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UC Summer Introductory Physics
Formula Sheet Module
Note: bold symbols represent vector quantities v = ๐ฃ๐ฃโ
MECHANICS
Trigonometry in a right-angled triangle
sinฮธ= opposite/hypotenuse
cosฮธ = adjacent /hypotenuse
tanฮธ= opposite/adjacent
Equations of motion under constant
acceleration:
v = v 0 + at
x = x 0 + v 0 t + 1/2 at 2
v^2 = v^20 +2a(x- x 0 )
Newtonโs second law
ฮฃ F =m a
Kinetic friction force
F fr = ฮผk F N
Work
W= F d cosฮธ
Kinetic energy
K= 1/2mv^2
Work-energy theorem
Wnet = KE 2 โ KE 1
Mechanical energy
E (^) mech = KE +PE
Gravitational force
F G = Gm 1 m 2 /r^2
Weight: F G = m g
PE G = mgh
Case of a spring
F s = -k x
PE (^) el = 1/2kx^2
Power
P (^) avg = W/โt
P= F โ v
Momentum
p =m v
Coordinates of center of mass
xCM = ฮฃi mi xi /M
Rotational motion
AR =v^2 /r
T = 2๐๐r/v
๐๐=2๐๐/T =v/r
K = ยฝ I ๐๐^2
Torque
๐๐ =Fr sin๐๐
ฮฃ๐๐=I๐ผ๐ผ
Angular momentum
L = r ร p
Angular momentum
L = I๐๐
Moment of inertia
Solid cylinder: I = ยฝ MR^2
Rectangular plate: I = ยฝ M(l^2 + w^2 )
Long thin rod:
- Rotation axis through center: I = 1/12ml^2
- Rotation axis through end: I= 1/3ml^2
Orbiting particle: I = mr 2
FLUID MECHANICS
Pressure
P=F/A
In a fluid at depth h : P = P 0 + ๐๐gh
Buoyant force
B= ๐๐F gVdisp
Continuity equation in a fluid
A 1 v 1 = A 2 v 2
Bernoulliโs equation
P 1 + 1/2๐๐v 12 + ๐๐gy 1 = P 2 + + 1/2๐๐v 22 +
๐๐gy 2
Poiseuilleโs law
Q = [๐๐r^4 (P 1 -P 2 ) /8๐๐L
WAVES
Simple harmonic motion
x(t) = Acos(๐๐t +๐๐)
E = 1/2kA^2
For a spring: ๐๐โ(๐๐/๐๐)
T = 2๐๐/๐๐
f= 1/T
Mechanical wave
y = Asin(kx โ ๐๐ t)
Wave number: k = 2๐๐/๐๐
v=๐๐f
Mechanical wave in a fixed string
v=๏ฟฝ ๐น๐น๐๐ ๐๐
Doppler effect
๐๐ โฒ^ = ๐๐ ๏ฟฝ ๐ฃ๐ฃ๐ ๐ ๐ ๐ ๐ ๐ โ๐ฃ๐ฃ๐๐๐๐๐ ๐ ๐ฃ๐ฃ๐ ๐ ๐ ๐ ๐ ๐ โ๐ฃ๐ฃ๐ ๐ ๐๐๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๏ฟฝ^ (both moving)
Standing wave
y = 2Asin(kx) cos(๐๐t)
In a string: fn =nv/2L, n=1,2,3...
