Physics Formula Sheet Memory Aid PDF

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Memory Aid that will help with remembering basic formulas for physics....

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Formulas and Constants for College Physics Stefan Bracher 04/17/17

Formulas and Constants for College Physics

Stefan Bracher

Math + Measurement Quadratic Equations Quadratic Equation

ax² + bx + c=0 → x12 =

−b± √ b² −4ac 2a

Trigonometry

Trigonometric Identitiess Soh-Cah-Toa opp sin (θ)= hyp adj cos(θ)= hyp opp sin(θ) = tan(θ)= adj cos(θ)

Vectors Dot-Product

sin (2 θ)=2 sin (θ) cos(θ)

Identities

2

2

sin (θ)+ cos (θ)=1 1 =tan 2 (θ)+ 1 2 cos (θ)

* *a⋅b=a x b x + a y b y + a z b z =a bcos( θ)

[ ][ ][

Cross-Product

]

a y b z− b y a z ax bx * a x b = a b −b a *a xb= z x z x y y a x b y −b x a y az bz

* a b sin (θ) ∣*a x b∣= (Applies to Right-Hand Coordinate Systems only)

Right hand coordinate system

Addition/Subtraction

Head-to-Tail method or by components

ln(xy)=ln(x)+ln(y)

ln(x/y)=ln(x)-ln(y)

Logarithms ln(xs)=sln(x)

ln(ex)=x

Linear Alegebra System of Equations

a 11 x1 +a12 x2 +a 13 x3 +...=c1 a 21 x1 +a22 x 2+a 23 x3 +...=c2 a 31 x1 +a32 x2 +a 33 x3 +...=c3 ...

→

A x=C→ x= A− 1 C

[

a11 a 12 a 13 a21 a 22 a 23 A= a31 a 32 a 33 ... ... ...

] [ ] []

... ... ... ...

x1

c1

x= x 2 x3 ...

C= c 2 c3 ...

Uncertainty Absolute

x= xavg ±Δ x

Relative

x= xavg ±

Uncertainty is obtained by:

Δx x

- Estimation (at least, ½ the lowest increment)* - Statistics (mean deviation, standard deviation) ** * Usually much more (uncertainty of method, object, tool and observer add up) ** typically more than 10 measurements are needed

Addition/Subtraction

( x avg ±Δ x)+ ( y avg ±Δ y)=( x avg + y avg )±(Δ x+ Δ y )

Multiplication/ Division

( x avg ±Δ x)∗( y avg ±Δ y )=x avg∗ yavg (1±[

Min-Max Method Estimated Digit

xbest ±

Δx Δy + ]) xavg yavg

xmax− xmin 2

The last written digit of a number is estimated. Only one written digit should be affected by the uncertainty

http://stefan.bracher.info/physics.php

Example:

1.50 ± 0.04

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Formulas and Constants for College Physics

Stefan Bracher

Excel / LibreOffice / OpenOffice General Typing formulas and equations

Click on a cell and type =.......

Fixing a cell reference

Add a $ in front of the row or column. Ex. $C$4

Average

AVERAGE(Range)

Highest, Lowest Value

MIN(Range) , MAX(Range)

Mean Deviation (Uncertainty)

AVEDEV(Range)

Standard Deviation

STDEV(Range)

Statistics

Slopes and Intercept Slope

INDEX(LINEST(Y_Values,X_Values, TRUE,TRUE),1,1)

Intercept

INDEX(LINEST(Y_Values,X_Values, TRUE,TRUE),1,2)

Uncertainty of Slope

INDEX(LINEST(Y_Values,X_Values, TRUE,TRUE),2,1)

Uncertainty of Intercept

INDEX(LINEST(Y_Values,X_Values, TRUE,TRUE),2,2)

Calculator Hints Equation Model

Sharp

Casio

EL-520X

EL-W516X

Quadratic Equation

Mode → 2 → 2

Mode → 6 → 2

System of equations

Mode → 2 → 0 / 1

Mode → 6 → 0/1

fx-991 ES

Mode → 5 → 2

Wolfram Alpha - http://www.wolframalpha.com/ Equation Example Quadratic Equation

3x^2 - 2x - 2= 0 → Enter

System of equations

3x+2y+7z=4, 3x+y=0, y+z=0 → Enter

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Formulas and Constants for College Physics

Stefan Bracher

Mechanics 1D Kinematics Position

2D/3D Kinematics s=x, s=y or s=z (sign gives direction) Δ s =s f − s i

Displacement Average velocity

vav =

Instantaneous velocity

v=

Average acceleration

ds dt

aav=

Instantaneous acceleration

a=

s f −s i Δt (slope of position)

v f −v i Δt

dv dt

Graphs

*r=( x , y , z )

