Formula Sheet For Physics Halliday

 

Formula Sheet: Physics 121

A. Carmichael

Position, velocity and acceleration x/∆ ∆x/ ∆t  = v  =  v av   Area under v under  v((t) = ∆x  = s  =  s   Area under a under  a((t) = ∆v  

v/∆t  = a ∆v/∆  =  a av  x (t) =  v(  v (t) slope of  x(  v (t) =  a(  a(t) slope of  v(

Uniformly accelerated linear motion  motion   (a=const.) v  = u  =  u + at v 2 =  u 2 + 2as 2as

s  = ut  =  ut +   21 at2 s  =   u + v  t 2

Friction (kinetic) Weight

−x )

 

0

x  = x  =  x0  +

v0 t +   21 at2

x  = x  =  x0  +

  v0  + v 2

F nnet  =  ma et  = ma f s f s,max  = µ  =  µ s n

 

 ≤

f k  =  µ k n w  = mg  =  mg

 

 

t

ay   = g  = const. const. vy (t) =  v y (0) gt

−

y (t) =  y(0)  y (0) + vy (0)t (0)t

−

 1 gt 2 2

Orbitall motion Orbita

x(t) =  x(0)  x (0) + vx (0)t (0)t

Kepler’s 2nd Law (K2) Orbit (circular) Escape velocity

−

− −

 1 gt 2 2

v =  GM/r v 2 = 2GM/r

   

ax  = 0 = const. const. vx  = v  =  v(0)cos (0)cos θ  = const. const.

Constants related to gravity

x  = x  =  x(0) (0) + v (0)t (0)t cos θ

Universal const. of gravitation   G = 6.67  

Earth surface gravity v 2 (y) =  v 2 (0)

− 2g(y − y ) vy (y) =  v y (0) − 2g (y − y ) 2

2

0

Earth mass & G &  G

0

Solar mass & G &  G

Trajectory and velocity equations for x for  x(0) (0) = y =  y(0) (0) = 0

−

 R,, height h Range R Range height  h,, flight time T  time  T  2

2

2

h  = v  =  v y (0)/ (0)/2g

h  = v θ/2g  =  v (0)sin θ/2

/g R  = 2vx (0) (0)/g (0)vvy (0) T  T    = 2v (0)/g 2 vy (0)/g

R  = v θ/g  =  v 2 (0)sin2 (0)sin2θ/g T   T   = 2v (0)sin θ/g

h  =

 R   tan θ 4

Circular Circul ar motion  

Centripetal acceleration    

GM E  3 .98 E   = 3.

N m2 /kg2

·

14

m3 /s2

20

m3 /s2

12

m3 /s2

× 10 GM   = 1.33 × 10 GM   = 4.91 × 10

Moon mass & G &  G

  Kinetic energy   Work done, const. force   Power   Instantaneous Power Power   Work-energy theorem Work done   b y con. forces     Mechanical energy

Spring (strain) P.E.

s  = rθ  =  rθ v  = rω  =  rω  = 2πr/T  at  =  rα

Mechanical energy

 

2

K  =  = 2 mv W   =  F s cos ϕ  = F   =  F  s P  P    = ∆E/ E/∆ ∆t  = W/  =  W/∆ ∆t P  P    =  F v cos ϕ  = F   =  F  v W nnet et  =  W c  + W nc nc  = ∆K  W c  = ∆U  E mech  =  K  +  + U  mech  = K   =  K f  Conservation of mech. energy   K i  + U i  + W nc nc  = K  f  f  + U f  Work done by non-con. forces   W nc nc  = ∆E mech mech   U  U ((y ) =  mgy  mgy + U (0) GPE uniform field  + U  (0) GPE uniform field ∆U grav  =  mgh  = mg  =  mg∆ ∆y grav.  = mgh

ar  =  v 2 /r = /r  = rω  rω 2

version: versi on: Monday Monday 6 th Mar March, ch, 2017 14:07

11

g  = 9.81 m/s2

 1



 − 

× 10−

Work and energy

vy (0) 1 g x2 x vx (0) 2 vx2 (0) g 1 x2 y (x) =  x tan θ 2 2 2 v (0)cos θ y (x) =

  Arc length Tangential speed Tangetial acceleration

π2 /GM )r3 T 2 = (4 (4π

 

2

ay   = g  = const. const.  v (0)sin θ gt vy  =  v(0)sin y  = y  =  y(0) (0) + v (0) (0)tt sin θ

g  = Gm/r  =  Gm/r2 U   = Gmm /r V  V    = Gm/r

Gravity field of mass m Gravity mass m   G.P.E. two masses m Grav. potential of   m

ax  = 0 = const. const. vx (t) =  v x (0) = const. const.

v (0) and θ In terms of   v(0) and  θ

−

F   = Gmm /r2

 

Force between masses

Projectile motion 2D 2D (uniform  (uniform field g field  g =const.)