Beat frequency: fb = |f 1 โ f 2 |
VECTORS
Vectors and Scalars
A: scalar, i.e a number
ฤ = A : vector, i.e, a set of numbers
A = (Ax, Ay , Az) = Ax รฎ + A y ฤต + Az k๏ฟฝ
รฎ = direction of the x axis, unit vector
ฤต = direction of the y axis, unit vector
k๏ฟฝ = direction of the z axis, unit vector
Magnitude of vectors
A = | A | = Magnitude of vector A
A = (Ax, Ay , Az) = Ax รฎ + A (^) y ฤต + Az k๏ฟฝ
A = โ(๐ด๐ด๐ฅ๐ฅ^2 + ๐ด๐ด๐ฆ๐ฆ^2 + ๐ด๐ด_๐ง๐ง^2 )
The magnitude of the unit vectors is 1
Sum of vectors
A = (Ax, Ay , Az) = Ax รฎ + A y ฤต + Az k๏ฟฝ
B = (Bx, By , Bz) = Bx รฎ + B y ฤต + Bz k๏ฟฝ
A + B = (Ax + B (^) x, Ay + By , Az + B (^) z)
=(Ax + Bx) รฎ +( Ay + By ) ฤต +(Az + Bz) k๏ฟฝ
Scalar product
A product between two vectors which is itself a scalar
It is zero if the two vectors are perpendicular one to the other
A = (Ax, Ay , Az) = Ax รฎ + A y ฤต + Az k๏ฟฝ
B = (Bx, By , Bz) = Bx รฎ + B y ฤต + Bz k๏ฟฝ
AโB = (AxBx + Ay By + Az Bz) = ABcos๐๐
Vector product
A product between two vectors which is itself a vector
It is zero if the two vectors are parallel to each other
A = (Ax, Ay , Az) = Ax รฎ + A (^) y ฤต + Az k๏ฟฝ
B = (Bx, By , Bz) = Bx รฎ + B (^) y ฤต + Bz k๏ฟฝ
| A ร B| = ABsin๐๐
A ร B = ABsin๐๐n
๐๐ ฬis a unit verctor perpendicular to the plane that contains A and B
A ร B = (Ay Bz - A zBy ) รฎ + (AzBx โ A xBz) ฤต + (AxBy - Ay Bx) k๏ฟฝ
RECOMMENDED VALUES OF PHYSICAL
CONSTANTS AND CONVERSION
FACTORS
Speed of light in vacuum c = 299792458 (exact) ms -
Electric (permittivity) constant ๐๐ 0 = 8.854187817... ร 10-12^ (exact) Fm-
Magnetic (permeability) constant ๐๐ 0 = 12.566370614... ร 10-7^ (exact) NA-
(Unified) Atomic mass unit u = 1.660538782(83) ร 10-27^ kg
Alpha particle mass (in u) m๐ผ๐ผ = 4.001 506 179 127(62) u
Atomic mass unit energy equivalent (J) u = 1.492417830(74) ร 10-10^ J
Atomic mass unit energy equivalent u = 931.494 028(23) MeV
Avogadro constant NA = 6.02214179(30) ร 10^23 mol-
Bohr radius a 0 = 0.52917720859(36) ร 10-10^ m
Bohr magneton ๐๐B = 927.400915(23) ร 10-26^ JT-
Boltzmann constant k = 1.3806504(24) ร 10 -23^ JK -
Classical electron radius re = 2.8179402894(58) ร 10-15^ m
Compton wavelength of the electron ๐๐c = 2.4263102175(33) ร 10-12^ m
Deuteron mass (in u) mD = 2.013 553 212 724(78) u
Electron mass (in u) me = 5.4857990943(23) ร 10-4^ u
Elementary charge e = 1.602176487(40) ร 10-19^ C
Molar mass of carbon- = 12 ร 10-3^ (exact) kg mol-
Neutron mass (in u) mn = 1.00866491597(43) u
Gravitational constant G = 6.67428(67) ร 10-11^ m^3 kg -1^ s-
Nuclear magneton ๐๐N = 5.05078324(13) ร 10-27^ JT-
Planck constant h = 6.62606896(33) ร 10-34^ Js
Proton mass (in u) mp = 1.00727646677(10) u
Rydberg constant R = 10973731.568527(73) m-
Stefan-Boltzmann constant ๐๐ = 5.670400(40) ร 10-8^ Wm-2K
Triton mass (in u) = 3.015 500 7134(25) u
