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Formula sheet-physics, Cheat Sheet of Physics

Physics formula sheet in 2d vectors, algebra & trigonometry, general kinematics, gravitation, simple harmonic motion, mechanical waves, momentum, impulse and system particles.

Typology: Cheat Sheet

2021/2022

Uploaded on 02/07/2022

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Formula Sheet - Physics 111
x
y
~
Ax
~
Ay
~
A
θ
2D Vectors:
A=Ax+Ay= (Ax, Ay)
A=A=qA2
x+A2
y
Ax=Acos θ Ay=Asin θ
tan θ=Ay
Ax
Algebra & Trigonometry:
ax2+bx +c= 0 x=b±pb24ac
2a
2 sin2x+ cos (2x) = 1 cos (x+y) = cos xcos ysin xsin y
2 cos2xcos (2x) = 1 sin (x+y) = sin xcos y+ cos xsin y
sin2x+ cos2x= 1 2 cos xcos y= cos (xy) + cos (x+y)
sec2xtan2x= 1 2 sin xsin y= cos (xy)cos (x+y)
csc2xcot2x= 1 2 sin xcos y= sin (xy) + cos (x+y)
Conversion factors:
femto- (f) = 1015
pico- (p) = 1012
nano- (n) = 109
micro- (µ) = 106
milli- (m) = 103
centi- (c) = 102
kilo- (k) = 103
mega- (M) = 106
giga- (G) = 109
tera- (T) = 1012
1 yd=3 ft = 36 in 1 in=2.54 cm
1 mi=1609 m 1 lb=4.448 N
1 gal=3.78541 L 1 m3=1000 L
1 atm=1.013 ×105Pa 1 cal=4.186 J
=760 mm Hg 1 Cal=1000 cal
Geometry
Circle: Perimeter = 2πr
Area =πr2
Sphere: Surface Area = 4πr2
Volume =4
3πr3
Cylinder: Surface Area = 2πr2+ 2πr h
Volume =πr2h
Few Physical Constants & Quantities:
REarth = 6.38 ×106mg= 9.80 m/s2
MEarth = 5.97 ×1024 kg G= 6.67 ×1011 Nm2/kg2
Msun = 1.99 ×1030 kg R= 8.314 J/mol ·K
rsunEarth = 1.496 ×1011 mkB= 1.381 ×1023 J/K
STP : 1 atm,273 KNA= 6.022 ×1023 mol1
vsound,air = 343 m/s σ= 5.67 ×108W/m2K4
ρwater = 1000 kg /m3
General Kinematics
vavg =r
t=r2r1
t2t1
v= lim
t0
r
t
aavg =v
t=v2v1
t2t1
a= lim
t0
v
t
constant acceleration 1D motion Kinematics
x=x0+v0t+1
2at2v=v0+at
v2=v2
0+ 2ax v =v+v0
2
Constant acceleration Kinematics 2D:
x=x0+v0xt+1
2axt2vx=v0x+axt v2
x=v2
0x+ 2axx
y=y0+v0yt+1
2ayt2vy=v0y+ayt v2
y=v2
0y+ 2ayy
vx=vx+v0x
2vy=vy+v0y
2
Projectile motion (ax= 0 and ay=g)
x=x0+vxt vx= constant.
y=y0+v0yt1
2gt2vy=v0ygt v2
y=v2
0y2gy
vy=vy+v0y
2v2=v2
x+v2
yv2=v2
02gy
Relative motion
ra,c=ra,b+rb,cva,c=va,b+vb,caa,c=aa,b+ab,c
Forces
XF=ma F AB =FBA Fspring =kx
Fg(W) = mgfk=µkN fsµsN
Circular motion
ω=θ
tα=ω
ts= v = atan =
a=atan +ararad =v2
R=ω2R a =qa2
tan +a2
r
Constant α:
ω=ω0+αt ω2=ω2
0+ 2αθ
ω=ω+ω0
2θ=θ0+ω0t+1
2αt2
Uniform Circular Motion(α= 0) : T=1
f=2π
ω=2πR
v
Work and Energy
W=F d cos θKE =Wtotal K E =1
2mv2=p2
2m
P=W
t=F v cos θP E =Wcons E=Wnoncons
E=P E +KE E= P E + K E
P Es=1
2kx2P Egrav = mgy
pf3

