Physics 121, Formula Sheet

Final Exam

Geometry/Trigonometry:

1 3 sin ( 30° ) = 2 1 cos ( 45° ) = 2 sin ( 45° ) = 2 1 cos ( 60° ) = sin ( 60° ) = 2 cos ( 30° ) =

1 2 1 2 2 1 3 2

tan ( 30° ) =

1 3 3

tan ( 45° ) = 1 tan ( 60° ) = 3

$1 ' cos & ! " # ) = sin (# ) %2 (

$1 ' sin & ! " # ) = cos (# ) %2 (

cos ( 2# ) = 1 " 2 sin 2 (# )

sin ( 2# ) = 2 sin (# ) cos (# )

circle circumference

2! r

(surface) area

!r2

volume

sphere

4! r 2 4 3 !r 3

Integrating and Differentiating:

d(x n ) = nx n !1 dx x n +1 n x dx = " n +1 Linear Motion in One Dimension (general):

dx dt dv d 2 x a= = dt dt 2 v=

- 1 -

Physics 121, Formula Sheet

Final Exam

Linear Motion in One Dimension (special case):

a(t) = a = constant v(t) = v0 + at x(t) = x0 + v0t +

1 2 at 2

Linear Motion in Two/Three Dimensions:

! ! dr v= dt ! ! ! dv d 2 r a= = dt dt 2 Circular Motion:

v2 aR = r dv atan = dt Force Laws:

!

!

! F = ma

Newton's Second Law of Motion

! ! F12 = " F21

Newton's Third Law of Motion

i

i

Friction:

Fs ! µ s N Fk = µ k N FD = "bv

Static Friction Kinetic Friction Dragg Force

- 2 -

Physics 121, Formula Sheet

Final Exam

Newton’s Gravitational Law:

! mm F12 = !G 12 2 rˆ21 r 21

Kepler’s Third Law (Law of Periods): 2

! T1 $ ! r1 $ #" T &% = #" r &% 2 2

3

Work Done by a Force:

! ! W = F•d ! b ! W = ! F • dl a

Constant Force Variable Force

Translation Kinetic Energy: K=

1 2 mv 2

Work-Energy Theorem: W = !K

Potential Energy: ! 2 ! !U = U 2 " U1 = " # F • dl = "W 1

General Definition of Potential Energy

! dU dU dU F=" xˆ " yˆ " zˆ dx dy dz U(h) = mgh Gravitational Potential Energy (Close to the Surface)

U(r) = !G U(x) =

mM E r

1 2 kx 2

Gravitational Potential Energy (r>rE ) Spring with Spring Constant k

- 3 -

Physics 121, Formula Sheet

Final Exam

Conservation of Energy:

!U + !K = 0 !U + !K = WNC

Conservation of Mechanical Energy Conservation of Energy

Power: P=

dW dt

Linear Momentum and Newton’s Second Law:

! ! p = mv ! ! dp = !F dt Collision Impulse:

! J=

!

t2

t1

! ! ! ! Fdt = p f " pi = #p

Elastic Collisions in One Dimension:

" m ! m2 % " 2m2 % v1 ' = v1 $ 1 + v2 $ ' # m1 + m2 & # m1 + m2 '& " 2m1 % " m ! m2 % v2 ' = v1 $ + v2 $ 1 ' # m1 + m2 & # m1 + m2 '& Center of Mass:

!

! rcm =

!m r

i i

i

!m

i

i

1 ! rcm = M

"

! rdm

- 4 -

Physics 121, Formula Sheet

Final Exam

Motion of the Center of Mass:

! ! Macm = ! Fi Rocket Equations:

! ! dv ! dM M = ! Fext + vrel dt dt Marocket = RU 0 "M % v f = vi + u ln $ i ' # Mf &

First Rocket Equation Second Rocket Equation

Angular Variables:

Definition Linear Variable Angular Position

!

