Chapter 8Chapter 8
Rotational MotionRotational Motion
Rotational MotionRotational Motion
Angular Distance ()o Replaces distance for rotational motiono Measured in
Degrees
Radians
Revolutions
Radian MeasureRadian Measure
r
r r
1 rad
1 rad = 57.3 degrees
2 rad in one circle
Windows CalculatorWindows Calculator
Rotational MotionRotational Motion
Speed of Rotation () = Angle covered/Time required
o = to Note similarity to v = x/t
o Measured in degrees/second radians/second revolutions/second
Rotational MotionRotational Motion
Angular Acceleration - Measures how angular velocity is changing () = /t Note similarity to a = v/to Measured in …
degrees/s2
radians/s2
revolutions/s2
Rotational InertiaRotational Inertia
Property of an object that resists changes in rotation• For linear motion mass was a measure of
inertia
• For rotational motion Moment of Inertia (I) is the measure of rotational Inertia
Moments of InertiaMoments of Inertia
Depends on …
o Mass of the Object
o Axis of Rotation
o Distribution of Mass in the Object
Moments of InertiaMoments of Inertia
Standard Shapes
Moment of InertiaMoment of Inertia
Inertia Bars Ring and Disk on Incline Metronome People walking Weighted Stick - Bare Stick
TorqueTorque
Product of Force and Lever Armo Torque = Force X Lever Arm
Examples:o Balanceo See-Sawo Wrench
W1d1 = W2d2
Sample Torque ProblemSample Torque Problem
(0.5 kg)(9.8 m/s(0.5 kg)(9.8 m/s22)(0.1 m) = (0.2 kg)(9.8 m/s)(0.1 m) = (0.2 kg)(9.8 m/s22)d)d
m 0.25 d
m) (0.1 kg 0.2kg 0.5
d
Line of Action
Lever Arm
F
Torque ExamplesTorque Examples
TorqueTorque
Just as unbalanced forces produce acceleration, unbalanced torques produce angular acceleration.
Compare:
F = ma = I
Center of MassCenter of Mass
Average position of the mass of an object
o Newton showed that all of the mass of the object acts as if it is located here.
o Find cm of Texas/USA
Finding the Center of MassFinding the Center of Mass
weight
Line of actionPivot point
Lever armTorque
No Torque
High JumperHigh Jumper
StabilityStability
In order to balance forces and torques, the center of mass must always be along the vertical line through the base of support.
Demo• Coke bottle• Chair pick-up
StabilityStability
Base of Support
StabilityStability
Which object is most stable?
Centripetal ForceCentripetal Force
Any force that causes an object to move in a circle.
Examples:• Carousel• Water in a bucket• Moon and Earth• Coin and hanger• Spin cycle
Centripetal forceCentripetal force
F = mac
= mv2/r
= mr2
Centrifugal forceCentrifugal force
Fictitious center fleeing forceo Felt by object in an accelerated reference
frame Examples:
o Car on a circular patho Can on a string
Space Habitat(simulated gravity)
r
Space Habitat(simulated gravity)
“Down” is away from the center The amount of “gravity” depends on
how far from the center you are.
Angular MomentumAngular Momentum
L = (rotational inertia) X (angular velocity)
L = I
Compare to linear momentum:
p = mv
Linear Momentum and ForceLinear Momentum and ForceAngular Momentum and TorqueAngular Momentum and Torque
Linear F =p/to Impulse p = F t
Rotational = L/ to Rotational Impulse L = t
Conservation of MomentumConservation of Momentum
Linearo If F = 0, then p is constant.
Angularo If = 0, then L is constant.
Conservation of Conservation of Angular MomentumAngular Momentum
Ice Skater Throwing a football Rifling Helicopters Precession
RiflingRifling
Football PhysicsFootball Physics
L
Helicopter PhysicsHelicopter Physics
Rotation of Rotor
Body Rotation
Tail rotor used to produce thrust in opposite direction
of body rotation
PrecessionPrecession
Age of AquariusAge of Aquarius
Linear - Rotational ConnectionsLinear - Rotational Connections
Linear Rotational x (m) (rad)
v (m/s) (rad/s)
a (m/s2) (rad/s2)
m (kg) I (kg·m2)F (N) (N·m)
p (N·s) L (N·m·s)