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Density: p 
m
V
Average speed: v 
Average velocity: v x 
x
t
Average acceleration: a x 
d
t
Instantaneous velocity: v x 
v x
t
dx
dt
Instantaneous acceleration: a x 
dv x
dt
4 equations of motion:
x  xo 
v  vo  a x t
x  xo  vo t 
1 2
at
2
Projectile motion:
1
(v  vo )t
2
2a( x  xo )  v 2  vo
2
v xi  vi cos  i
v yi  vi sin  i
ac 
Centripetal acceleration:
at 
Tangential acceleration:
v2
r

dv
dt
  
a  a r  at
Total acceleration:
Period:
T
2r
v
Radial acceleration:
 v2
a r  a c 
r
Relative velocity:
v po  v po  voo
Newton’s Laws
1st Law:
-Object at rest will remain at rest
-Object in motion remains in motion (law if inertia)
2nd Law:
-Acceleration of an object is directly proportional to the net force acting on it and
F
inversely proportional to its mass a
m
rd
3 Law:
-If two objects interact, the force F12 exerted by object 1 on object 2 is equal in magnitude


but opposite in direction to the force F21 exerted by object 2 on object 1 F12   F21
Net Force:


 F  ma
Force of static friction:
Resistive force


Low speed:
R  bv
f s  s n
Force of kinetic friction:
High speed:


R  cv 2
f k  k n
Differential form:
Terminal speed:
Law of gravitation:
dv
b
g vv
dt
m
mg
mg
vT 


b
c
mg
(1  e bt / m )
b
2mg
DA
m1m2
r2
Fg  G
Coulomb’s law:
xf 
Work: W  Fd cos    Fx  dx
Work & KE:
KE 
1 2
mv
2
W  KE f  KEi  KE
KE   f k x   Wotherforces
Instantaneous Power: P 
Average power:
P
Conservation of energy:
Mechanical energy:
where Eint  f x x
 
dE dW

 F v
dt
dt
W
t
Gravitational potential energy:
Electric potential energy:
q1q2
r2
Hooke’s law: Fs  kx
xi
Kinetic energy:
Fe  k e
U g  mgy  G
m1m2
r
q1q2
r
KEi  U i  KE f  U f
U e  ke
Emech  KE  U
General conservation of energy:
Elastic potential:
Us 
1 2
kx
2
KE  U  Eint  cons tan t
xf
Potential energy from force: U f    Fx dx  U i
xi
du
Force from potential energy: Fx  
dx


p  mv
Linear Momentum:

ptot  cons tan t
Net Force:
 dp
F
  dt
Impulse:

tf
I 


F
 dt  p

tf
I 


F
 ext dt  ptot
ti
ti
Inelastic collision: (KE is not conserved)
Perfect Inelastic: (Momentum is conserved & stick together)
m1v1i  m2 v 2i  (m1  m2 )v f
Elastic collision: (Momentum & KE conserved)
m1v1i  m2 v 2i  m1v1 f  m2 v 2 f
1
1
1
1
2
2
2
2
m1v1i  m2 v 2i  m1v1 f  m2 v 2 f
2
2
2
2
Center of mass:
m x  m2 x 2
xcm  1 1
m1  m2

i mi ri
rcm 
M
CM of extended objects:

1 
rcm 
r  dm
M

drcm

1
vcm 

dt
M
Velocity of CM:

dri
1
i mi dt  M
i
dv
1

Acceleration of CM: acm  cm 
dt
M
i
i

dvi
1
i mi dt  M
Rocket Propulsion:
m
v f  vi  ve ln  i
m
 f
Rocket Thrust:
Thrust  ma  m

i i




Mvcm   mi vi   pi  ptot
Momentum of system:
Radian:

m v
s
 t
r
Average angular speed:
i
i




dv
dM
 ve
dt
dt
Arc length:
 

m a
 f  i
t f  ti


t
s  r
i
Instantaneous angular speed:  
average angular acceleration:  
d
dt
 f  i
t f  ti
Instantaneous angular acceleration:  


t
d
dt
Rotational Motion (fixed axis/constant ):
1
2
  o  t
  o 
   o   o t  t 2
2    o    2  o
1
   o t
2
Tangential velocity:
v
ds
d
r
 r
dt
dt
Tangential acceleration:
at 
dv
d
r
 r
dt
dt
Centripetal acceleration:
ac 
v2
 r 2
r
Moment of Inertia:
2
I   mi ri   r 2 dm
2
i
KE of rotating rigid body:
KE R 
1 2
I
2
Moment of inertia for extended continuous object:
Torque:
I   pr 2 dv

dL
 
  Fr  Fr sin   r  F  I 
dt
Power by torque:
f
f
i
i
W   d   Id 
Work done by torque:
P
d
 
