MidTerm 3

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 = (Θ `f` - Θ `i` ) / (t `f` -t `i` )	ω = "velocity"
1 rev/s = 2π rad/s	1 rev/min = 1 rpm = 2π/60 rad/s
Angular Acceleration	(2nd Derivative)
α = (ω `f` - ω `i` ) / (t `f` -t `i` )	α = "acceleration"
Rotation w/ Constant Angular Acceleration
α `f` = (ω `f` - ω `i` ) / (t - 0)	α `f` = constant
ω `f` = ω `i` + α `f` ⋅ t
Θ `f` - Θ `i` = 1/2(ω `i` +ω `f` ) ⋅ t
Θ `f` = Θ `i` + (ω `i` ⋅ t ) + 1/2 (α `f` ⋅ t²)
ω `f` ² = ω `i` ² + 2⋅α `f` (Θ `f` - Θ `i` )
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	a `tan` = r ⋅ α
Centripetal Component of Acceleration	a `rad` = (v²/r) = ω²⋅r
Energy in Rotational Motion	K `E` : 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⋅y `cm`
Moment of Inertia	I `p` =I `cm` +Md²

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(R `i` ²+R `f` ²)
Solid Cylinder	I = 1/2MR²
Hollow Cylinder (Thin)	I = MR²
Solid Sphere	I = 2/5MR²
Hollow Sphere (Thin)	I = 2/3MR²

1st Condition of Equilibrium (at rest)	ΣF = 0
2nd Condition of Equilibrium (nonrotating)	Στ = 0
Center of Gravity	r `cm` = (m `1` ⋅ r `1` ) / m `1`
Solving Rigid-Body Equilibrium Problems	ΣF `x` = 0
1st Condition	ΣF `x` = 0 ΣF `y` = 0
2nd Condition (Forces xy-plane)	Στ `z` = 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	(l `f` - l `i` ) / (l `i` )
Young Modulus	(Tensile Stress) / (Tensile Strain)
Pressure	p = F (Force Fluid is Applied) / A (Area which force is exerted)
Bulk Stress	(p `f` - p `i` )
Bulk Strain	(V `f` - V `i` ) / (V `i` )
Bulk Modulus	Bulk Stress / Bulk Strain

Torque
F = Magnitude of F	\|\| \|\| = Magnitude Symbol
τ = F⋅l = r⋅F⋅sinΘ = F `tan` r	L = lever arm of F
τ = \|\|r\|\| x \|\|F\|\|
Torque and Angular Acceleration for a Rigid Body
Newtons 2nd Law of Tangential Component	F `tan` = m `1` ⋅a `1`
Rotational analog of Newton's second law for a rigid body
Στ `z` = l⋅α `z`	`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	Στ `z` = l⋅α `z`
Work and Power in Rotational Motion	F = M ⋅ a
When it rotates from Θ `i` to Θ `f`	W = ∫ (Θ `f` to Θ `i` ) τ `f` dΘ
When the torque remains constant while angle changes	W = τ `f` (Θ `f` to Θ `i` )
Total WorkDone on rotating rigid body	W = 1/2(ω `f` ²) - 1/2(ω `i` ²)
Power due to torque on rigid body	P = τ `z` ⋅ ω `z`
Angular Momentum	L = r x p (r x m ⋅ v)
Angular Momentum of a Rigid Body	L = m `i` ⋅r `i` ²⋅ω

	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

MidTerm 3 by brandenz1229