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Formula Sheet - Physics 111

x

y

A^ ~

x

A y

A^ ~

θ

2D Vectors:

A = A x + A y = (A x , A y ) ∣ ∣ ∣ A

∣ = A =

A

2 x +^ A

2 y

A x = A cos θ A y = A sin θ

tan θ =

A y

A x

Algebra & Trigonometry:

ax

2

  • bx + c = 0 ⇒ x =

−b ±

b 2 − 4 ac

2 a

2 sin

2 x + cos (2x) = 1 cos (x + y) = cos x cos y − sin x sin y

2 cos

2 x − cos (2x) = 1 sin (x + y) = sin x cos y + cos x sin y

sin

2 x + cos

2 x = 1 2 cos x cos y = cos (x − y) + cos (x + y)

sec

2 x − tan

2 x = 1 2 sin x sin y = cos (x − y) − cos (x + y)

csc

2 x − cot

2 x = 1 2 sin x cos y = sin (x − y) + cos (x + y)

Conversion factors:

femto- (f) = 10

− 15

pico- (p) = 10

− 12

nano- (n) = 10

− 9

micro- (μ) = 10

− 6

milli- (m) = 10

− 3

centi- (c) = 10

− 2

kilo- (k) = 10

3

mega- (M) = 10

6

giga- (G) = 10

9

tera- (T) = 10

12

1 yd=3 ft = 36 in 1 in=2.54 cm

1 mi=1609 m 1 lb=4.448 N

1 gal=3.78541 L 1 m

3 =1000 L

1 atm= 1. 013 × 10

5 Pa 1 cal=4.186 J

=760 mm Hg 1 Cal=1000 cal

Geometry

Circle: Perimeter = 2πr

Area = πr

2

Sphere: Surface Area = 4πr

2

Volume =

πr

3

Cylinder: Surface Area = 2πr

2

  • 2πrh

Volume = πr

2 h

Few Physical Constants & Quantities:

REarth = 6. 38 × 10

6 m g = 9. 80 m/s

2

MEarth = 5. 97 × 10

24 kg G = 6. 67 × 10

− 11 Nm

2 /kg

2

Msun = 1. 99 × 10

30 kg R = 8. 314 J/mol · K

rsun−Earth = 1. 496 × 10

11 m kB = 1. 381 × 10

− 23 J/K

STP : 1 atm, 273 K NA = 6. 022 × 10

23 mol

− 1

vsound , air = 343 m/s σ = 5. 67 × 10

− 8 W/m

2 K

4

ρwater = 1000 kg/m

3

General Kinematics

v avg =

r

∆t

r 2 − r 1

t 2 − t 1

v = lim ∆ t → 0

r

∆t

a avg =

v

∆t

v 2 − v 1

t 2 − t 1

a = lim ∆ t → 0

v

∆t

constant acceleration 1D motion Kinematics

x = x 0 + v 0 t +

at

2 v = v 0 + at

v

2 = v

2 0 + 2a∆x^ v^ =^

v + v 0

Constant acceleration Kinematics 2D:

x = x 0 + v 0 x t +

a x t

2 v x = v 0 x + a x t v

2 x =^ v

2 0 x + 2a x ∆x

y = y 0 + v 0 y t +

a y t

2 v y = v 0 y + a y t v

2 y =^ v

2 0 y + 2a y^ ∆y

v x =

v x + v 0 x

v y =

v y + v 0 y

Projectile motion ( ax = 0 and ay = − g )

x = x 0 + v x t v x = constant.

y = y 0 + v 0 y t −

gt

2 v y = v 0 y − gt v

2 y = v

2 0 y − 2 g∆y

v y =

v y + v 0 y

v

2 = v

2 x +^ v

2 y v

2 = v

2 0 −^2 g∆y

Relative motion

r a , c = r a , b + r b , c v a , c = v a , b + v b , c a a , c = a a , b + a b , c

Forces

F = m a F (^) AB = − F (^) BA Fspring = −kx

F (^) g (≡ W ) = m g fk = μkN fs ≤ μsN

Circular motion

ω =

∆θ

∆t

α =

∆ω

∆t

s = Rθ v = Rω atan = Rα

a = a tan + a r arad =

v

2

R

= ω

2 R a =

a

2 tan +^ a

2 r

Constant α :

