
Motion of simple objects: PositionMotion of complex objects: Position & Rotation
SIMPLE COMPLEX
Axis ofRotation
Physics 10114-Sep-10
Mass is a measure of inertia for linear motion.
Rotational inertia is similar concept for rotation.
Rotational Inertia
M mGold brick Normal brick
Difficult to start moving Easy to accelerate
x xWood Bat Plastic Pee-wee Bat
Difficult to Start Rotating 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 10114-Sep-10
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
Physics 101
Rotational InertiaDepends 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 InertiaDepends 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 B
Physics 10114-Sep-10
Demo: Long Legs
Long legs have greater rotational inertia than short legs so long legged animals have a slow walking stride.
Physics 10114-Sep-10
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
Physics 10114-Sep-10
Demo: Hoop & Disk Racing
Roll a hoop and a disk down a ramp; which wins the race?
Physics 10114-Sep-10
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
Torque
Physics 10114-Sep-10 Physics 1 (Garcia) SJSU
Lever ArmLever arm is perpendicular distance from axis of rotation to the direction of the force.
Physics 10114-Sep-10
Check Yourself
Identify the lever arm for each wrench.
ForceForce
Which wrench design gives more torque for the same force?
Physics 101
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.
Push onFloor
ReactionForce
LeverArm
Feet apart Feet together
Physics 10114-Sep-10
Check Yourself
In which case are you exerting more torque?
A
BLever ArmLever Arm
almost zero
Physics 10114-Sep-10
Balance of TorquesTorques clockwise and counter-clockwise
balance in both cases since(250 N) x (3 m) = (500 N) x (1.5 m)
Her lever armis still 3 meters
Physics 10114-Sep-10
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.
Physics 10114-Sep-10
Locating Center of Gravity
Balance an object to find center of gravity
Centerof
Gravity
Physics 101
Key Points of Lecture 9Key Points of Lecture 9
Before next lecture, read Hewitt through 1st half Chap.8
Homework Assignment #5 is due before 11:00 PM on Thursday, Sept. 16.
Homework Assignment #6 is due before 11:00 PM on Sunday, Sept. 19.
• Rotational Motion• Angular position• Angular (rotational) velocity• Angular acceleration• Rotational Inertia• Torque• Torque Balance• Center of Mass
Physics 10114-Sep-10
Stability
Object is stable if CG is above the base.
CG CG
Wei
ght
Wei
ght
BASEAxis
AxisBASE
STABLE
UNSTABLE