−

 

Grav. fields due to point or spherical sources

v  = v  =  v 0  + at

−

ω  = 2πf   πf   = 2π/T  f   = 1/T  at  = 0, α  = 0

Forces Newton’s 2nd Law Newton’s Friction (static)

Alternative form

v 2 =  v 02 + 2a 2 a(x

  Angular frequency Frequency and time period     Uniform circular motion

Page 1

   

·

·

−

2  1 U spring spring  = 2 kx

E mech  + U total mech.  =  K  + total

CalPoly Department of Physics

 

Formula Sheet: Physics 121

A. Carmichael

Substitutions for rotational dynamics

Momentum  

Linear Momentum

 p =  p  = m  mv  aavv  = ∆ p/∆ F   p/   ∆t

 

Newton’s second law Impulse for constant force

 

Impulse for variable force

 

s  = u  =

 

⇒ ∆θ ⇒ω v  =⇒ ω a  =⇒ α

   = ∆ p =  ∆t J   p  =  F ∆ F     = ∆ p =  aavv ∆t J   p  =  F 

   = F  m  =

⇒  Γ ⇒ I 

 

0

K  =  =   21 mv2 = K r   =   12 I ω2  p  p =   = m  mv  =    L  = I    =  I ω

⇒

⇒

Moments of inertia Theorems for variable forces    = ∆ p =  p   = Area under F  under  F nnet Impulse-momentum   J  et (t)  

Work-energy

W nnet  F nnet Areaa under under F  et  = ∆K   = Are et, (s)

Moment I  =  =  M R2

Ob ject Unif Un ifor orm m ri ring ng/t /tube ube

Axis Thro Throug ugh h cent centre re

I  =  =   21 M R2

Uniform Unifor m disk/c disk/cylinde ylinderr

Through Through centre centre

Uniform rod

Through centre

Uniform rod

Through end

Uniform sphere

Through centre

Hollow sphere

Through centre

I  =  =

Centre of mass x-coordinate

 

y -coordinate

 

I  =  =

  m1 x1  + m2 x2  + ... xcm  = m1  + m2  + ...   m1 y1  + m2 y2  + ... ycm  = m1  + m2  + ...

I  =  = I  =  = I  =  =

Types of collision

 1 M L2 12  1 M L2 3  2 M R2 5  2 M R2 3  1 M a2 3

ω  = ω  =  ω0  + αt

•  inelastic: Some loss of K.E., 0 < 0  < e <  1 •   completely inelastic:   v  =  v  =  v,  v ,  e =  e  = 0 Max K.E. loss  = v 1

2

Collision conservation laws (1D & 2D)  

K.E. (elastic only)

m1 u1  + m2 u2  = m  =  m1v1  + m2v2   1 m1 u21  +   21 m2 u22  =   12 m1 v12 +   21 m2 v22 2

Newton’s collision law (1D only) Newton New ton’s ’s col collisi lision on la law w (1D) (1D)

  m1

− em

(v2

− v ) = −e(u − u ) 1

2

u1

 

v2  =

2

  (1 + e)m1

u1

m1  + m2

m1  + m2 loss of K.E.

Along edge (door)

∆K  = (1 K i

 

− e ) m  m + m

1

∆θ  = ω  =  ω0 t +   21 αt2

 

ω 2 =  ω02 + 2α 2 α∆θ

Hooke’s Law

 

Acceleration

 

Velocity

F  F ((x) =

−kx a(x) = −ω x  = −n x v (x) = ±ω A − x 2

 

Total energy  x((t) Position x Position  v((t) Velocity v Velocity

2

E  =  =   12 kA 2 =   12 mω2 A2 x(t) =  A sin( ωt + ϕ) sin(ωt v (t) =  Aω cos( ωt + ϕ) cos(ωt

       

a(t) =  

Period, mass-spring

ωt + ϕ) Aω2 sin( sin(ωt

  1   2π T   T   =   =   = 2π ω f 

−

T   = Period, Peri od, simple pendulum   T  

  1   2π   =   = 2π f  n

K r  =   12 I ω2

T   = Period, physical pendulum   T  

  1   2π   =   = 2π f  n

  Moment of inertia Magnitude of torque Work done by a torque   Rotational power   N2 for rotation

I  =  = Σ mr2  rF  sin Γ =  rF   sin ϕ  = rF   =  rF ⊥ W   = Γ ∆θ  = ∆K r

Elasticity

P   P   = Γω Γω Γ =  I α

Hooke’s law (cables) Tensile stress Tensile strain

  Angular momentum   Conservation of   L Rolling without slipping  

L  = I  =  I ω I i ωi  =  I ff  ωf  vcm  =  Rω, acm  = Rα  =  Rα

Young’s modulus Strain energy

1

2

Rotational motion Rotational K.E.

