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Chapter 9Angular 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 Elasticity1st 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 MotionTorque  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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