Chapter 8 Rotational Motion

Angular Distance (  ) o oReplaces distance for rotational motion o oMeasured in Degrees Radians Revolutions 

Radian Measure r r r 1 rad 1 rad = 57.3 degrees 2 rad in one circle

Windows Calculator

Rotational Motion Speed of Rotation (  )   = Angle covered/Time required o o =  t o o Note similarity to v =  x/  t o oMeasured in degrees/second radians/second revolutions/second 

Rotational Motion   Angular Acceleration - Measures how angular velocity is changing (  )   =  /  tNote similarity to a =  v/  t o oMeasured in … degrees/s 2 radians/s 2 revolutions/s 2

Rotational 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 Inertia Depends on … o oMass of the Object o oAxis of Rotation o oDistribution of Mass in the Object

Moments of Inertia Standard Shapes

Moment of Inertia   Inertia Bars   Ring and Disk on Incline   Metronome   People walking   Weighted Stick - Bare Stick

Torque Product of Force and Lever Arm o oTorque = Force X Lever Arm Examples: o oBalance o oSee-Saw o oWrench

W 1 d 1 = W 2 d 2

Sample Torque Problem (0.5 kg)(9.8 m/s 2 )(0.1 m) = (0.2 kg)(9.8 m/s 2 )d

Line of Action Lever Arm F

Torque Examples

Torque Just as unbalanced forces produce acceleration, unbalanced torques produce angular acceleration. Compare: F = ma= I

Center of Mass Average position of the mass of an object o oNewton showed that all of the mass of the object acts as if it is located here. o oFind cm of Texas/USA

Finding the Center of Mass weight Line of action Pivot point Lever arm Torque No Torque

High Jumper

Stability 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

Stability Base of Support

Stability Which object is most stable?

Centripetal 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 force F = ma c = mv 2 /r = mr  2

Centrifugal force Fictitious center fleeing force o oFelt by object in an accelerated reference frame Examples: o oCar on a circular path o oCan on a string

Space Habitat (simulated gravity) r 

“Down” is away from the center The amount of “gravity” depends on how far from the center you are.

Angular Momentum L = (rotational inertia) X (angular velocity) L = I  Compare to linear momentum: p = mv

Linear Momentum and Force Angular Momentum and Torque Linear  F =  p/  t o oImpulse  p =  F  t Rotational  =  L/  t o oRotational Impulse  L =  t

Conservation of Momentum Linear o oIf  F = 0, then p is constant. Angular o oIf  = 0, then L is constant.

Conservation of Angular Momentum Ice Skater Throwing a football Rifling Helicopters Precession

Rifling

Football Physics L

Helicopter Physics Rotation of Rotor Body Rotation Tail rotor used to produce thrust in opposite direction of body rotation

Precession

Age of Aquarius

Linear - Rotational Connections LinearRotational x (m)  (rad) v (m/s)  (rad/s) a (m/s 2 )  (rad/s 2 ) m (kg) I (kg·m 2 ) F (N)  (N·m) p (N·s) L (N·m·s)