Displacement

Δ *r= r*f −*r i=(Δ x , Δ y , Δ z )

Average velocity

v*av =

Instantaneous velocity

*v=

Average acceleration Instantaneous acceleration

(slope of velocity)

Δ v=at Δ s=1/ 2( vi+ v f )t Δ s=vi t + 1/2 a t² Δ s= v f t − 1/ 2 a t² v 2f =v i2 + 2a Δ s

Five equations for CONSTANT acceleration

Position

Five equations for CONSTANT acceleration

→ area v-t slope ←

v*f − v*i Δ v x Δ v y Δ v z ) =( , , Δt Δt Δt Δt

d *v =( dv x , dv y , dv z ) dt dt dt dt

v f = v*i+ *a t * r*f =*ri + 1/2(* v i+ v*f )t ri + *vi t+ 1/2 a* t² r f =* * r*f =*ri + v*f t−1/ 2*a t² 2 v 2fxyz =vixyz + 2a xyz Δ r xyz

Projectile Motion

→ area

a-t

dx dy dz d *r ) =( , , dt dt dt dt

a*av = a= *

ri r* f −* Δx Δ y Δz =( , , ) Δt Δt Δt Δt

s-t slope ←

Path r i=0 Trajectory for * r i=0 *

Range for

*r (t )=* ri + * vi t+

1 2 * at 2

y=tan (Θ i) x−

gx 2 2( vi cos(Θi ))

2

2

Δ x=

vi sin (2Θ i) g

Circular motion Angular position

s θ= r

(Consider CCW of +x as positive)

Angular displacement*

Δ θ=θ f −θi

Average angular velocity

ωav =

Instantaneous velocity

ω=

Average angular acceleration

Conversion to linear entities

[rad]

θ f −θi Δt

[rad] [rad/s]

dθ (slope of angular position) dt ω f −ω ωi α av= [rad/s2] Δt dω dt

Instantaneous acceleration

α=

Five equations for CONSTANT angular acceleration

Δ ω=α t Δ θ=1/2(ω i+ ω f )t Δ θ=ω i t+ 1/ 2 αt² Δ θ=ω f t −1/ 2α t² 2 2 ω f =ω i + 2α Δ θ

Graphs

→ area slope ←

ϴ-t slope ←

Circumference:

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v=ω r

Tangential acc.

at =α r

Centripetal acc.

ac =

v 2 =ω 2 r r

a= a*t+ a*c *

2π r

Frequency: f [Hz]

f=1/T

(how many cycles per second)

Period: T *While angular velocity and angular acceleration are vectors, angular displacement Δ ϴ is not. (Vector addition does not work)

Speed

Uniform Circular Motion (α=0)

→ area ω-t

s=Δ θ r

Linear acceleration:

(slope of velocity)

α-t

Distance travelled

(time to go around once)

Speed v =

2πr =ωr T

Centripetal acc.

Velocity

*v=−v sin (θ(t )) 6i + v cos(θ( t )) j6

Acceleration

˙ *a=*v=− ac cos (θ( t )) 6i −ac sin(θ( t )) 6j

v2 ac = =ω 2 r r

θ( t )=θi + ω t

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Formulas and Constants for College Physics

Stefan Bracher

Classical (Galilean) Relativity For two inertial* reference frames A and B * + r*BA r*PA =r PB

Position of a point P: Velocity of a point P:

v*PA=v*PB+v*BA

Acceleration of a point P:

a*PA= a*BA

* Inertial Reference frame : A reference frame in which all laws of physics hold (typically a reference frame that itself is not accelerated)

Statics +Dynamics Linear

Rotational

Newton's 1st Law

F*net=0⇔ *a=0

τ*net=0⇔ α * =0

Torque

τ= r F sin (α )

Newton's 2nd Law

F*net =m*a

τ*net = I *α

Rotational Inertia in General

I =∫ x² dm

Newton's 3rd Law

F*AB =− F*BA

τ*AB = − τ*BA

Parallel axis theorem

I = I com+ m h

Mass - distance L from axis.

