Preliminary formula Sheet for LSU Physics 2101, Exam 3, Fall ’12

Units:1m = 39.4 in = 3.28 ft 1 mi = 5280 ft 1min = 60 s, 1 day = 24 h 1 rev = 360 = 2π rad

1 atm = 1.013×105 Pa T =(

1K

1C

)TC + 273.15K TF =

(9 F5C

)TC + 32F

Constants:g = 9.8 m/s2 REarth = 6.37× 106 m MEarth = 5.98×1024 kgG = 6.67×10−11 m3/(kg·s2) RMoon = 1.74× 106 m MMoon = 7.36×1022 kgEarth-Sun distance = 1.50×1011 m MSun = 1.99×1030 kg Earth-Moon distance = 3.82×108 m

Quadratic formula: for ax2 + bx + c = 0, x1,2 =−b ± √

b2 − 4ac

2a

Magnitude of a vector: |~a| =√

a2x + a2

y + a2z

Dot Product: ~a ·~b = axbx + ayby + azbz = |~a|∣∣∣~b

∣∣∣ cos(φ) (φ is smaller angle between ~a and ~b)

Cross Product: ~a ×~b = (aybz − azby)ı + (azbx − axbz) + (axby − aybx)k,∣∣∣~a ×~b

∣∣∣ = |~a|∣∣∣~b

∣∣∣ sin(φ)

Equations of Constant Acceleration:

linear equation along x missing missing rotational equationvx = vox + axt x − xo θ − θo ω = ωo + αxt

x − xo = voxt +1

2axt2 vx ω θ − θo = ωot +

1

2αt2

v2x = v2

ox + 2ax(x − xo) t t ω2 = ω2o + 2α(θ − θo)

x − xo =1

2(vox + vx)t ax α θ − θo =

1

2(ωo + ω)t

x − xo = vxt − 1

2axt2 vox ωo θ − θo = ωt − 1

2αt2

Vector Equations of Motion for Constant Acceleration: ~r = ~ro + ~vot +1

2~at2, ~v = ~vo + ~at

Projectile Motion: (with + direction pointing up from Earth)

x − xo = (vo cos θo)t y − yo = (vo sin θo)t − 1

2gt2

vx = vo cos θo vy = (vo sin θo) − gt

y = (tan θo)x − gx2

2(vo cos θo)2v2

y = (vo sin θo)2 − 2g(y − yo)

Newton’s Second Law:∑

~F = m~a

Uniform circular motion: Fc = mac =mv2

r, T =

2 π r

v

Force of Friction: Static: fs ≤ fs,max = µsFN , Kinetic: fk = µkFN

Spring (elastic) Force: Hooke’s Law Fspring = −kx (k = spring (force) constant)

Kinetic Energy: Translational K =1

2mv2

Work: W =∫ xf

xi

F (x)dx (variable 1D force), W =∫ rf

ri

~F (~r) · d~r (variable 3D force),

W = ~F · ~d (constant force)

Work - Kinetic Energy Theorem: W = ∆K = Kf − Ki

Work done by weight (gravity close to the Earth surface): W = m ~g · ~d = −m g ∆y

Work Done by a Spring Force: Ws = −k

(x2

f

2− x2

i

2

)

Power: Average: Pavg =W

∆t, P = ~F · ~vavg (constant force)

Instantaneous: P =dW

dt, P = ~F · ~v (constant force)

Potential Energy Change: ∆U = −W

Gravitational Potential Energy: Ug(y) = mgy

Elastic (Spring) Potential Energy: Us =1

2kx2 (relative to the relaxed spring)

Potential-Force Relation: F (x) = −dU(x)

dxMechanical Energy: Emec = K + U

Change of Mechanical Energy due to Non-Conservative (nc) Forces:

Wnc = ∆Emec = (Kf + Uf) − (Ki + Ui)

