Significant Mechanics Equations

Displacement

∆ x⃗= x⃗2− x⃗1∨∆ r⃗=r⃗ 2−r⃗1

Average velocity

v⃗avg=∆ r⃗∆ t

Average speed

vavg=distance∆ t

Instantaneous velocity

v⃗= lim∆t→ 0

∆ r⃗∆ t

=d r⃗dt Graphs!!!

Instantaneous acceleration

a⃗= lim∆t →0

∆ v⃗∆ t

=d v⃗dt

=d2 r⃗d t2

average acceleration is secant line... algebra!

Constant acceleration equations

v⃗ f= v⃗o+a⃗ t equivalent ¿ a⃗=v⃗ f−v⃗ o∆ t

∆ r⃗=v⃗ ot+12a t2

∆ r⃗=v⃗ avgt ⋮ if a isCONSTANT v⃗avg=12 ( v⃗o+ v⃗ f ) ⋮ then∆ r⃗=

12 ( v⃗o+v⃗ f ) t

v⃗ f2= v⃗o

2+2 a⃗∆ r⃗

Projectile Motion For max range off a cliff, angle is less than 45 degrees

∆ r⃗=∆ x⃗+∆ y⃗

∆ x⃗= v⃗x t ∆ y⃗=v⃗oy t+12a⃗ t 2

Circular Motion

a⃗c=v⃗2

r v⃗=2πr

T

Newton1. Inertia (rest or constant velocity)

2. F⃗net=ma⃗ (object or system of objects)

3. Action↔Reaction (touching or stacked blocks)

Weight

Fg=W=mg depends on location/planet

Friction

f⃗ s , max≤μs N⃗ f⃗ k=μk N⃗ D⃗=12CρA v⃗2

Work

W net=∫xo

x f

F⃗ ° d r⃗=∆K K=12mv2

If F⃗ is constant thenW=F⃗∆ r⃗ cosθ hanging rope dF=λgdy

Potential Energy

∆U=U f−U i=−W=−∫xo

x f

F⃗ °d r⃗ so F⃗=−dUdr

U spring=12k x2 if F⃗=−k x⃗ U gravitational=mghif g is constant

1J=1Nmconservative∧nonconservative net forces will accelerate objects

Conservation of Mechanical Energy

Etotal=U instant+K instant Sameinstant of time!

W=∫xo

x f

F⃗ ° d r⃗=−∆U=∆ K so potential energy lossis kinetic energy gained

If nonconservative forces

Etotal=U instantlater+K instant later+W NC

Power

Rate of energy transfer P=dWdt

=F⃗ ° v⃗ Energy=∫0

t

Pdt

if power averagethen velocitymust be average, if power is instant then velocitymust be instant

Center of mass

F⃗net=ma⃗center of mass xcenterof mass=∑m1 x1

M total=∫

0

M xdmM

=∫0

x xλdxM

vcenterof mass=∑ m1 v1

M total

Momentum

p⃗=mv⃗ F⃗net=ma⃗=d p⃗dt

Collisions

if F⃗net , external=0 then p⃗before= p⃗after

Elastic : Momentum and Energy is conserved!

if v2i=0 thenv1 f=(m1−m2)v1i

(m1+m2)v2 f=

(2m1)v1 i

(m1+m2) v1i+v1f=v2i+v2f

Inelastic: only momentum conserved!

Completely inelastic: momentum conserved but max ∆ K because stuck together

Impulse

J⃗=∫tinitial

t final

F⃗ dt=∆ p⃗ if F constant thenJ⃗=F⃗ ∆ t=m∆ v⃗

Impulse increases when object bounces due to change of direction

Rotation

displacement ∆θ⃗ angular velocity ω⃗=d θ⃗dt

angular accelerationα⃗=d ω⃗dt

=d2 θ⃗d t 2

Tangential displacement s=θr tangential velocity v⃗=ω⃗ r tangential accelerationa⃗ t= α⃗ r

Tangential acceleration and angular acceleration are zero if angular velocity is constant

c entripetal /radial acceleration ac=ω2r= v

2

ra⃗ total

2 =a⃗c2+a⃗ t

2

Constant angular acceleration equations

ω⃗f=ω⃗o+ α⃗ t equivalent ¿ α⃗=ω⃗f−ω⃗o∆ t

∆ θ⃗=ω⃗ot+12α t2

∆ θ⃗=ω⃗avg t ⋮ if α isCONSTANT ω⃗avg=12 ( ω⃗o+ω⃗ f ) ⋮ then∆ θ⃗=

12 ( ω⃗o+ ω⃗f ) t

ω⃗f2=ω⃗o

2+2 α⃗ ∆θ⃗

Moment of inertia

I=∫r2dm=∫r2 λdr I=∑mi ri2

¿axis theorem I=I center of mass+mh2

Some¿ know :