dt

1
2
2
I  f  i
2

  
L  r  p  mvr sin   I
Angular momentum: 
Ltot  cons tan t
ds
d
R
 R
dt
dt
dv
d
 cm  R
 R
dt
dt
vcm 
Pure rolling motion:
a cm
KE of rolling object: KE 
1 2 1
2
I  Mv cm
2
2
Kepler’s laws of planetary motion:
1st: Each planet in the solar system moves in an elliptical orbit with the Sun at one focus
a2  b2  c2
eccentricity : e  c
a
2nd: The radius vector drawn from the Sun to any planet sweeps out equal areas in equal
time intervals
3rd: The square of the orbital period of any planet is proportional to the cube of the semimajor axis of the elliptical orbit.
 4 2  3
2
a  k s a 3
T  
 GM s 
2GM
R
Speed of light: c  f 
Escape velocity:
vesc 
Rydberg formula:
 1
1 
 RH  2  2 
n

ni 
 f
1
Frequency of emitted radiation:
Total energy of hydrogen atom:
E
n 2 2
me k e e 2
13.606eV
En  
n2
Radius of Bohr orbits in H:
Energy levels of H:
Ei  E f  hf 
rn 
kee
2r
hc

2
n = 1, 2, 3,…
n = 1, 2, 3,…
E
1

B
o  o
Frequency of radiation emitted:
k e e 2  1
1 
f 

2ao h  n f 2 ni 2 
Energies of quantum states for H:
kee 2  1 
En  
 
2a o  n 2 
n = 1, 2, 3,…
Fs  kx
Hooke’s Law:
xt   A cost   
v max  A
Simple harmonic motion:
a max   2 A

Angular frequency:
Period:
T
Frequency:
f 
2

k
2
 2f 

m
T
m
L 2r
 2

k
g
v
 2
1
1

T 2

Physical pendulum:
g v

L r
k
m
Total energy: E 
mgd
I
T  2
I
mgd
bt
Position damped oscillation (small): x   Ae 2 m  cost   



Angular frequency damped oscillation (small):
Fo
A
Amplitude damped oscillator:
Speed of traveling wave:
v

T


k

 f
2
 o
2
k  b 


m  2m 
m

 b 


 m 
2
2
1 2
kA
2
k
Angular wave number:
2

y  A sin kx  t 
Wave function sinusoidal wave:
d2y 1 d2y

dx 2 v 2 dt 2
Linear wave equation:
Velocity of wave (stretched): v 
Wave power:  
T
where  

m

1
 2 A 2 v
2
Doppler effect:
 v  vo
f   f 
 v  vs
Standing wave:
y  2 A sin kxcos t
Normal modes:
n 
2L
n
where n = 1, 2, 3,…
fn 
Frequencies of normal modes:
Fundamental frequency:



f1 
v
n

n
n T
v
2L
2L 
1 T
2L 
Natural frequency
Air column open on both ends:
v
4L
Closed on one end:
fn  n
Beat frequency:
f b  f 2  f1
New frequency (beat):
Pressure:
P
F
A
Archimedes’s principle:
f 
fn  n
v
2L
where n = 1, 2, 3,…
where n = 1, 3, 5,…
f1  f 2
2
Pressure with depth:
B   f gV  Mg
P  Po  gh
Continuity equation: A1v1  A2 v2
1
1
2
2
Bernoulli’s equation: P1  v1  gy1  P2  v 2  gy 2
2
2


 Fe


q
E  k e 2 rˆ
Electric field: E 
Electric field at a point:
 Fe  qo E
qo
r

q
Electric field due to a group of charged particles:
E  k e  2i rˆi
i ri
Q
Q

 
Volume charge density:
Surface charge density:
V
A
Q

Linear charge density:
Electric flux:  E  E  A  EA cos

 
 
Electric flux:  E   E  dA   E  dA
surface
  q
 E   E  dA  in
o
B 

U  U B  U A  qo  E  ds
Electric flux (closed surfaces):
Change in potential energy:
A
Potential difference between two points:
V 
B 
U

   E  ds
A
qo
Potential difference between two points in a uniform E field:
Electric potential:
V 
V  ke
q
r
Electric potential due to continuous charge distribution:
Q
V
A
U
qo
Electric potential due to a point charge:
Capacitance: C 
B
V   E  ds   Ed
V  ke 
Parallel-plate capacitor:
dq
r
C
o A
d
Q2 1
1
2
 QV  C V 
2c 2
2
2
Q q
Q
Work required to charge a capacitor: W  
dq 
0 C
2C
Q
Capacitor with dielectric:
C   o  Co
Vo
dQ
I
I
 nqv d A
J   nqv d
Current:
Current density:
dt
A
V


R
 
Resistance:
I
A A
Energy stored in a charged capacitor: U 
Power:
  IV  I 2 R 
Conservation of charge:
V 2
R
I1  I 2  I 3
qt   C 1  e

 t
I t   e RC
R
q vs. t for a charging capacitor:
I vs. t for a charging capacitor:
t
RC
  Q 1  e  t RC 



qt   Qe RC
dq
t
I t  
  I o e RC
I vs. t for a discharging capacitor:
dt

 
Magnetic force on moving charged particle: FB  qv  B
t
q vs. t for a discharging capacitor:



b 
Magnetic force on a current carrying conductor:
FB  I  B  I  ds  B
a


Magnetic dipole moment of a current loop:   IA
Torque on current loop:
Biot-Savart law:






    B  N  B  NIA  B



Ids  rˆ  o Ids  rˆ
dB  k m

4 r 2
r2
 oI ds  rˆ
B
Total magnetic field at a point due to a current:
4  r 2
  I   I I
Magnetic force between two wires: F1  I 1 o 2   o 1 2
2a
 2a 
 II
F
Force per unit length: 1  o 1 2

2a
 
o I
Ampere’s law (steady currents):
 B  ds  B ds  2r 2r   o I
 I 
Ampere’s law (interior to R):
B   o o2 r
 2R 
 NI
N
B   o I   o nI
Toroid:
Solenoid:
B o

2r
 
 m   B  dA
Magnetic flux:
 
d m
dB
  E  ds   A
dt
dt
d m
d
Induced emf through coil:
  N
  BA cos  
dt
dt
Voltage across a conductor moving through a magnetic field:
Faraday’s law:
 
V  El  Blv
Motional emf:   
d m
d
dx
  Bx    B
  Bv
dt
dt
dt

I
Power delivered by applied force:
  Fapp v  IB v 
R

B v
R
Magnitude of induced current:
R
d


1
r dB
m
E


2r
2r dt
2 dt
Tangential electric field:
d m
dI
 L
dt
dt
N m
Inductance of an N-turn coil:
L
I

Inductance:
L
dI
dt
1
Energy stored in an inductor: U m  LI 2
2
Um
B2
Magnetic energy density:
m 

A 2 o
d E
Displacement current:
I d o
dt
Maxwell’s equations:
 
d B
  Q
E
 ds  
E

d
A



dt
o
 
 
d E
 B  ds   o I  o  o dt
 B  dA  0
Self induce emf:
  N
Resonance frequency of an LC circuit:
Poynting vector:
Intensity:
fo 
1
2 LC
 1   EB
S
EB 
o
o
I  S av 
Bv 2
Emax Bmax
2 o
Radiation pressure (complete absorption):
P
I
c
Polarization by selective absorption: I  I o cos 
2
i  r
sin  2 v2
Snell’s law:

 cons tan t
sin 1 v1
SOLvacuun c
Index of refraction: n 

SOLmedium v
Reflection of light:
n1 sin  1  n 2 sin  2
1 n1   2 n 2

n o
n
Snell’s law of refraction:
n1 sin 1  n 2 sin  2 
Critical angle ( for n1 > n2 ) :
Magnification:
M 
sin  c 
n2
n1
Im ageheight
Object height

h
h
Mirror equation\thin lens:
1 1 2 1
  
p q R f
Focal length of a mirror:
f 
R
2
Refraction through a single curved surface:
n1 n 2 n 2  n1


p q
R
Flat refractive surface:
q   n2  p
 n1 
Lens makers’ equation:
 1
1
1 

 n  1 
f
 R1 R2 
Path difference:
  r2  r1  d sin 
Constructive interference for two slits:
  d sin  bright  m m = 0, 1, 2,…
Destructive interference for two slits:
  d sin  dark   m   m = 0, 1, 2,…
Phase difference:



 2


1
2
 min 
Limiting angle of resolution for a slit:

a
Limiting angle of resolution for a circular aperture:  min  1.22

D
Lorentz transformations:
x     x  vt 
y  y
z  z
 vx 
t   t  2 
 c 
Gamma:
 v2 
  1  2 
 c 

1
2

1
1
Time dilation:
t  t p
Length contraction:
L
Spacetime interval:
s 
v2
c2
Lp

2
 ct   x 
2
2
Doppler effect:
Source/observer approaching:
f 
1 
fo
1 
Observer/source approaching:
f 
1 
fo
1 
Relativistic momentum:
Rest energy:
Kinetic energy:
where  


p  mu
E R  mc 2
Total energy: E  mc 2
K  mc 2  mc 2    1mc 2
Energy-momentum relationship:

E 2  p 2 c 2  mc 2
Energy-momentum relationship for a photon:
:

2
E  pc
 u v
ux  x
u v
1  x2
c
v  t1  t 0

c  xb  xa

c

Lorentz velocity transformation for S´→S:
ux 
 dp
F
dt
Force in relativity:
Blackbody radiation
Stefan-Boltzmann law:
Wien’s displacement law:
R  T 4
mT  2.898  10 3 m  k
Plank’s radiation law: u   
8hc5
hc
e kT  1
Photoelectric effect: eVo  hf  
h
1  cos  
2  1 
Compton effect:
mc
f E
h
De Broglie relations:
 hp
1
xp  
2
Heisenberg uncertainty principle:
1
Et  
2
2

Particle in a box:
E
2mL2
Radioactivity: R  
Half-life:
t1 
2
dN
  N o e   t  Ro e   t
dt
ln 2



ux  v
u v
1  x2
c
0.693

Rectangle:
Circle:
Cylinder:
A  bh
A  r 2
C  2r
V  r 2 
S  2r  2r 2
Triangle:
V
bh
3
1
bh
2
V  wh
Parallelepiped:
Sphere:
A  r  r 2
Cone:
A
Cube:
4
V  r 3
3
S  4r 2
A  6s 2
V  s3