 

·

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Page 2

 

   

2

U  U ((x) =   21 kx2

 

 a((t) Acceleration a Acceleration

2

 

2

2

  t ∆θ  =   ω0  + ω 2

 

Simple harmonic motion (SHM)

SPE for a spring

1D collision, stationary target (u (u2  = 0) v1  =

 

Rotational motion with ( with  (α α  = con const. st.))

•  totally elastic: No loss of K.E. ,  e  = 1

Momentum

Slab width a width  a

m k

     

l g   I  mgr

F   = (Y δl  = k  k δl = δl  = λδl/l  λδl/l ( Y A/l) δl = stress =  F /A strain =  δl/l

·

·

Y  Y  =  = stress/ stress/strain W   =  U   U    =   21 F  δl = δl  =   12 kx 2

 · ·

CalPoly Department of Physics

 

Formula Sheet: Physics 121

A. Carmichael

Inverse trig functions where  where   α  = principal value

Trigonometry sin θ  =

  opp. opp.   adj. adj.   opp opp..   sin θ   cos θ  =   tan θ  =  = hyp.. hyp hyp.. hyp adj.. adj cos θ

θ(rad)   2π   π  π   2π  =   =   1 rpm rp m = rad/s = rad/s θ(deg) 360o 180o 60 30

cos θ  = cos α sin θ  = sin α tan θ  = tan α

(1 + x)n = 1 + nx +

π/6) π/33 = sin(2π/ 6) = sin π/ sin(2π/3) 3) = 3/2 cos( π/ π/3) π/66 = sin(5π/ cos( π/ 3) = sin π/ sin(5π/6) 6) = 1/ 1 /2

2

2

a =  b + c 2bc cos A a   b   c  =  = sin A sin B sin C 

−

sin(θ sin(θ cos(θθ cos( sin(θ sin(θ cos(θθ cos( π sin(π sin( π cos(π cos(

Law of cosines cosines

 

Law of sines

sin θ + cos θ   = 1 sin2θθ  = 2 sin sin2 sin θ cos θ cos2θθ  = cos2 θ cos2

2

± ±

± ∓

± ±

cos(ωt cos(ωt

π) =

±

n

C r  =

 n integer n integer

  n! r !( n r)! !(n

n

−

P r  =

  n! (n r )!

−

√ b − 4ac x  = −  ± 2a x  = −

 

max, min at

2

 b 2a b/ b/22a

 

Roots at

− θ∓φ

±     −     θ

φ

 

− y  = m  =  m((x − x ) y −y y − y  = x −x y

1

1

1



2

1

2

1



(x

−x ) 1

Exponential behaviour y (t) =  y0 e−t/τ  =  y0 e−λt T 1/2  = τ   =  τ   ln 2

Exponential decay Half life

−  e−t/τ 

y (t) =  ymax 1

 

Exponential growth

Percent difference between quantities   A,   B % diff (A, (A, B ) =

  |A − B|  × 100 = |A − B|  × 200 av(A, B ) av(A,

A+B

Percent error % err error or =

 |measured − true   | × 100 true

Mathematical constants e  = 2.71828 ... 71828...

 

π  = 3.14159 14159... ...

 

... 434... log10 e  = 0.434 ... ln10 = 2. 2.3025 3025... ln2 = 0. 0.693 693....

− − −

 1 (in−radians) cos θ ≈ 1 − θ /2 tan θ ≈ θ

1o = 1.745

× 10− rad 1 = 2.9089 × 10− rad 1 = 4.8481 × 10− rad

   

6

1 ra rad d = 57. 57.296o π/6 ra rad d = 30o π/3 ra rad d = 60o

− e− ) = 0.632 632.... √ 3√ /2 = 0.866 ... 866...  