hoop M R2 , disk∨cylinder 12M R2 , solid sphere 2

5M R2 , shell sphere 2

3M R2Rod 1

12M L2

disk or cylinder: I=∫0

R

r2σ 2πrdr Rod: 2∫0

L/2

r2 λdr

Angular Momentum

L⃗=I ω⃗∨l⃗=m(r⃗ × v⃗) if∑ τ external=0 then L⃗before=L⃗after∨∆ L⃗=0

If pinned down Fnet , external❑ means linear momentum not conserved, but no net external torque means angular

momentum is conserved

Torque

τ⃗=r × F⃗ τnet=d L⃗dt

=I α

Rotational Kinetic Energy

K=12I ω2

Rolling

v⃗centerof mass=ω⃗ R K=12I ω2+ 1

2mv2

pure rolling: f(R)=Iα could be tension instead of frictiondown Incline: mgsinθ−f=ma a=Rα slipping a≠αr v≠ωr f=-ma

Work

W net=∫ τ⃗ ° d θ⃗=∆K

Power

P=dWdt

=τ⃗ ° ω⃗

Static Equilibrium

∑ F⃗net=0∧∑ τ⃗net=0 Hanging signs, ladders

Gravity

Kepler 1. Elliptical orbits, sun/planet at focus (faster sun/planet at near focus, slower sun/planet at far focus)

2. Radius vector sweeps out equal areas in equal times ∆ L⃗=0 mvr=mvr

3.T 2

R3 =constant

Gravitational Potential Energy

U=−GMmr

U=0when r=∞ K=GMm2 r

Total Energy=U+K=−GMm2 r

Escape velocity

∆U+∆ K=0 Gmmr

=12mv2 vescape=√ 2GM

rif v final=0

Newton’s Universal Law of Gravitation

F⃗=GMmr2 =mv

2

R R=radius of circle r=center to center Inside planet F=GMmr

R3 R= Radius of Planet

ρ=MV

Acceleration due to gravity

F=mg=GMmr2 g=GM

r2

Oscillations

Definition of simple harmonic motion a=−ω2 x ω=√ kmT=2π √mk f= 1T

=ω

2π

F⃗net=m a⃗=−kx x=Acos (ωt+ϕ ) v=−Aωsin (ωt+ϕ )a=−Aω2cos (ωt+ϕ)

Energy

Etotal=12k A2=1

2k x2+ 1

2mv2=U+K

Pendulum

T=2π √ Lg T=2π √ Imgh

small angle approximation! T=2π √ Iκ τ=Iα=−mghsinθ

τ=Iα=mL2α=−mgLsinθ α=−ω2θ

Electricity and Magnetism Significant Equations

Point Charges

Coulomb’s Law: Force between two charges

F⃗=k q1q2

r2

Electric field

created by point charge E⃗= kqr2 created byrin g flat∈ page E⃗=2∫

0

θ kλRdθR2 cosθ

created by ring on z axis E⃗=∫0

Q kdqr2 :r2=x2+R2

created by disk E⃗=∫0

R kσ 2πzdzr 2 cosθ :r2=x2+z2

Force on a point charge due to Electric Field

E⃗= F⃗q

N/C or V/m

Gauss’s Law

Φ=∫ E⃗∘d A⃗=qencε o

solidqenc=∫ pdV surface∫σdA line∫ λdlElectric field for line , cylindrical shell , solid cylinder , spherical shell , solid shell

Subtract for empty space A=2πrL A=4πr2

Electric Potential Energy due to electric field

ΔU=Ub−Ua=−∫a

b

q E⃗∘d r⃗

Electric Potential Energy between charges

U= kqqr

Work to bring charge distribution together

U=W=k q1q2

r12+k q1q3

r13+…

Electric Potential due to Electric Field (Constant inside conductors)

ΔV=V b−V a=−∫a

b

E⃗∘d r⃗ : start at infinity: If constant E then ΔV=Ed E⃗=−dVdr

Electric potential due to a point charge

ΔV= ΔUq

= kqr (equipotential lines=circles around a point charge, scalar...add!)

Electric potential due to many point charges

V=∑ kdqr

=∫ kdqR scalar quantity

Capacitance

C= QΔV Farads dielectric κ increasesC

Energy stored in capacitors

U=12QV=1

2CV 2=1

2Q2

C

Energy density

u= energyvolume

=12ε0E

2

Parallel Plate

C=κ ε o AdV=Ed E= σ

εo This is true only because E=uniform between plates

Battery connected Voltage doesn’t changeBattery not connected charge on capacitor doesn’t change

Capacitors in Series

1Ceq

= 1C1

+ 1C2……

Capacitors∈¿C eq=C1+C2+…

Cylindrical/spherical capacitors

∫ E⃗∘d A⃗=qencε0

ΔV=−∫a

b

E⃗∘d r⃗ (outside in) C= QΔV

Electric Current

I=Qt=dqdtamperes

Current Density

J= iA used for Ampere's Law

Drift velocityi=nqA vdn=number of charges per volume

Ohm’s Law

V=IR

Resistivity

ρ=EJ

Resistanceof an object

R=ρLAResistance∧resistivity increaseswithtemperature

Power

P=IV=I 2R=V2

R Watts (rate of energy) Energy =∫Pdt

Resistors in Parallel

1R eq

= 1R1

+ 1R2… . .

