Bedford & Fowler — Chapter 18, Planar Dynamics of Rigid Bodies
Chapter 18 — Planar Dynamics of Rigid Bodies: Decision Map
Forces and moments on a rigid body — how does it move?
General selection
Moment of inertia
Translation
Fixed-axis rotation
General plane motion
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Planar Equations of Motion — §18.1–18.2
ΣF
x
= m(a
G
)
x
, ΣF
y
= m(a
G
)
y
ΣM
G
= I
G
α
Free-body diagram = kinetic diagram (m
a
G
at G, plus I
G
α)
Details
Or, about any point P: ΣM
P
= m(a
G
)
x
d
y
− m(a
G
)
y
d
x
+ I
G
α (kinetic moment).
Mass Moment of Inertia — Ch. 18 Appendix
I = ∫ r² dm
I = I
G
+ m d²
k = √(I/m)
Details
Rod (about G): 1/12 ml²; rod (about an end): 1/3 ml².
Thin disk (about G, ⊥): ½ mR²; ring: mR²; sphere: 2/5 mR².
Composite body: locate G, then add I
G,i
+ m
i
d
i
² for each part.
The parallel-axis theorem works only from an axis through G.
How Does the Body Move?
Classify the motion (Chapter 17), then specialize the equations
Translation — §18.2
α = 0
ΣF = m
a
G
, ΣM
G
= 0
Details
All forces must balance their moments about G: this sets the wheel loads of an accelerating cart.
Slide or tip?
Assume it slides (F = μ
k
N), solve for where N acts, check it lies inside the base.
General Plane Motion — §18.2
Translation and rotation together
Rotation About a Fixed Axis — §18.2
ΣF
n
= m r
G
ω²
ΣF
t
= m r
G
α
ΣM
O
= I
O
α, I
O
= I
G
+ m r
G
²
Details
Use n (toward O) and t at G. The pin reactions and m r
G
ω² pass through O, so ΣM
O
gives α directly.
α depends on position? Integrate ω dω = α dθ to get ω.
ΣM
O
= I
O
α only when O is the fixed axis.
Rolling Without Slipping
a
G
= rα
F
f
≤ μ
s
N (static, unknown)
ΣM
IC
= I
IC
α
Check F
f
≤ μ
s
N at the end.
Details
ΣM
IC
= I
IC
α holds for a round wheel with G at its center, rolling without slipping — not in general.
Slipping
F
f
= μ
k
N (kinetic, known)
a
G
≠ rα
Details
a
G
and α are independent: use ΣF
x
for a
G
and ΣM
G
= I
G
α for α.
Reached when the no-slip check fails (required friction > μ
s
N).
Other Bodies and Linked Points
a
A
=
a
G
+
α
×
r
A/G
− ω²
r
A/G
Details
Three equations of motion plus Chapter 17 kinematics when more unknowns remain.
Released from rest: ω = 0 at that instant, so the −ω²r term vanishes.