π/4 ra rad d = 45o

1

... 707... 1/ 2 = 0.707

2

4

e−1 = 0.368 368.... (1

Small angle formulae for small   θ  

 | | |  1

Quadratic equation  equation   y  = ax  =  ax2 + bx + c

2cos θ cos ϕ  = cos(θ cos(θ ϕ) + cos(θ cos(θ + ϕ) 2sin θ sin ϕ  = cos(θ cos(θ ϕ) cos( cos(θθ + ϕ) 2sin θ cos ϕ  = sin(θ sin(θ + ϕ) + sin(θ sin(θ ϕ)

≈θ

+ ...   if  x

Cominatorics

Product to sum

sin θ

2

C r an−r br

cos ωt

cos 2 2 θ φ θ + φ cos cos θ  + cos φ  = 2 cos cos 2 2 θ + φ θ + φ sin cos θ cos φ = 2sin 2 2

−

− 1) x 1)x

± −

r=0

∓

± π/2) π/2) = ± cos ωt ± π/ π/2) 2) = ∓ sin ωt sin(ωt ± π) = − sin ωt sin(ωt

Sum to product

−

 1 n(n 2!

Given (x (x1 , y1 ), (x2 , y2 )  

sin(ωt cos(ωt cos(ωt

θ  

± sin φ = 2 sisinn

n n



 m,, (x1 , y1 ) Given m Given

π/2) = cos θ sin(θθ π/2) sin( π/2) cos(θ π/ cos(θ 2) = sin θ sin(π/22 θ ) = cos θ sin(π/ cos(π/22 θ) = sin θ cos(π/

       

 

− sin

θ  = α + 2nπ 2nπ θ  = ( 1)n α + nπ θ  = α  =  α +  + nπ

Linear Equation  Equation   y  = mx  =  mx + b

2

sin θ

(a + b)n =

±± φφ)) == sin θ cos φ ± cos θ sin φ cos θ cos φ ∓ sin θ sin φ

± π) = − sin θ ± π) = − cos θ ± θ) = ∓ sin θ ± θ) = − cos θ

2

 

     

⇒ ⇒ ⇒

Binomial formulae

√ 

±± √  π/4) π/44 = sin(3π/ cos(±π/ 4) = sin π/ sin(3π/4) 4) = 1/ 1/ 2 √  π/6) = sin(−π/3) π/3) = sin(−2π/3) π/3) = − 3/2 cos(±5π/6) π/3) = sin(−π/6) π/6) = sin(−5π/6) π/6) = −1/2 cos(±2π/3) √  π/4) = sin(−π/4) π/4) = sin(−3π/4) π/4) = −1/ 2 cos(±3π/4)

= = =

     

 

1 rp rpm m = 0.104 10477 rad/ rad/s 1rad//s = 9. 1rad 9 .549 rpm

2

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Page 3

CalPoly Department of Physics

 

Formula Sheet: Physics 121

A. Carmichael

Let Lette terr Alpha Beta Gamma Delta Epsilon Zeta Eta Theta

Uppe Upperr A B Γ ∆ E Z H Θ

cas asee                

Lower ca case se α β  γ  δ  ,  ε ζ  η θ

IKoatappa Lambd bdaa Mu Nu Xi Omic Om icrron Pi Rho Sigma Tau Upsilon Phi Chi Psi

I K Λ M N Ξ O Π P Σ T Y Φ X Ψ

                            

ι κ λ µ ν  ξ  o π ρ σ τ  υ φ,  ϕ χ ψ

Omega

Ω

 

ω

Abbreviations used: SHM = Simple Harmonic Motion GPE = Gravitational Potential Energy SPE = Strain (Spring) Potential Energy EMF = ElectroMotive Force (voltage) EM or E&M = ElectroMagnetism PE = Potentia Potentiall Energy Energy PD = Potential Difference AC = Alternating Current DC = Direct Current (or Detective Comics) con. = conservative (force) STP = Standard Temperature and Pressure (20o C, 1 atm) N1,N2,N3= N1,N2,N 3= Newton’s Newton’s laws of motion motion T0,T1,T2,T3= the laws of thermal physics K1,K2,K3= K1,K2,K 3= Kepler’s Kepler’s laws of planetary planetary motion

Metric Prefixes tera

T

1012

giga

G

109

mega

M

106

kilo

k

103

hecto

h

102

deci

d

10−1

centi

c

10−2

milli

m

10−3

 

micro nano pico

µ n

 

10−6 10−9 10−12

p

SI units and derived units Quantity Mass   Length   Time   Force

   

Energy Power

 

Sy Symbol m   l   t  

Un U ni t kg m s

Name kilogram meter second

Basic Units kg m s

F 

 

N

Newton

kg ms−2

E 

 

J

Joule

k kgg m2 s−2

P 

 

