PDF version of the Equation Sheet

Miscellaneous
If ax 2 + bx + c = 0, then x =
Rectilinear (1-D) Motion
Equation Sheet for EMch 112H
Mechanics of Motion
−b ± √ b2
− 4ac 2a.
Position: s(t); Velocity: v = ˙s = ds/dt; Acceleration: a =¨s = dv/dt = d2s/dt2 = vdv/ds. F or
constant acceleration ac:
v 2 = v 2 0 +2ac(s − s0) v = v0 + act s = s0 + v0t + 1 2
2act There are corresponding equations for constant angular acceleration α.
2D Motions—Cartesian Coordinates
Position: r = x î+yˆj; Velocity: v = dr/dt =˙x î+ ˙yˆj; Acceleration: a = dv/dt = d 2 r/dt 2 =¨x î+¨yˆj
2D Motions—Normal-Tangential (Path) Coordinates
v = v êt = ρ ˙ β êt
2D Motions—Polar Coordinates
Relative Motion
a =˙v êt +(v 2 /ρ)ên,
r = r êr v =˙r êr + r ˙ θ êθ a =(¨r − r ˙ θ 2 )êr +(r ¨ θ +2˙r ˙ θ)êθ
Newton’s Second Law
rB = rA + r B/A vB = vA + v B/A aB = aA + a B/A
F = ma; Fx = max; Fy = may; Fn = man; Ft = mat; Fr = mar; Fθ = maθ
Work-Energy Principle
U1–2 =
path
F · dr T = 1
2 mv2
Vg = mgh Ve = 1
2 kδ2
T1 + U1–2 = T2
T1 + Vg1 + Ve1 + U ′ 1–2 = T2 + Vg2 + Ve2
Linear Impulse-Momentum Principle
G = mv F ˙
= G
t2
mv1 + F dt= mv2
Angular Impulse-Momentum Principle
HO = r P/O × mvP
MO = ˙ HO
t1
HO1 +
t2
t1
MO dt = HO2
Last modified: March 25, 2001

Impact of Smooth Particles
m1(v1)n + m2(v2)n = m1(v ′ 1)n + m2(v ′ 2)n; (v1)t =(v ′ 1)t; (v2)t =(v ′ 2)t; e = (v′ 2 )n − (v ′ 1 )n
Systems of Particles
F = maG
T = 1
2mv2 G + 1
MP = ˙ HG
+ r G/P × maG (M&K)
T1 + V1 + U ′ 1–2 = T2 + V2
(v1)n − (v2)n
2mi| ˙ ρ| 2 G = mvG
MP =( ˙ HP
)rel + r G/P × maP (M&K)
MP = ˙ HP
+ m˙ rP × ˙ rG (GLG)
MP = ˙ hP + mρG × ¨ rP (GLG)
Under the appropriate conditions, the GLG moment-angular momentum relations simplify to:
MP = ˙ HP
or MP
= ˙ hP Rigid Body Kinematics
Moments of Inertia
(IG)disk = 1
2 mr2
vB = vA + v B/A
vB = vA + ω × r B/A
(IG)rod = 1
12 ml2
aB = aA + a B/A
aB = aA + α × r B/A + ω × (ω × r B/A)
aB = aA + α × r B/A − ω 2 r B/A
(IG)plate = 1
12 m(a2 + b 2 ) (IG)sphere = 2
5 mr2
Parallel Axis Theorem: IA = IG + md 2 ; Radius of Gyration: IA = mk 2 A
Angular Momentum and Equations of Motion for a Rigid Body
Angular Momentum
Equations of Motion
HG = IGω
The translational equations are given by
F = maG
HO = IGω + r G/O × mvG
The rotational equation. For the mass center G: MG = IGα and for a fixed point O: MO = IOα.
For an arbitrary point A:
MP = IGα + r G/P × maG
Work-Energy for a Rigid Body
MP = IP α + r G/P × maP
MA
FBD =
MA
KD
The work-energy principle is the same as that for particles. The kinetic energy of a rigid body is:
T = 1 2
2IOω T = 1
2mv2 G + 1
2
IGω 2
Last modified: March 25, 2001