I =m L²

Beam, hinged at one side

1 I= mL² 3

Thin rod – perpendicular central axis

I=

Solid sphere– central axis

2 I = m r² 5

Spherical shell – central axis

2 I = m r² 3

Solid cylinder – central axis

1 I = m r² 2

m 1 m2

Universal Law of Gravity

F G =G

Hooke's Law

* F =−k *x

r

2

,

* *τ= *r × F

2

1 m L² 12

Work+Energy Conservation of Energy

E final= E initial + W by NCext

Potential gravitational Energy:

PE=mgh

Work done by a force

W= *F⋅*s =F s cos(Θ)

Potential spring Energy

PE=1/2kx²

Linear Kinetic Energy

KE= ½mv²

Rotational Kinetic Energy

KE= ½Iω²

Mechanical Energy

ME=PE+KE

NCext: Non Conservative External force

Power [Watt]

P=

ΔW * =F⋅*v Δt

Momentum Linear

Rotational

Momentum

*p=mv*

* L =*r ×* p =m (*r ×* v )= I ω *

Newton's 2nd Law

Δ *p F*net = Δt

τ*net =

Conservation of Momentum

p*f = *pi+ F*ext Δ t

L*f = *L i+ τ*ext Δ t

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Δ *L Δt

Impulse

* J =Δ p*

Inelastic collision :

Only momentum is conserved

Elastic Collision

Momentum and Mechanical energy is conserved

4

Formulas and Constants for College Physics

Stefan Bracher

Waves and Optics Simple Harmonic Motion Frequency f [Hz]

F=1/T=ω/2π

Position:

x ( t )= xm cos(ω t +Φ)

Angular Frequency ω [rad/s]

ω=2π/T=2πf

Velocity:

v (t )=

dx =−vm sin ( ω t +Φ) dt

Period T [s]

T=1/f=2π/ω

Acceleration:

a(t )=

dv =−a m cos(ω t +Φ ) dt

Mass and Spring

ω=

restoring force

√

k m

,

T =2 π

√

m k

Torsion Pendulum

√

I k

T =2 π

F= m a(t) = - m ω² x(t)

Kinetic / Potential Energy

2 1 K = m v ( t) 2

2 1 U = k x( t) 2

,

restoring force Simple Pendulum

τ=−κΘ

Physical Pendulum

τ=−L F g sin(Θ )=I α

restoring force

√

… for small angles

ω=

√

g L

, T =2 π L g

restoring force … for small angles

τ=−h c F g sin(Θ)= I α

√

ω=

mgh I

√

,

T =2 π

I mgh

Mechanical Waves Δ x=λ f = ω = λ k T Δt

Wavelength [m]

λ=

2π k

Wave speed [m/s]

v=

Angular wave number [1/m]

k=

2π λ

Superposition principle

y ( x , t )= y1 ( x , t )+ y 2( x ,t )

Soft reflection (free end, denser to less dense)

No phase shift

Hard reflection (fixed end, π less dens to denser medium

Transverse waves

f (x , t)=h (kx±ω t ) f (x , t )= y m sin( kx±ω t +Φ) +: moving left - : moving right

Longitudinal waves (Sound)

s( x , t )=sm cos(kx ±ω t+Φ) Δ p( x ,t)=Δ p m sin (kx±ω t+Φ)

Sound Level (Decibel) I 0=10−12 W /m ²

I β=10dB⋅log 10 ( ) I0

Particle velocity / acceleration:

u=

δ y ( x ,t ) δt

v = τμ

√

Wave speed:

,

a=

δu( x ,t) δt

τ=Tension, μ= linear density

Intensity I Wave speed:

I=

,

+1dB= x 10

β

P A

I =1010⋅I 0

√

B v= ρ

, B: Bulk modulus

−Δ p B= Δ V /V

...Interference identical waves, shifted by Φ(=Δx/λ), traveling in the same direction y '( x , t )=[2 ym cos( Φ )]sin (kx−ω t+ Φ ) 2 2 (for different amplitudes, use the Phasor-Diagram) Constructive:

ϕ = 2πm or Δx=mλ, m=0, 1, 2, ...

s '( x , t )=[2 sm cos( Φ )]cos( kx±ωt + Φ) 2 2 Deconstructive: Beats Doppler Effect (classic)

ϕ = πm or Δx=mλ/2, m= 1, 3, 5,... f beat = f 1−f 2

f '=f

v±v d v∓ vs

...Standing waves (identical waves traveling in opposite directions)

y '( x , t )=[ 2 ym sin (kx)] cos ( ωt ) nodes: no amplitude antinodes: max amplitude String, both ends fix

λ n=

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2L , n=1,2,3, 4... n

Pipe, both ends open

λ n=

2L , n=1, 2,3,4... n

Pipe, open-closed

λ n=

4L , n=1,3, 5. .. n

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Formulas and Constants for College Physics

Stefan Bracher

Geometrical Optics Snells Law (Law of refraction)

Law of reflection

n1 sin(Θ1 )=n 2 sin (Θ2 ) ,

n1 v2 λ2 = = n2 v1 λ1

Θi=Θr

Ray diagrams for curved mirrors: 1. 2. 3. f=radius/2

Any ray reflects according to the law of reflection at the tangent to the curve A ray parallel to the axis goes through the focal point. A ray through the focal point becomes parallel to the axis.

