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Miscellaneous<br />
If ax 2 + bx + c = 0, <strong>the</strong>n x =<br />
Rectilinear (1-D) Motion<br />
<strong>Equation</strong> <strong>Sheet</strong> for EMch 112H<br />
Mechanics <strong>of</strong> Motion<br />
<br />
−b ± √ b2 <br />
− 4ac 2a.<br />
Position: s(t); Velocity: v = ˙s = ds/dt; Acceleration: a =¨s = dv/dt = d2s/dt2 = vdv/ds. F or<br />
constant acceleration ac:<br />
v 2 = v 2 0 +2ac(s − s0) v = v0 + act s = s0 + v0t + 1 2<br />
2act There are corresponding equations for constant angular acceleration α.<br />
2D Motions—Cartesian Coordinates<br />
Position: r = x î+yˆj; Velocity: v = dr/dt =˙x î+ ˙yˆj; Acceleration: a = dv/dt = d 2 r/dt 2 =¨x î+¨yˆj<br />
2D Motions—Normal-Tangential (Path) Coordinates<br />
v = v êt = ρ ˙ β êt<br />
2D Motions—Polar Coordinates<br />
Relative Motion<br />
a =˙v êt +(v 2 /ρ)ên,<br />
r = r êr v =˙r êr + r ˙ θ êθ a =(¨r − r ˙ θ 2 )êr +(r ¨ θ +2˙r ˙ θ)êθ<br />
Newton’s Second Law<br />
rB = rA + r B/A vB = vA + v B/A aB = aA + a B/A<br />
F = ma; Fx = max; Fy = may; Fn = man; Ft = mat; Fr = mar; Fθ = maθ<br />
Work-Energy Principle<br />
<br />
U1–2 =<br />
path<br />
F · dr T = 1<br />
2 mv2<br />
Vg = mgh Ve = 1<br />
2 kδ2<br />
T1 + U1–2 = T2<br />
T1 + Vg1 + Ve1 + U ′ 1–2 = T2 + Vg2 + Ve2<br />
Linear Impulse-Momentum Principle<br />
G = mv F ˙<br />
= G<br />
t2<br />
mv1 + F dt= mv2<br />
Angular Impulse-Momentum Principle<br />
HO = r P/O × mvP<br />
MO = ˙ HO<br />
t1<br />
HO1 +<br />
t2<br />
t1<br />
MO dt = HO2<br />
Last modified: March 25, 2001
Impact <strong>of</strong> Smooth Particles<br />
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<br />
Systems <strong>of</strong> Particles<br />
F = maG<br />
T = 1<br />
2mv2 G + 1<br />
<br />
MP = ˙ HG<br />
+ r G/P × maG (M&K)<br />
T1 + V1 + U ′ 1–2 = T2 + V2<br />
(v1)n − (v2)n<br />
2mi| ˙ ρ| 2 G = mvG<br />
<br />
MP =( ˙ HP<br />
)rel + r G/P × maP (M&K)<br />
<br />
MP = ˙ HP<br />
+ m˙ rP × ˙ rG (GLG)<br />
<br />
MP = ˙ hP + mρG × ¨ rP (GLG)<br />
Under <strong>the</strong> appropriate conditions, <strong>the</strong> GLG moment-angular momentum relations simplify to:<br />
<br />
MP = ˙ HP<br />
or MP<br />
= ˙ hP Rigid Body Kinematics<br />
Moments <strong>of</strong> Inertia<br />
(IG)disk = 1<br />
2 mr2<br />
vB = vA + v B/A<br />
vB = vA + ω × r B/A<br />
(IG)rod = 1<br />
12 ml2<br />
aB = aA + a B/A<br />
aB = aA + α × r B/A + ω × (ω × r B/A)<br />
aB = aA + α × r B/A − ω 2 r B/A<br />
(IG)plate = 1<br />
12 m(a2 + b 2 ) (IG)sphere = 2<br />
5 mr2<br />
Parallel Axis Theorem: IA = IG + md 2 ; Radius <strong>of</strong> Gyration: IA = mk 2 A<br />
Angular Momentum and <strong>Equation</strong>s <strong>of</strong> Motion for a Rigid Body<br />
Angular Momentum<br />
<strong>Equation</strong>s <strong>of</strong> Motion<br />
HG = IGω<br />
The translational equations are given by<br />
F = maG<br />
HO = IGω + r G/O × mvG<br />
The rotational equation. For <strong>the</strong> mass center G: MG = IGα and for a fixed point O: MO = IOα.<br />
For an arbitrary point A:<br />
MP = IGα + r G/P × maG<br />
Work-Energy for a Rigid Body<br />
MP = IP α + r G/P × maP<br />
<br />
MA<br />
FBD =<br />
<br />
MA<br />
KD<br />
The work-energy principle is <strong>the</strong> same as that for particles. The kinetic energy <strong>of</strong> a rigid body is:<br />
T = 1 2<br />
2IOω T = 1<br />
2mv2 G + 1<br />
2<br />
IGω 2<br />
Last modified: March 25, 2001