Angular Velocity

"=

Angular Acceleration # tan

d! dt d 2! = 2 dt

l = R! v = R" atan = R# tan

Equations of Motion for Constant Acceleration:

Rotationional Motion

Linear Motion

Acceleration

! (t) = !

a(t) = a

Velocity

" (t) = " 0 + ! t

v(t) = v0 + at

Position

1 # = # 0 + " 0t + ! t 2 2

x(t) = x0 + v0t +

- 5 -

1 2 at 2

Physics 121, Formula Sheet

Final Exam

Moment of Inertia:

I = ! mr ri 2 (individual point masses)

Moment of Inertia:

i

I=

"

r 2 dm (continuous mass distribution)

Voume

Parallel-axis Theorem

I = I cm + Mh 2

Perpendicular-axis Theorem

Iz = Ix + Iy

Torque:

Definition: ! =r "F Newton's Second Law for Rotational Motion: ! = I# Angular Momentum:

Definition:

L=r!p

Rotating rigid object:

L = I"

Relation between torque and angular momentum:

dL = $# dt

Rotational Energy: 1 2 I! 2

Kinetic energy:

K=

Work:

W = $ " d#

Power:

P = "!

Work-Energy Theorem: W = %K =

1 1 I! f 2 & I! i 2 2 2

Precession: !=

Mgrcm L

- 6 -

Physics 121, Formula Sheet

Final Exam

Moments of inertia of various objects of uniform composition.

- 7 -

Physics 121, Formula Sheet

Final Exam

Conditions for Equilibrium:

!F

=0

!"

x

=0

!F

=0

!"

y

=0

!F

=0

!"

z

=0

x

y

z

Hooke’s Law: F = k!L

Stress: stress = force/area = F/A Strain: strain = change in length / original length = ΔL/L0 Young’s Modulus E: E = stress/strain Simple Harmonic Motion:

x(t) = A cos(! t + " ) A = Amplitude ! = angular frequency " = phase Force Requirement: F(x) = #m! 2 x Period: T = 2$ / ! Frequency: f = 1 / T = ! / 2$ Definition:

The Physical Pendulum: T = 2!

I mgh

- 8 -

Physics 121, Formula Sheet

Final Exam

Damped Harmonic Motion (damping force = -bv):

Solution: x(t) = Ae!" t cos(# t) A = Amplitude k b2 #= ! m 4m 2 b "= 2m Forced Harmonic Motion (external force Fext = F0 cos(ω t) and damping force = -bv):

Solution: x(t) = A0 cos(! t + "0 ) F0 A0 = m

(!

2

# ! 02

)

2

+

b 2! 2 m2

$ ' 2 2) & ! #! "0 = tan #1 & 0 ) & ! $& b ') ) % % m( (

!0 =

k m

Thermal Expansion:

!L = " L0 !T

Linear Expansion

!V = #V0 !T # $ 3"

Volume Expansion

Ideal Gas Law:

PV = nRT PV = NkT Average Translational Kinetic Energy for an Ideal Gas: K=

3 kT 2

- 9 -

Physics 121, Formula Sheet

Final Exam

The Maxwell Distribution: 3

2

1 mv " m % 2 2 ( 2 kT f ( v ) = 4! N $ v e # 2! kT '&

Mean free path in a gas:

lM =

1 4! r ( N / V ) 2

Specific Heat c:

Q = mc!T Molar Specific Heats for Gases: Q = nCV !T

Constant Volume

Q = nCP !T

Constant Pressure

CP " CV = R CV =

3 R 2

Ideal Monatomic Gas

Latent Heat L:

Q = mL First Law of Thermodynamics:

!U = Q " W Adiabatic Expansion of a Gas:

PV ! = constant

- 10 -

Physics 121, Formula Sheet

Final Exam

Work Done during Volume Changes of an Ideal Gas:

W = nRT ln

VB VA

" V % W = nRTB $ 1 ! A ' # VB &

Isothermal Process Isobaric Process

Heat Transfer: !Q T " T2 = kA 1 !t l

Efficiency of a Heat Engine: e=

W QH

Coefficient of Performance of Refrigerators and Air Conditioners: QL W

CP =

Coefficient of Performance of Heat Pumps: QH W

CP =

Carnot Efficiency:

e = 1!

TL TH

Entropy: dS =

dQ T

- 11 -