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Chapter 9
Angular Velocity and Acceleration 
Θ = angle (radians) 
s = length 
r = radius 
90^{o} = π/2 rad 
Θ = (s/r) 
s = r ⋅ Θ 
1 rad = (360^{o} / 2π) = 57.3^{o} 
180^{o} = π rad 
Angular Velocity 
(1st Derivative) 
ω = (Θf  Θi ) / (tf ti ) 
ω = "velocity" 
1 rev/s = 2π rad/s 
1 rev/min = 1 rpm = 2π/60 rad/s 
Angular Acceleration 
(2nd Derivative) 
α = (ωf  ωi ) / (tf ti ) 
α = "acceleration" 
Rotation w/ Constant Angular Acceleration 
αf = (ωf  ωi ) / (t  0) 
αf = constant 
ωf = ωi + αf ⋅ t 
Θf  Θi = 1/2(ωi +ωf ) ⋅ t 
Θf = Θi + (ωi ⋅ t ) + 1/2 (αf ⋅ t^{2}) 
ωf ^{2} = ωi ^{2} + 2⋅αf (Θf  Θi ) 
Relating Linear and Angular Kinematics 
K = 1/2(m ⋅ v^{2}) 
Linear Speed in RigidBody Rotation 
s = r ⋅ Θ 
Linear Speed 
v = r ⋅ ω 
Linear Acceleration in RigidBody Rotation 
atan = r ⋅ α 
Centripetal Component of Acceleration 
arad = (v^{2}/r) = ω^{2}⋅r 
Energy in Rotational Motion 
KE : 1/2⋅m⋅v^{2} = 1/2⋅m⋅r^{2}⋅ω^{2} 
K = 1/2⋅m⋅r^{2}⋅ω^{2} 
I = m⋅r^{2} 
Gravitational Potential Energy for an Extended Body 
U = M⋅g⋅ycm 
Moment of Inertia 
Ip =Icm +Md^{2} 


Chapter 9 Cont:
Rotational Kinetic Energy 
K = Joules 
K = 1/2⋅I⋅ω^{2} 
R = Radius 
M = mass pivoted about an axis 
Perpendicular to the Rod 
I = (M⋅L^{2}) / 3 
Slender Rod (Axis Center) 
I = 1/12M⋅L^{2} 
Slender Rod (Axis End) 
I = 1/3M⋅L^{2} 
Rectangular Plate (Axis Center) 
I = 1/12M⋅(a^{2}+b^{2}) 
Rectangular Plate (Axis End) 
I = 1/3M⋅(a^{2}) 
Hallow Cylinder 
I = 1/2M(Ri ^{2}+Rf ^{2}) 
Solid Cylinder 
I = 1/2MR^{2} 
Hollow Cylinder (Thin) 
I = MR^{2} 
Solid Sphere 
I = 2/5MR^{2} 
Hollow Sphere (Thin) 
I = 2/3MR^{2} 
Chapter 11: Equilibrium and Elasticity
1st Condition of Equilibrium (at rest) 
ΣF = 0 
2nd Condition of Equilibrium (nonrotating) 
Στ = 0 
Center of Gravity 
rcm = (m1 ⋅ r1 ) / m1 
Solving RigidBody Equilibrium Problems 
ΣFx = 0 
1st Condition 
ΣFx = 0 ΣFy = 0 
2nd Condition (Forces xyplane) 
Στz = 0 
Stress, Strain, and Elastic Moduli 
Stress = Force Applied to deform a body Strain = how much deformation 
Hooke's Law 
(Stress / Strain) = Elastic Modulus 
A = Area 
F = Magnitude of Force 
Tensile Stress 
F / A 
1 Pascal = Pa = 1 N/m^{2} 
1 psi = 6895 Pa 
l = length 
1 Pa = 1.450 ⋅ 10^{4} 
Tensile Strain 
(lf  li ) / (li ) 
Young Modulus 
(Tensile Stress) / (Tensile Strain) 
Pressure 
p = F (Force Fluid is Applied) / A (Area which force is exerted) 
Bulk Stress 
(pf  pi ) 
Bulk Strain 
(Vf  Vi ) / (Vi ) 
Bulk Modulus 
Bulk Stress / Bulk Strain 


Chapter 10: Dynamics of Rotational Motion
Torque 
F = Magnitude of F 
  = Magnitude Symbol 
τ = F⋅l = r⋅F⋅sinΘ = Ftan r 
L = lever arm of F 
τ = r x F 
Torque and Angular Acceleration for a Rigid Body 
Newtons 2nd Law of Tangential Component 
Ftan = m1 ⋅a1 
Rotational analog of Newton's second law for a rigid body 
Στz = l⋅αz 
z = rigid body about zaxis

Combined Translation and Rotation: Energy Relationships 
K = 1/2M⋅v^{2} + 1/2⋅I⋅ω^{2} 
Rolling without Slipping 
v = R⋅ω 
Combined Translation and Rotation: Dynamics 
Rotational Motion about the center of mass 
Στz = l⋅αz 
Work and Power in Rotational Motion 
F = M ⋅ a 
When it rotates from Θi to Θf 
W = ∫ (Θf to Θi ) τf dΘ 
When the torque remains constant while angle changes 
W = τf (Θf to Θi ) 
Total WorkDone on rotating rigid body 
W = 1/2(ωf ^{2})  1/2(ωi ^{2}) 
Power due to torque on rigid body 
P = τz ⋅ ωz 
Angular Momentum 
L = r x p (r x m ⋅ v) 
Angular Momentum of a Rigid Body 
L = mi ⋅ri ^{2}⋅ω 
Chapter 11: Equilibrium and Elasticity (cont.)

F = Force acting tangent to the surface divided by the Area 
Shear Stress 
F / A 
h = transverse dimension [bigger] 
x = relative displacement (empty) [smaller] 
Shear Strain 
x / h 

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