Phys 1301 Final Exam Formula Sheet

tva

txv

sm.g

ΔtΔva

ΔtΔxv

vv

vvv

yBAxBABA

AA

yAxAA

t

t

avg

avg

yyxx

yx

yx

ΔΔ

=

ΔΔ

=

=

=

=

=

−=

+=

+++=+

+=

+=

→Δ

→Δ

→→

→→→

→→

→

0

0

2

2112

231213

2/122

lim

lim

timedistancespeed

819

ˆ)(ˆ)(

)(A

ˆˆ

Equations General

motion Relative

itionVector Add

Vectors

tvvyy

tvvxx

tatvyy

tatvxx

yyavv

xxavv

tavvtavv

yy

xx

yy

xx

yyy

xxx

yyy

xxx

)(21

)(21

2121

)(2

)(2

00

00

200

200

020

20

20

2

0

0

++=

++=

++=

++=

−+=

−+=

+=

+=

Motion of uationsqE

Projectile motion: make ax=0, ay= - 9.81m/s2

θ⋅=

θ⋅=

sincos

00

00

vvvv

y

x

centripetal force: Fc= mv2/r equilibrium conditions à ΣFx=0, ΣFy=0

non-equilibrium conditions à ΣFx=max, ΣFy=may static friction force fs≤μsN kinetic friction force fk=μkN spring force: Fs=-kx

€

Work and Energy equationsW = (F cosθ)dWTotal =W1 +W2 +W3 + ...WTotal = (FTotal cosθ)dWTotal = ΔK

K =12mv 2

Wspring =12kx 2

P =Wt

P = Fv cosθ

Momentum .vmp

=

 

Conservation of momentum: m1v1i+m2v2i= m1v1f+m2v2f

Inelastic collision:

of vmmmmv ⎟⎟

⎠

⎞⎜⎜⎝

⎛

+

−=

21

211 (for a 1-dimensional elastic collision)

of vmmmv ⎟⎟

⎠

⎞⎜⎜⎝

⎛

+=

21

12

2 (for a 1-dimensional elastic collision)

Mmx

XCM∑=

Mm

CM∑= v

V

Mm

CM∑= a

A

€

Wc = −ΔU

€

Ugravity = mgyi 2

21 kxUspring =

€

E =U +K

€

Wnc = ΔE = E f − Ei

Rotational Kinematics and Energy Average angular velocity: ω θ

av =Δ

Δt Tangential speed:  v! = rω

Instantaneous angular velocity: ω

θ=

→limΔ

Δ

Δt t0

Period of rotation: T = 2π

ω

Average angular acceleration: α

ωav =

Δ

Δt

Instantaneous angular acceleration: α ω=

→limΔ

Δ

Δt t0

Angular kinematic equations:

ω ω α= +0 t

θ θ ω ω= + +012 0( )t

θ θ ω α= + +0 012

2t t

ω ω α θ θ202

02= + −( )

Centripetal acceleration: a rcp = ω 2=v2/r

Tangential acceleration: a rt = α

Rotational kinetic energy: K I= 12

2ω

Moment of inertia: I m r=∑ i i2

Kinetic energy of rolling motion:  K =!!mv! + !

!Iω!

Mo

Moment of inertia of objects:              𝐼!"#$ =!!𝑀𝑅!

𝐼!"# =!!"𝑀𝐿! axis through midpoint

𝐼!"# =!!𝑀𝐿! axis at one end

Rotational Dynamics and Static Equilibrium Torque (Tangential force): τ = rF

Torque (general definition):τ θ= rF sin

Newton’s law for rotational motion: τ α∑ = =I L

tΔ

Δ

Angular momentum: L I= ω

Equilibrium conditions: ΣFx=0; ΣFy=0; Στ=0

Work done by torque: W = τ θΔ

Work-energy theorem: W K K K= = −Δ f i

Angular momentum for point particle: L rmv= sinθ

Gravity

Gravitational force between m1 and m2: 221

rmmGF =

Gravitational acceleration on the surface of the Earth: 2E

E

RGM

g =

Gravitational acceleration on the surface of a planet P: g = GMP

RP2

Period of rotation around the sun: T = 2π

GMS

!

"##

$

%&&(r)

32

Gravitational potential energy: rmmGU 21−=

Escape speed for the Earth: E

Ee R

GMv

2=

Mechanical energy: UKE +=

2211 /1067.6 kgNmxG −= kgxM Earth

241097.5= kgxM sun

301000.2= kmxREarth31037.6=

Oscillations

Frequency of oscillation: Tf 1=

Position versus time in Simple Harmonic Motion: ( )tAx ωcos=

Velocity in Simple Harmonic Motion: v =- Aωsin(ωt) ; vmax=Aω

Acceleration in Simple Harmonic Motion: a =- Aω2cos(ωt); amax= Aω2

Angular frequency of a mass on a spring: mk

=ω

Period of a mass on a spring: kmT π2=

Tf ππω

22 ==

Potential energy in oscillatory motion: )(cos

21 22 tkAU ω=

Kinetic energy in oscillatory motion: )(sin

21 22 tkAK ω=

Total energy in oscillatory motion: 2

21 kAE =

Period of a Pendulum: gLT π2=

Sound and Waves

speed of a wave: v=λf

speed of a wave on a string:

harmonics for string or pipe with both end open: , n=1,2,3…

harmonics in a pipe, one end open: , n=1,3,5…

Doppler effect: f ' = f (1± vo / v1 vs / v

)

Constructive interference: d2-d1=nλ, n=1,2,3…

Destructive interference: d2-d1=(2n+1) λ2

, n=0,1,2,3…