Particle vs. Rigid-Body Mechanics What is the difference between particle and rigid-body mechanics? Rigid-body can be of any shape Block Disc/wheel Bar/member Etc. Can determine motion of any single particle (pt) in body particle Rigid-body (continuum of particles)

Types of Rigid-Body Motion Kinematically speaking… Translation Orientation of AB constant Rotation All particles rotate about fixed axis General Plane Motion (both) Combination of both types of motion B A B A B A B A

Kinematics of Translation B A y Kinematics Position Velocity Acceleration True for all points in R.B. (follows particle kinematics) rB rA x

Rotation about a Fixed Axis – Angular Motion Point P travels in circular path (whether “disk” or not) Angular motion Angular position, θ Angular displacement, dθ Angular velocity ω=dθ/dt Angular Acceleration α=dω/dt r Axis of rotation  

Rotation about a Fixed Axis – Angular Motion Point P travels in circular path (whether “disk” or not) Angular motion Angular position, θ Angular displacement, dθ Angular velocity ω=dθ/dt Angular Acceleration α=dω/dt Angular motion Equations r Axis of rotation   In 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 )

Rotation about a Fixed Axis – Motion of Point Point P travels in circular path Position of P Defined by r If body rotates some dθ, then displacement is ds = r dθ Velocity (tangent to path) Acceleration (2 components) r an ∆v v a ∆v  a an  at an v

Example Problem When 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)

Example Problem The 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.