Resistors in SeriesReq=R1+R2+….

Kirchhoff Rules

∑ i¿=∑ iout at junctions ∑ ΔV loop=0 RC charging( graphs!!! initially capacitor acts like a wire)

ε−iR−QC

=0 i=dq/dt q=Cε (1−e−tRC ) i= ε

Re

−tRC τ=RC

V= qC

=ε (1−e−tRC )

RC discharging (maintain voltage) ( graphs!!! initially capacitor acts like battery)

−qc

−IR=0 i=dqdtq=Cε e

−tRC i= ε

Re

−tRC

Magnetic Force on a point charge due to a magnetic field

F⃗=q v⃗× B⃗ F⃗=mv2

r

Magnetic force on a wire due to a magnetic field

F⃗=I L⃗× B⃗ teslas : electric motors

Magnetic force between two wires

F⃗=μ0 IIL2πd

currents the same direction attract

Velocity selector

qvB=qE v=EB mass spectrometers

Torque on a loop due to a magnetic field

τ⃗= μ⃗× B⃗=NiA×B motors Hall Effect- piece of metal/measuring voltage across, use left hand for electrons!!!

V=Ed=vBd= iBdnqA

= iBnqlwherel=thickness

B-S Law (rings of wire)

B⃗=μo Ids×r

4 π r3 =μo Ids sin 90

4 π r2 =∫0

θ μo IRdθ4 π r2 :R=Radius of ring , r=distance¿ ring ¿ point∈space

Solenoid

B=μ0∋¿ Amper e' sLaw : n= number of turns per unit length of solenoid

Ampere’s Law

∮ B⃗∘d s⃗=μ0 I enc I enc=∫ JdA=∫ J 2πrdr inside vs. outside wires

Torrid

¿amper e' sLaw B=μ0∋¿

2πr¿ : N=number of turns

Magnetic Induction

Φ=∫ B⃗∘d A⃗ Webers CALCULUS

Faraday’s / Lenz’s Law

∮ E⃗∘d s⃗=ε=iR=−N dΦdt ds=2πr : generators

Straight wiremoving throughmagnetic field

ε=Blv

Inductance

Φ=LiNHenry

Inductor

ε=−dΦdt

=−L didt

Solenoid inductorLl=μ0n

2 A

LR charging

ε−iR−L didt

=0 I= εR

(1−e−RtL ) τ=L/R

LRdischarging (inductors maintain current)

−iR−L didt

=0 I= εRe

−RtL

Energy Stored in an inductor

U=12L i2

LC circuit

L d2Qdt 2

+QC

=0q=Qcos (ωt+ϕ )ω= 1√LC

Maxwel l' s Equations

Gauss’ Law ∫ E⃗∘d A⃗=qencε0

Gauss’ Law for Magnetism ∫ B⃗∘d A⃗=0

Faraday’s Law ∮ E⃗∘d s⃗=−N dΦdt

Ampere’s Law with Maxwell’s displacement current ∮ B⃗∘d s⃗=μ0 I enc+μo ε0d Φdt

Transformer V sV p

=N sN p

1. Design experiments

Students should understand the process of designing experiments, so they can:

a) Describe the purpose of an experiment or a problem to be investigated. b) Identify equipment needed and describe how it is to be used. c) Draw a diagram or provide a description of an experimental setup. d) Describe procedures to be used, including controls and measurements to be taken.

2. Observe and measure real phenomenaStudents should be able to make relevant observations, and be able to takemeasurements with a variety of instruments (cannot be assessed via paper-and-pencilexaminations).

3. Analyze dataStudents should understand how to analyze data, so they can:

a) Display data in graphical or tabular form. b) Fit lines and curves to data points in graphs. c) Perform calculations with data. d) Make extrapolations and interpolations from data.

4. Analyze errorsStudents should understand measurement and experimental error, so they can:

a) Identify sources of error and how they propagate. b) Estimate magnitude and direction of errors.c) Determine significant digits.

d) Identify ways to reduce error.

5. Communicate resultsStudents should understand how to summarize and communicate results, so they can:

a) Draw inferences and conclusions from experimental data. b) Suggest ways to improve experiment. c) Propose questions for further study.