W = Js−1

Watt

kg m2 s−3

 

Pa = N.m2

Pascal

kg/ms2

Pressure   p

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Page 4

CalPoly Department of Physics

 

Formula Sheet: Physics 121

A. Carmichael

Unit Conversions Quantity Units Length in inch, cm Length foot, cm Length

mile, km

Conversion or value 1 in.  = 2.54cm 1 ft = 30.48cm 1 mile = 1.609 609 km

Energy Energy

electron-volt, Joule calorie, Joule

1 eV eV = 1.602 10−19 J 1 ca cal = 4.1868J 1868J

En Ener ergy gy Energy Energy Power

Brit Britis ish h th ther erma mall unit unit,, Jo Joul ulee foot-pound, Joule kilowatt-hour, Joule horsepower, Watt

1 Btu Btu = 1055J 1055J 1 ft ft lb = 1. 1.356J 1 k W h = 3.600 600 MJ 1 hp h p = 746 W Waatt

Mass Force

atomic unit, kg pound, Newton

1 u = 1.6605 10−27 kg 1 lb lb = 4.442N

Density

g/cm3

Pressure Pressure

Pascal, psi atmosphere, Pascal

/m2 = 1.450 10−4 psi 1 Pa Pa = 1 N N/ 1atm 1atm = 101, 325Pa = 760Torr 760Torr = 14. 14.7psi

Pressure

psi, Pascal

1 ps psi = 6.895

Pressure

mm Hg

1 torr = 1 mm Hg = 0.0394inHg = 1. 1.333

→ kg kg//m

×

·

·

×

3

1 g/cm3 = 1000 1000 kg/ kg/m3

×

3

× 10

Pa

5

Pressure Volume Volume Volume Angle

bar lilitre quart (US) gallon (US) rev, rad rad,, deg

2

× 10

Pa

1 bar = 10 Pa 1 l = 103 cm3 = 10 −3 m3 = 1.057 057 qt (U (US) S) 1 qt qt (U (US) = 946 m mll 1 gal.(US) = 3. 3.758l o 1 rev = 360 = 2π 2 π rad

Astrophysical Data

Body Bod y surf surfac ace e   g   Mass 2

(m/ (m/s ) Sun Earth Moon

  −− 9.81

1.62

 

3

1.99 5.97

× 10 × 10 7.36 × 10

2

(m /s )

kg  

 

GM 

30 24 22

1.33 3.98

× 10 × 10 4.91 × 10

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20 14 12

Radiu adiuss

Orbit rbit Radiu adiuss

Orbit rbit Perio eriod d

m

m

Earth years

6.96 6.37

8

× 10 − − × 10 1.50 × 10 −− 1.74 × 10 6

11

6

 

Page 5

−−

1.00

 

−−

 

Sym Symbol bol

 ♁



CalPoly Department of Physics

 

Formula Sheet: Physics 121

A. Carmichael

Symbols used in mechanics:

Andrew Carmichael California Polytechnic University San Luis Obispo Monday 6th March, 2017 Updates on my profile at:

A   Amplitude Amplitude for SHM A,  A 1 ,  A 2   Cross sectional sectional area area of of pipe a   Acceleration at   Tangential component of acceleration ar   Radial component of acceleration e   Coefficient of resitution E    Total energy F , F ,  F aavv   Force, aver average age force f    Frequency (rev/second or cycles/second) f    Friction (force) G   Universal gravitation constant g   Gravitational field strength h   depth or height height I    Moment of inertia    = ∆ p)    p ) J    Impulse (change in momentum  J  K    Kinetic energy K r   Rotational kinetic energy k   Spring constant k   wavenumber 2π/λ   L   Angular momentum l   Length M , M ,  m   Mass n P  P aavv  p r s T  T  U  u v W  W c W nc nc W nnet et Y  α ∆ µk µs ω ω0 ∆θ θ0 Γ ρ

                                              

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Normal Power force Average power Momentum radius Displacement Time period/ time of fligh flightt tension Potential Potential energy velocity at time t time  t =  = 0 velocity at time t time  t Work Work done by a con. force(s) force(s) Work done by non-con. force(s) Work done by net force Young’s modulus /s2 ) Angular acceleration (rad (rad/ change in... Coefficient of kinetic friction Coefficient of static friction  t  (rad /s) Angular Angular speed at time time t  (rad/  t = Angular Angular speed at time time t  = 0 (rad/ (rad/s) angular displacement ∆θ  = θ  =  θ θ0  t = Angular Angular position position at time time t  = 0 Torque density (mass/volume)

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CalPoly Department of Physics