Thin lens equation: Diopters D = 1/f

h −q 1 1 1 1 1 = + =(n−1)( − ) , m= I = hO p r 1 r2 f p q p: Object distance q: Image distance (0: Real) f: focal distance (f0: Converging)

Ray diagrams for Lenses: 1. 2. 3.

A ray through the center goes straight through. A ray parallel to the axis goes through the focal point. A ray through the focal point becomes parallel to the axis.

Wave Optics Phase difference due to path traveled in different medium (number of wavelengths)

L N 2−N 1 = ( n2 −n 1 ) λ0

Double Slit Interference

Thin film interference

Double Slit Diffraction

Maxima:

d⋅sin (Θ)=m λ , m=0,1, 2...

Maxima

Minima

d⋅sin (Θ)=( m+0.5)λ , m=0, 1,2...

Minima

Diffraction by a circular aperture

sin(Θ )=1.22 λ d

Rayleigh's Criterion for resolvability

ΘR=arcsin (1.22 λ ) d

Diffraction Grating Maxima Lines

Complicated a⋅sin (Θ)=m λ , m=1,2, 3... Grating spacing: d=width/(# of gratings N) d⋅sin (Θ)=m λ , m=0, 1,2...

… First minimum

N d sin(Δ Θ)= λ

… Half line width

Θhw =

Resolving power

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Add phase shift due to reflection and path difference to determine constructive and destructive interference.

λ N d cos(Θ)

R= λ avg /Δ λ=Nm

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Formulas and Constants for College Physics

Stefan Bracher

Modern Physics Nuclear Physics A Z

X

r=r 0 A 1 /3

Radius of the nucleus

A: Mass number (Protons+Neutrons)

r0: 1.2 fm (1.2 x 10-15 m)

Z: Atomic number (charge of the nucleus) X: Chemical Symbol Nuclear Equation

Mass number and Charge is conserved (Mass is not)

Binding Energy:

E BE=Δ m⋅c²=[(Z⋅(m p+me )+ N⋅mn )−mtot ]⋅c²

Z: N: mtot: mp+me=mH-1: mn: c²:

Number of protons Number of neutrons Total mass of the isotope Mass of a proton + an electron ( 1.007 825 u) Mass of a neutron (1.008 665 u) 931.494013 MeV/u

Radiation / Decay Event in Nucleus α-Decay

− He

β-Decay

1 0

Positron-Decay

1 +1

He

1 +1

0 −1

n → p + e+ νD e 1 0

p → n+

γ-Decay

0 +1

e+ ν e

Becomes stable N (t )=N 0 e−λt

Number of radioactive nuclei:

Radiated particles 4 2

4 2

0 −1

e + νDe e + νe

0 +1

γ

0 0

R (t)=λ N (t )= R 0 e−λ t

Decay rate [Bq]:

Units of radiation Activity

Decay events per second

→ Becquerel [Bq] = disintegrations/ s

1 Curie = 3.7 x 1010 Becquerel

Absorbed Dose

Absorbed energy

→ Gray [Gy] = Joules / kg

1 Gray = 100 rad

Biological Damage

Effect on humans

→ Sievert [Sv] = Gray x Factor

1 Sievert = 100 rem

Quantum Physics Energy of a photon Momentum of a photon De Broglie Wavelength

E =h f p=

hf h = c λ

λ=

h p

Photoelectric Effect:

K max=e V stop , h V stop =( )f − Φ e e

Heisenbergs Uncertainty principle:

Δ x⋅Δ p x ⩾ℏ Δ y⋅Δ p y⩾ℏ Δ z⋅Δ p z ⩾ℏ

ℏ=h /( 2 π )

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h f =K max +Φ

,

7

Formulas and Constants for College Physics

Stefan Bracher

Special Relativity First postulate: The laws of physics are the same for observers in all inertial reference frames. No one frame is preferred over any other.

Simultaneity is relative, it depends on the motion of the observer

Second postulate: The speed of light in vacuum has the same value c in all directions and in all inertial reference frames.

Lorentz Factor:

Two observers in relative motion will in general not agree whether two events are simultaneous or not. γ=

1

√ 1−(v /c) ²

Proper time t 0

Time interval between two events occurring at the same location in an inertial reference frame.

Time dilation:

Δt =γ Δt 0

Proper length L0:

The distance between two points measured in an inertial reference frame at rest relative to both points.

Length contraction:

L L = γ0

Rest Mass

The mass of an object as measured in a reference frame not moving relative to the object (the only mass that is directly measurable)

Momentum:

p=m v⋅γ

Kinetic Energy

KE =γm c²−mc²

Rest Energy

E 0 =m c²

Lorentz Transformation A system x'y'z't' moving at a speed in x-direction relative to the system xyzt.

Velocities

u− v u' = uv 1− c²

Doppler Effect