ω = ω 0 + αt ω

2 = ω

2 0 + 2α∆θ

ω =

ω + ω 0

θ = θ 0 + ω 0 t +

αt

2

Uniform Circular Motion (α = 0) : T =

f

2 π

ω

2 πR

v

Work and Energy

W =F d cos θ ∆KE = Wtotal KE =

mv

2

p

2

2 m

P =

W

t

= F v cos θ ∆P E = −Wcons ∆E = Wnon−cons

E =P E + KE ∆E = ∆P E + ∆KE

P Es =

kx

2 P Egrav = mgy

Gravitation

F (^) g = −

GmM

r 2

ˆ r g =

GM

r 2

T

2

r^3

4 π

2

GM

vcirc orbit =

GM

r

Momentum, Impulse, System of particles

F (^) net =

p

∆t

= mtot a cm p = m v J = ∆ p = F (^) net , avg∆t

r cm =

m i r i ∑ m i

v cm =

m i v i ∑ m i

a cm =

m i a i ∑ m i

p i = mtot v cm If F (^) net = 0 ⇒ ∆ p = 0 ⇒ p = p

Elastic Collision:

KEbefore = KE

′ after vA , x − vB , x = −

v

′ A , x − v

′ B , x

Rigid-body and Rotational motion

KE = KEtr + KErot KErot =

2 KEtr =

mv

2 cm

τ = rF sin θ = r⊥F = rF⊥

τ = I~α W = τ ∆θ

L = rp sin θ = r⊥p = rp⊥ L = Iω P = τ ω

τ =

∆ L

∆t

if

~τ = 0 ⇒ ∆ L = 0 ⇒ L = L

Icm for few objects about symmetry axis:

Isolid sphere =

mr

2 Isolid cylinder =

mr

2

Ihollow sphere =

mr

2 Ihollow cylinder = mr

2

Irod =

m`

2 Irectangle =

m

a

2

  • b

2

a b

Elasticity

E =

F/A

/ 0

B = −

∆P

∆V /V 0

G =

F||/A

x/h

Fluids

P =

F

A

P = P 0 + ρgh ρ =

m

V

FB = ρFVdisplg

A 1 v 1 = A 2 v 2 = const P + ρgy +

ρv

2 = const

Simple Harmonic Motion

x = A cos(ωt) v = −Aω sin(ωt) a = −Aω

2 cos(ωt)

ωspring =

k

m

T =

f

2 π

ω

Tpendulum = 2π

`

g

Mechanical Waves

v = λf ω = 2πf v =

F

m/`

f =

T

〈P 〉 =

ω

2 A

2

μF I =

P

4 πr 2

sound:

v =

γRT

M

γair = 1. 40 β = (10 dB) log 10

I

I 0

fbeat = |f a − f b | I 0 = 10

− 12 W/m

2

fobs =

vsnd ± vobs

vsnd ± vsrc

fsrc

Interference:

d 2 − d 1 = n

λ

n = 0, ± 2 , ± 4 ,... (constructive)

n = ± 1 , ± 3 , ± 5 ,... (destructive)

Standing wave normal modes:

f n = nf 0 =

nv

2 `

n = 1, 2 , 3 ,... (both ends either fixed or free)

nv

4 `

n = 1, 3 , 5 ,... (one end fixed)

Temperature and Heat

TK = TC + 273.15 K TC =

(TF − 32

◦ C) ∆= α 0 ∆T

Qrad

t

=  σA

T

4 − T

4 s

) Q

cond

t

=kA

T H − T C

`

∆V = βV 0 ∆T

Q = mc∆T Q =nCmolar∆T Q = mL

Ideal Gas

pV = nRT = N kT vrms =

3 kT

m

3 RT

M

n =

N

N A

m

M

〈KE〉 =

kT =

mv

2 rms

k =

R

N A

= 1. 381 × 10

− 23 J/K U =

nRT (monoatomic)

Thermodynamics

∆U = Q − W W = p∆V W = |QH| − |QL|

eengine =

W

QH

COPrefrig =

QL

|W |

COPideal pump

TH

TH − TL

eideal = 1 −

TL

TH

COPpump =

QH

W

COPideal refrig

TL

TH − TL

∆S =

Q

T

ideal :

TL

TH

QL

QH