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PHY2048 Spring 2015
PHY2048 Exam 2 Formula Sheet
Work (W), Mechanical Energy (E),r Kinetic Energy (KE), Potential Energy (U)
r
r r
r r
r r
dW
2
r→F ⋅d
Kinetic Energy: KE = 12 mv
Work: W = F ⋅ dr ⎯⎯ ⎯ ⎯
Power: P =
=
F
⋅v
⎯
Cons tan t F
∫rr
dt
2
1
r
r2
r
∫
r
Potential Energy: ΔU = − F ⋅ dr
Work-Energy Theorem: KE f = KEi + W
r
r1
Work-Energy: W(external) = ΔKE + ΔU + ΔE(thermal) + ΔE(internal)
Fx ( x) = −
dU ( x)
dx
Work: W = -ΔU
U ( y ) = mgy
Gravity Near the Surface of the Earth (y-axis up): Fy = − mg
Spring Force: Fx ( x) = − kx
U ( x) = 12 kx 2
Mechanical Energy: E = KE + U Isolated and Conservative System: ΔE = ΔKE + ΔU = 0
E f = Ei
Linear Momentum, Angular Momentum, Torque
r
t
r
r
r r dp
r f r
p2
Linear Momentum: p = mv F =
Kinetic Energy: KE =
Impulse: J = Δp = ∫ F (t )dt
dt
2m
ti
Center of Mass (COM): M tot =
N
∑ mi
i =1
r
r
r
dPtot
Net Force: Fnet =
= M tot aCOM
dt
r
1
rCOM =
M tot
r
∑ mi ri
N
i =1
r
N
∑p
i =1
i
N
r
r
r
Ptot = M tot vCOM = ∑ pi
i =1
N
Moment of Inertia:
r
1
vCOM =
M tot
I = ∑ mi ri 2 (discrete) I = ∫ r 2 dm
i =1
(uniform)
Parallel Axis: I = I COM + Mh 2
r
θf
r r dL
Torque: τ = r × F =
Work: W = ∫ τ dθ
dt
θi
r
r
r
r
r
dp
Conservation of Linear Momentum: if Fnet =
= 0 then p = constant and p f = pi
dt r
r
r
r
r
dL
Conservation of Angular Momentum: if τ net =
= 0 then L = constant and L f = Li
dt
Rotational Varables
r r r
Angular Momentum: L = r × p
Angular Position:
r
2
θ (t ) Angular Velocity: ω (t ) = dθ (t ) Angular Acceleration: α (t ) = dω (t ) = d θ 2(t )
dt
dt
Torque: τ net = Iα Angular Momentum: L = Iω
Arc Length: s = Rθ
Rolling Without Slipping: xCOM = Rθ
L
Power: P = τω
2I
Tangential Acceleration: a = Rα
Kinetic Energy: Erot = 12 Iω 2 =
Tangential Speed: v = Rω
dt
2
vCOM = Rω aCOM = Rα
2
KE = 12 MvCOM
+ 12 I COM ω 2
Rotational Equations of Motion (Constant Angular Acceleration α)
ω (t ) = ω0 + αt
θ (t ) = θ 0 + ω0t + 12 αt 2
ω 2 (t ) = ω02 + 2α (θ (t ) − θ 0 )
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