Physics 121, Formula Sheet Exam # 3

- 1 -

Geometry/Trigonometry:

cos 30°( ) =1

23 sin 30°( ) =

1

2tan 30°( ) =

1

33

cos 45°( ) =1

22 sin 45°( ) =

1

22 tan 45°( ) = 1

cos 60°( ) =1

2sin 60°( ) =

1

23 tan 60°( ) = 3

cos1

2! "#$

%&'()= sin #( ) sin

1

2! "#$

%&'()= cos #( )

cos 2#( ) = 1" 2sin2 #( ) sin 2#( ) = 2sin #( )cos #( )

circle sphere

circumference 2!r

(surface) area !r2 4!r2

volume4

3!r

3

Integrating and Differentiating:

d(xn)

dx= nx

n!1

xndx" =

xn+1

n +1

Linear Motion in One Dimension (general):

v =dx

dt

a =dv

dt=d2x

dt2

Physics 121, Formula Sheet Exam # 3

- 2 -

Linear Motion in One Dimension (special case):

a(t) = a = constant

v(t) = v0+ at

x(t) = x0+ v

0t +

1

2at

2

Linear Motion in Two/Three Dimensions:

!v =

d!r

dt

!a =

d!v

dt=d2!r

dt2

Circular Motion:

aR=v2

r

atan

=dv

dt

Force Laws:

!Fi

i

! = m!a Newton's Second Law of Motion

!F

12= "!F

21Newton's Third Law of Motion

Friction:

Fs! µ

sN Static Friction

Fk= µ

kN Kinetic Friction

FD= "bv Dragg Force

Physics 121, Formula Sheet Exam # 3

- 3 -

Newton’s Gravitational Law:

!F12= !G

m1m2

r2

21

r21

Kepler’s Third Law (Law of Periods):

T1

T2

!

"#$

%&

2

=r1

r2

!

"#$

%&

3

Work Done by a Force:

W =!F •

!d Constant Force

W =!F • d

!l

a

b

! Variable Force

Translation Kinetic Energy:

K =1

2mv

2

Work-Energy Theorem:

W = !K Potential Energy:

!U =U2 "U1 = "!F • d!l

1

2

# = "W General Definition of Potential Energy

!F = "

dU

dxx "

dU

dyy "

dU

dzz

U(h) = mgh Gravitational Potential Energy (Close to the Surface)

U(r) = !GmME

rGravitational Potential Energy (r>rE )

U(x) =1

2kx

2 Spring with Spring Constant k

Physics 121, Formula Sheet Exam # 3

- 4 -

Conservation of Energy:

!U + !K = 0 Conservation of Mechanical Energy

!U + !K =WNC

Conservation of Energy

Power:

P =dW

dt

Linear Momentum and Newton’s Second Law:

!p = m

!v

d!p

dt=

!F!

Collision Impulse:

!J =

!Fdt

t1

t2

! =!pf "

!pi = #

!p

Elastic Collisions in One Dimension:

v1' = v

1

m1! m

2

m1+ m

2

"

#$%

&'+ v

2

2m2

m1+ m

2

"

#$%

&'

v2' = v

1

2m1

m1+ m

2

"

#$%

&'+ v

2

m1! m

2

m1+ m

2

"

#$%

&'

Center of Mass:

!rcm

=

mi

!ri

i

!

mi

i

!

!rcm

=1

M

!rdm"

Physics 121, Formula Sheet Exam # 3

- 5 -

Motion of the Center of Mass:

M!acm

=!Fi!

Rocket Equations:

Md!v

dt=

!Fext! +

!vrel

dM

dt

Marocket = RU0 First Rocket Equation

vf = vi + u lnMi

M f

"

#$

%

&' Second Rocket Equation

Angular Variables:

Definition Linear Variable

Angular Position ! l = R!

Angular Velocity " =d!

dtv = R"

Angular Acceleration # tan =d

2!

dt2

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 # = #0 +"0t +1

2!t

2x(t) = x0 + v0t +

1

2at

2

Physics 121, Formula Sheet Exam # 3

- 6 -

Moment of Inertia:

Moment of Inertia: I = mrri2

i

! (individual point masses)

I = r2dm

Voume

" (continuous mass distribution)

Parallel-axis Theorem I = Icm + Mh2

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:

Kinetic energy: K =1

2I! 2

Work: W = "d#$

Power: P = "!

Work-Energy Theorem: W = %K =1

2I! f

2 &1

2I! i

2

Precession:

! =Mgr

cm

L

Physics 121, Formula Sheet Exam # 3

- 7 -

Moments of inertia of various objects of uniform composition.

Physics 121, Formula Sheet Exam # 3

- 8 -

Conditions for Equilibrium:

Fx! = 0 "

x! = 0

Fy! = 0 "

y! = 0

Fz! = 0 "

z! = 0

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:

Definition: x(t) = Acos(!t + ")

A = Amplitude

! = angular frequency

" = phase

Force Requirement: F(x) = #m! 2x

Period: T = 2$ /!

Frequency: f = 1 /T =! / 2$

The Physical Pendulum:

T = 2!I

mgh

Physics 121, Formula Sheet Exam # 3

- 9 -

Damped Harmonic Motion (damping force = -bv):

Solution: x(t) = Ae!" t cos(#t)

A = Amplitude

# =k

m!

b2

4m2

" =b

2m

Forced Harmonic Motion (external force Fext = F0 cos(ωt) and damping force = -bv):

Solution: x(t) = A0cos(!t + "

0)

A0=

F0

m ! 2 #!0

2( )2

+b2! 2

m2

"0= tan

#1 !0

2 #! 2

!b

m

$%&

'()

$

%

&&&

'

(

)))

!0=

k

m