Embed Size (px)
Citation preview
Equation Sheet Physics 141
Page 1 of 8
The momentum principle:
d!p =!F
netdt
!p =
m!v
1!v
2
c2
!p
new=!p
old+!F
net!t
!r
new=!r
old+
1
1+p
mc
!"#
$%&
2
!p
m
!"#
$%&'t
Equations of motion in 1D for constant acceleration and low velocities (v << c):
x t( ) = x0+ v
0t +
1
2at
2
v t( ) =dx t( )
dt= v
0+ at
a t( ) =dv t( )
dt= a = constant
Requirement for uniform circular motion:
Fr=
mv2
r
Rotational motion: d = !r
v = !r ! =d"
dt
a = !r ! =d"
dt
Gravitational force:
!F = G
m1m
2
r2
r̂
!F = m
!g (close to the surface of the Earth)
Electrostatic force:
!F = !
1
4"#0
q1q
2
r2
r̂
Harmonic motion: F = !kx
x t( ) = xmax
cos !t + "( ) where ! =k
m
T =2!
"
Damped harmonic motion:
x t( ) = x
me!
bt
2meit
k
m
Driven harmonic motion:
x t( ) =F
0
!0
2"! 2
cos !t + #( )
Stress and strain:
F
A= Y
!L
L
Y =k
s
d
Work done by a force:
W =!Fi
!d constant force
=!Fid!r
!r1
!r2
! variable force
Power:
P =dW
dt=!Fi!v
Equation Sheet Physics 141
Page 2 of 8
Work-energy theorem:
!E
system= W
E
system= E
1+ E
2+ E
3+ .....( ) +U
Relativistic energy relations:
E =mc
2
1!v
2
c2
= mc2+ K
E
2! pc( )
2
= mc2( )
2
K =mc
2
1!v
2
c2
! mc2"v!c
1
2mv
2
Potential energy:
!U = "W
internal= "
!Fid!r#
!F = !
!"U =
!#U
#x
!#U
#y
!#U
#z
$
%
&&&&&&&
'
(
)))))))
U
gravity= !G
m1m
2
r
U
gravity= mgh
Uelectric
=1
4!"0
q1q
2
r
U
spring=
1
2kx2
Heat capacity:
!E
thermal= mC!T
Friction forces:
f
s! µ
sN
f
k= µ
kN
Drag force (air):
!F
air= !
1
2C"Av
2v̂
Energy levels for the Hydrogen atom:
EN=!13.6
N2
eV, N = 1,2,3,...
Vibrational energy levels:
EN= E
0+ N!!
0= E
0+ N!
ks
m, N = 0,1,2,..
Energy and wavelength of light:
Ephoton
=hc
!light
Center of mass:
!r
cm=
mi
!r
i
i
!
mi
i
!
=1
Mm
i
!r
i
i
!
!r
cm=
!rdm!dm!
=1
M
!rdm!
Motion of the center of mass:
M!a
cm=!F
net ,ext
Gravitational potential energy of a multi-particle system:
U = Mgy
cm
Equation Sheet Physics 141
Page 3 of 8
Kinetic energy of a multi-particle system:
K = Ktrans
+ Krel
=1
2Mv
cm
2+ K
rel
Impulse of a force:
!J =
!Fdt!
Momentum and impulse:
!J =!p
f!!p
i
Conservation of linear momentum:
!!P
system+ !!P
surroundings= 0
Elastic collision in one dimension (mass 2 at rest before the collision):
v1 f
=
m1! m
2
m1+ m
2
v1i
v2 f
=
2m1
m1+ m
2
v1i
Completely inelastic collision in one dimension (mass 2 at rest before the collision):
vf=
m1
m1+ m
2
vi
Moment of inertia:
I = mir
i
2
i
! Discreet mass distribution
I = r2dm
Volume
! Continuous mass distribution
Kinetic energy of a rotating rigid object:
K =1
2I!
2
Torque:
!! =!r "!F
Newton’s “second” law for rotational motion:
!! = I
!"
Angular momentum of a single particle:
!L =!r !!p
Angular momentum of a rotating rigid object:
!L = I
!!
The angular momentum principle:
d!L
dt=!r !!F( )
net, ext=!"
net, ext
Number of micro states:
! =q + N "1( )!q! N "1( )!
Definition of entropy S: S = k ln! Definition of temperature T:
1
T=
dS
dEint
The Boltzmann distribution:
P !E( ) " e
#!E /kT
The Maxwell-Boltzmann velocity distribution:
P v( ) = 4!M
2!kT
"#$
%&'
3
2
v2e(
1
2Mv
2/ kT( )
Root-mean-square speed:
vrms
= v2=
3kT
M
Average translational kinetic energy of an ideal gas:
Ktrans
=3
2kT
Equation Sheet Physics 141
Page 4 of 8
Mean-free path d:
N
V! R + r( )
2
d"#$
%&'( 1
Number of gas molecules hitting an area A per second:
molecules/s =1
4
N
VAv
Ideal gas law:
pV = NkT
Work done by a gas:
W = pdV
V1
V2
!
First law of thermodynamics:
!E
system= W + Q
Isothermal compression:
W = NkT lnV
1
V2
!
"#$
%&
Adiabatic compressions:
pV
!= pV
Cp
/CV= constant
Heat capacity C defined:
!Q = C!T
Heat capacities per molecule for ideal gasses:
CV=
3
2k monatomic gas
CV!
3
2k other gases
C
p= C
V+ k
Rate of thermal energy transfer:
dQ
dt= ! A
TH" T
L
L
Efficiency of a heat engine:
efficiency =W
QH
= 1!T
L
TH
Quality factor of a heat pump:
K =Q
C
W
Scratch Paper Physics 141
Page 5 of 8
Scratch Paper Physics 141
Page 6 of 8
Scratch Paper Physics 141
Page 7 of 8
Scratch Paper Physics 141
Page 8 of 8
