2Particle vs. Rigid-Body Mechanics What is the difference between particle and rigid-body mechanics?Rigid-body can be of any shapeBlockDisc/wheelBar/memberEtc.Can determine motion of any single particle (pt) in bodyparticleRigid-body (continuum of particles)
3Types of Rigid-Body Motion Kinematically speaking…TranslationOrientation of AB constantRotationAll particles rotate about fixed axisGeneral Plane Motion (both)Combination of both types of motionBABABABA
4Kinematics of Translation BAyKinematicsPositionVelocityAccelerationTrue for all points in R.B. (follows particle kinematics)rBrAx
5Rotation about a Fixed Axis – Angular Motion Point P travels in circular path (whether “disk” or not)Angular motionAngular position, θAngular displacement, dθAngular velocityω=dθ/dtAngular Accelerationα=dω/dtrAxis of rotation
6Rotation about a Fixed Axis – Angular Motion Point P travels in circular path (whether “disk” or not)Angular motionAngular position, θAngular displacement, dθAngular velocityω=dθ/dtAngular Accelerationα=dω/dtAngular motion EquationsrAxis of rotationIn solving problems, once know ω & α, we can get velocity and acceleration of any point on body!!! (next slide)(Or can relate the two types of motion if ω & α unknown )
7Rotation about a Fixed Axis – Motion of Point Point P travels in circular pathPosition of PDefined by rIf body rotates some dθ, then displacement is ds = r dθVelocity (tangent to path)Acceleration (2 components)ran∆vva∆vaanatanv
8Example ProblemWhen the gear rotates 20 revolutions, it achieves an angular velocity of ω = 30 rad/s, starting from rest. Determine its constant angular acceleration and the time required. (F16-1, 3.58 rad/s2, 8.38 s)
9Example ProblemThe gear A on the drive shaft of the outboard motor has a radius of rA = 0.5 in and the meshed pinion gear B on the propeller shaft has a radius rB = 1.2 in. Determine the angular velocity of the popular in t = 1.5 s, if the drive shaft rotates with an angular acceleration a = (400t3) rad/s2 , where t is in seconds. The propeller is originally at rest and the motor frame does not move.