Conservation of Energy: Wext = ∆K + ∆U + ∆Eth + ∆Eint, Wext is the net, external work done byexternal forces on the system, ∆Eth = −Wfk is the thermal energy change, ∆Eint is the internal energy change

Center of mass: M =N∑

i=1

mi, xcom =1

M

N∑

i=1

mixi, ycom =1

M

N∑

i=1

miyi, zcom =1

M

N∑

i=1

mizi

~rcom =1

M

N∑

i=1

mi~ri ~vcom =1

M

N∑

i=1

mi~vi ~acom =1

M

N∑

i=1

mi~ai =1

M

N∑

i=1

~Fi

Definition of Linear Momentum: one particle: ~p = m~v, system of particles: ~P =N∑

i=1

~pi = M~vcom

Newton’s 2nd Law for a System of Particles: ~Fnet = M~acom =d ~P

dt

Conservation of Linear Momentum of an Isolated System:∑

~pi =∑

~pf

Impulse - Linear Momentum Theorem: ∆~p1 = ~J12 =∫ t2

t1

~F12(t)dt = ~Favg,12∆t

Elastic Collision (1 Dim): v1f =m1 − m2

m1 + m2v1i +

2m2

m1 + m2v2i v2f =

2m1

m1 + m2v1i +

m2 − m1

m1 + m2v2i

Linear and Angular Variables Related:

s = rθ v = ωr at = αr ar =v2

r= ω2r (magnitude of the radial or centripetal acceleration)

Rotation: Rotational Inertia (Icom) for Simple Shapes: see next page

Rotational Inertia: Descrete particles: I =N∑

i=1

Ii Continuous object: I =∫

r2dm

Parallel Axis Theorem: I = Icom + Mh2

Torque: ~τ = ~r × ~F τ = rFt = r⊥F = rF sin φ

Angular Momentum: rigid body, fixed axis: ~L = I~ω point-like particle: ~L = ~r × ~p

Newton’s 2nd Law: ~τnet = I~α ~τnet =d~L

dtConservation Law (isolated system,

∑τ = 0):

∑ ~Li =∑ ~Lf

Rotational Work: W =∫ θf

θi

τdθ = τavg∆θ Kinetic Energy: K =1

2Iω2

Rotational Power: Instantaneous: P =dW

dt= τω Average: Pavg =

W

t= τavgωavg

Rolling: vcom = ωR acom = αR

Kinetic Energy of Rolling: K =1

2mv2

com +1

2Icomω2

Static equilibrium: ~Fnet = 0 ~τnet = 0

Gravity:

Newton’s law: |~F | = Gm1m2

r2Gravitational acceleration (planet of mass M): ag =

GM

r2

Law of periods: T 2 =

(4π2

GM

)r3 Potential Energy: U = −G

m1m2

r

Potential Energy of a System (more than 2 masses): U = −(

Gm1m2

r12+ G

m1m3

r13+ G

m2m3

r23+ ...

)

Static Fluids:

Density: ρ =∆m

∆VPressure: p =

∆F

∆AAbsolute Pressure: p = po + ρgh

Pressure Variation with Height or Depth: p2 = p1 + ρ g (y1 − y2) Gauge pressure: p − po

Archimedes’ Principle: Fb = ρfVdisplacedg = mfg weightapparent= mg − Fb

Simple Harmonic Motion (SHM): T =1

f=

2π

ωLinear: x(t) = xm cos(ωt + φ) Angular: Θ(t) = Θm cos(ωt + φ)

v(t) = −xmω sin(ωt + φ) Ω(t) = −Θmω sin(ωt + φ)a(t) = −xmω2 cos(ωt + φ) = −ω2x(t) α(t) = −Θmω2 cos(ωt + φ) = −ω2Θ(t)

Linear Oscillator: Spring-Block: ω =

√k

mHorizontal Spring-Block: Emec =

1

2kx2

m

Pendulums: Torsion: ω =√

κ

ISimple: ω =

√g

LPhysical: ω =

√mgh

ITorsion torque: τ = −κΘ