### Rigid-Body Mechanics

```Particle vs. Rigid-Body Mechanics
• What is the difference between particle and rigid-body
mechanics?
– Rigid-body can be of any shape
particle
• Block
• Disc/wheel
• Bar/member
• Etc.
• Can determine motion
of any single particle (pt)
in body
Rigid-body (continuum of particles)
Types of Rigid-Body Motion
• Kinematically
speaking…
B
– Translation
A
• Orientation of AB
constant
– Rotation
B
• All particles rotate
– General Plane Motion
(both)
A
B
• Combination of both
types of motion
A
B
A
Kinematics of Translation
• Kinematics
– Position
  
rB  rA  rB / A
y
B
A
rB
rA
– Velocity


vB  v A
– Acceleration


aB  aA
• True for all points in R.B.
(follows particle kinematics)
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
r
• Angular motion Equations
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)
r
an
v
∆v
a
v  r
  
v r
– Acceleration (2 components)
∆v

an

a
at
an
v
v2
an 
  2r
r
at  r
 
  

a  at  an    r    2r
 
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.
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  = (400t3) rad/s2 , where t is in seconds. The
propeller is originally at rest and the motor frame does not move.
```