Physics 121 Final Exam: Sample Formula Sheet (ALL material)vave. = ∆x/∆t ; v = dx/dt; aave. = ∆v/∆t ; a = dv/dt

Constant linear acceleration:x = x0 + v0t+ (1/2)at2 ; v = v0 + at ; v2 = v20 + 2a(x− x0)

|F| =√

F 2x + F 2

y

θ = arctan(Fy/Fx) (+π rad or 180 deg., if necessary)Fx = |F| cos θ; Fy = |F| sin θsin = opp./hyp. cos = adj./hyp. tan = opp./adj.

a = g = −9.8m/s2 j gives:x = x0 + vx0t; vx = vx0y = y0 + vy0t− 1

2gt2; vy = vy0 − gt

Normal force between surfaces always points perpendicular to the surfacesAngle in radians = (arc length of part of circle)/(radius)ω = dθ/dt ; α = dω/dt = d2θ/dt2

||acirc.|| = v2/r

Constant angular acceleration:θ = θ0 + ω0t+ (1/2)αt2 ; ω = ω0 + αt ; ω2 = ω2

0 + 2α(θ − θ0)

FIRST LAW: A body moving at constant velocity will continue to do sounless acted upon by a nonzero net outside force. In particular, a body at restwill remain at rest unless a net force acts on it.

SECOND LAW: Force acts to change momentum p (see definition below)according to the equation ΣF = dp/dt. In the special case that the object isnot losing mass, this reduces to ΣF = ma.

momentum p = mvweight w = mgperiod of circular motion T = 2π/ωFrictional forces: fs ≤ µs|n| ; fk = µk|n| ; fr = µr|n|Fx = |F| cos θ ; Fy = |F| sin θ if θ CCW from +x axissin = opp./hyp. ; cos = adj./hyp. ; tan = opp./adj. = sin/cos.

|F| =√

F 2x + F 2

y + F 2z ; θ = arctan(Fy/Fx)(possibly,+ π rad or 180 deg.)

acirc. = v2t /rat = αr vt = ωrv ·w = |v| |w| cos θ = vxwx + vywy + vzwz

W =∫ B

AF · dx

K = 12mv2

Wtotal = ∆K (work-energy theorem)

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P = F · v = dW/dt1 J = 1Nm = 1kgm2/s2 1W = 1J/sWloop =

∮

F · dx = 0 defines a conservative forceU = −

∫

F · dx Fx = −dU/dxUspring = 1

2kx2 Fspring = −kx

Ugravity = mgyTotal energy = constant (= K + U if all forces are conservative)Ki + Ui = Kf + Uf (if all forces are conservative)g = 9.8m/s2

THIRD LAW: For every force of action, there is an equal and oppositereaction.

Orbital speed at radius R: v =√gR.

rCM =

∑

i rimi∑

i mi

P =∑

i mivi = MvCM Fnet external = dP/dtP = constant when net external force vanishesImpulse =

∫

Fdt; net impulse = ∆pElastic collision: total kinetic energy is constantInelastic collision: objects stick together

Rotational motion: ω = dθ/dt α = dω/dt = d2θ/dt2

If α = constant:ω = ω0 + αt θ = θ0 + ω0t+ (1/2)αt2 ω2 = ω2

0 + 2α(θ − θ0)I =

∑

i mir2i I = ICM +Md2 τ = r× F

|τ | = |F| |r| sin θ ; τ net = Iα if I is constant

Krot. =1

2Iω2 ; Ktotal =

1

2Mv2CM +

1

2ICMω2

CM

v = ωr (actually special case of v = ω × r)Right-hand convention:

R.H. fingers curling along rotation direction (or change of rotation for the vectorα) gives thumb at right angles along ω, (or α) by convention.

Cross product magnitude: |v ×w| = |v| |w| sin θRight-hand rule for directon of cross product: If R.H. fingers curl

from v toward w along smallest angle, R.H. thumb pointing at right angles isdirection of v ×w.Components for cross product: (v ×w)x = vywz − vzwy;(v ×w)y = vzwx − vxwz; (v ×w)z = vxwy − vywx

L = r× p L = Iω τ net external = dL/dt in generalL = constant if net external τ vanishesQuadratic Formula: solution of Ax2 +Bx+ C = 0 is given by

x = [−B ±√

B2 − 4AC]/(2A)

|F| = GMm

r2attractive

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U = −GMm

r

Ki + Ui = Kf + Uf = Etotal

G = 6.67× 10−11 Nm2/kg2

vcirc =√

GM/r vesc. =√

2GM/rf = 1/T ω = 2πfFx = −kx comes from F = −∇U for U = kx2/2Simple Harmonic Motion:x(t) = A cos

(

ωt+ φ)

ω =√

k/m (linear spring) ω =√

g/l (ideal pendulum)

ω =√

MgR/I (general pendulum)Etotal = mv2/2 + kx2/2 = kA2/2 = constantω =

√

κ/I for τ = −κθTF = (9/5)TC + 32; TC = TK − 273.15C ≡ (∆Q)/(∆T )1 kcal = 4184 J; 1 cal = 4.184 J.c = C/m = ∆Q/(m∆T )cwater = 4184 J/(kg◦C)∆Q = LmIf no heat is lost while exhanging: m1c1∆T1 +m2c2∆T2 = 0, which gives:

Tfinal =m1c1T1 +m2c2T2

m1c1 +m2c2

∆Q = LmpV = NkBT = nRTkB = 1.38× 10−23 J/K R = 8.31 J/(molK)12mv2 = 3

2kBT for monatomic gas

First Law of Thermodynamics: ∆Einternal = ∆Q− p∆V∆W = −p∆V (specifies work done on the system);

∆W ′ = p∆V (specifies work done by the system)Second Law of Thermodynamics: Heat may not be converted entirely into

work; total entropy can never decrease with time.emax. = 1− Tc/TH

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