2 Particle 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)
3 Types of Rigid-Body Motion Kinematically speaking…TranslationOrientation of AB constantRotationAll particles rotate about fixed axisGeneral Plane Motion (both)Combination of both types of motionBABABABA
4 Kinematics of Translation BAyKinematicsPositionVelocityAccelerationTrue for all points in R.B. (follows particle kinematics)rBrAx
5 Rotation 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
6 Rotation 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 )
7 Rotation 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
8 Example 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)
9 Example 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.