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Chapter 9

Angular Velocity and Accele­ration
Θ = angle (radians)
s = length
r = radius
90o = π/2 rad
Θ = (s/r)
s = r ⋅ Θ
1 rad = (360o / 2π) = 57.3o
180o = 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(ωif) ⋅ t
Θf = Θi+ (ωi⋅ t ) + 1/2 (αf⋅ t2)
ωf2 = ωi2 + 2⋅αff- Θi)
Relating Linear and Angular Kinematics
K = 1/2(m ⋅ v2)
Linear Speed in Rigid-Body Rotation
s = r ⋅ Θ
Linear Speed
v = r ⋅ ω
Linear Accele­ration in Rigid-Body Rotation
atan = r ⋅ α
Centri­petal Component of Accele­ration
arad = (v2/r) = ω2⋅r
Energy in Rotational Motion
KE: 1/2⋅m⋅v2 = 1/2⋅m⋅r2⋅ω2
K = 1/2⋅m⋅r2⋅ω2
I = m⋅r2
Gravit­ational Potential Energy for an Extended Body
U = M⋅g⋅ycm
Moment of Inertia
Ip=Icm+Md2
 

Chapter 9 Cont:

Rotational Kinetic Energy
K = Joules
K = 1/2⋅I⋅ω2
R = Radius
M = mass pivoted about an axis
Perpen­dicular to the Rod
I = (M⋅L2) / 3
Slender Rod (Axis Center)
I = 1/12M⋅L2
Slender Rod (Axis End)
I = 1/3M⋅L2
Rectan­gular Plate (Axis Center)
I = 1/12M⋅(a2+b2)
Rectan­gular Plate (Axis End)
I = 1/3M⋅(a2)
Hallow Cylinder
I = 1/2M(Ri2+Rf2)
Solid Cylinder
I = 1/2MR2
Hollow Cylinder (Thin)
I = MR2
Solid Sphere
I = 2/5MR2
Hollow Sphere (Thin)
I = 2/3MR2

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 Rigid-Body Equili­brium Problems
ΣFx = 0
1st Condition
ΣFx = 0
ΣFy = 0
2nd Condition (Forces xy-plane)
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/m2
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 = Ftanr
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 z-axis
Combined Transl­ation and Rotation: Energy Relati­onships
K = 1/2M⋅v2 + 1/2⋅I⋅ω2
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
When the torque remains constant while angle changes
W = τff to Θi)
Total WorkDone on rotating rigid body
W = 1/2(ωf2) - 1/2(ωi2)
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⋅ri2⋅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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