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Angular Velocity and Acceleration
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Θ = angle (radians)
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s = length
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r = radius
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90o = π/2 rad
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Θ = (s/r)
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s = r ⋅ Θ
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1 rad = (360o / 2π) = 57.3o
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180o = π rad
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Angular Velocity
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(1st Derivative)
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ω = (Θ
f- Θi) / (tf-ti) |
ω = "velocity"
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1 rev/s = 2π rad/s
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1 rev/min = 1 rpm = 2π/60 rad/s
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Angular Acceleration
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(2nd Derivative)
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α = (ω
f- ωi) / (tf-ti) |
α = "acceleration"
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Rotation w/ Constant Angular Acceleration
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α
f = (ωf- ωi) / (t - 0) |
α
f = constant |
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ω
f = ωi + αf ⋅ t |
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Θ
f- Θi = 1/2(ωi+ωf) ⋅ t |
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Θ
f = Θi+ (ωi⋅ t ) + 1/2 (αf⋅ t2) |
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ω
f2 = ωi2 + 2⋅αf(Θf- Θi) |
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Relating Linear and Angular Kinematics
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K = 1/2(m ⋅ v2)
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Linear Speed in Rigid-Body Rotation
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s = r ⋅ Θ
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Linear Speed
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v = r ⋅ ω
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Linear Acceleration in Rigid-Body Rotation
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a
tan = r ⋅ α |
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Centripetal Component of Acceleration
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a
rad = (v2/r) = ω2⋅r |
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Energy in Rotational Motion
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K
E: 1/2⋅m⋅v2 = 1/2⋅m⋅r2⋅ω2 |
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K = 1/2⋅m⋅r2⋅ω2
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I = m⋅r2
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Gravitational Potential Energy for an Extended Body
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U = M⋅g⋅y
cm |
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Moment of Inertia
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I
p=Icm+Md2 |
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Rotational Kinetic Energy
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K = Joules
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K = 1/2⋅I⋅ω2
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R = Radius
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M = mass pivoted about an axis
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Perpendicular to the Rod
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I = (M⋅L2) / 3
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Slender Rod (Axis Center)
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I = 1/12M⋅L2
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Slender Rod (Axis End)
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I = 1/3M⋅L2
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Rectangular Plate (Axis Center)
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I = 1/12M⋅(a2+b2)
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Rectangular Plate (Axis End)
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I = 1/3M⋅(a2)
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Hallow Cylinder
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I = 1/2M(R
i2+Rf2) |
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Solid Cylinder
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I = 1/2MR2
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Hollow Cylinder (Thin)
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I = MR2
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Solid Sphere
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I = 2/5MR2
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Hollow Sphere (Thin)
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I = 2/3MR2
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1st Condition of Equilibrium (at rest)
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ΣF = 0
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2nd Condition of Equilibrium (nonrotating)
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Στ = 0
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Center of Gravity
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r
cm = (m1⋅ r1) / m1 |
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Solving Rigid-Body Equilibrium Problems
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ΣF
x = 0 |
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1st Condition
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ΣF
x = 0 ΣF y = 0 |
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2nd Condition (Forces xy-plane)
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Στ
z = 0 |
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Stress, Strain, and Elastic Moduli
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Stress = Force Applied to deform a body
Strain = how much deformation |
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Hooke's Law
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(Stress / Strain) = Elastic Modulus
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A = Area
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F = Magnitude of Force
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Tensile Stress
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F / A
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1 Pascal = Pa = 1 N/m2
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1 psi = 6895 Pa
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l = length
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1 Pa = 1.450 ⋅ 10-4
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Tensile Strain
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(l
f- li) / (li) |
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Young Modulus
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(Tensile Stress) / (Tensile Strain)
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Pressure
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p = F (Force Fluid is Applied) / A (Area which force is exerted)
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Bulk Stress
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(p
f - pi) |
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Bulk Strain
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(V
f- Vi) / (Vi) |
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Bulk Modulus
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Bulk Stress / Bulk Strain
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Torque
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F = Magnitude of F
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|| || = Magnitude Symbol
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τ = F⋅l = r⋅F⋅sinΘ = F
tanr |
L = lever arm of F
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τ = ||r|| x ||F||
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Torque and Angular Acceleration for a Rigid Body
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Newtons 2nd Law of Tangential Component
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F
tan = m1⋅a1 |
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Rotational analog of Newton's second law for a rigid body
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Στ
z = l⋅αz |
z = rigid body about z-axis |
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Combined Translation and Rotation: Energy Relationships
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K = 1/2M⋅v2 + 1/2⋅I⋅ω2
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Rolling without Slipping
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v = R⋅ω
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Combined Translation and Rotation: Dynamics
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Rotational Motion about the center of mass
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Στ
z = l⋅αz |
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Work and Power in Rotational Motion
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F = M ⋅ a
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When it rotates from Θ
i to Θf |
W = ∫ (Θ
f to Θi) τf dΘ |
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When the torque remains constant while angle changes
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W = τ
f(Θf to Θi) |
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Total WorkDone on rotating rigid body
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W = 1/2(ω
f2) - 1/2(ωi2) |
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Power due to torque on rigid body
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P = τ
z⋅ ωz |
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Angular Momentum
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L = r x p (r x m ⋅ v)
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Angular Momentum of a Rigid Body
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L = m
i⋅ri2⋅ω |
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F = Force acting tangent to the surface divided by the Area
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Shear Stress
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F / A
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h = transverse dimension [bigger]
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x = relative displacement (empty) [smaller]
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Shear Strain
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x / h
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