Equation Sheet – PHYS–2350
1
Kinematics – One & two dimensions
Average speed
Average acceleration 1-D Kinematics
Relative velocity:
#
∆𝑥
∆𝑣!
𝑥 = 𝑥" + 𝑣"! 𝑡 + $𝑎! 𝑡 $
$
'
⇒ 𝑣!$ = 𝑣"!
+ 2𝑎! ∆𝑥
𝐯-⃗%# = 𝐯-⃗%$ + 𝐯-⃗$#
𝑣! =
𝑎! =
∆𝑡
∆𝑡
𝑣! = 𝑣"! + 𝑎! 𝑡
Quadratic Formula
Projectile with ∆𝑦 = 0
2𝑣" sin 𝜃
𝑣"$ sin 2𝜃
𝑣"$ sin$ 𝜃
−𝑏 ± √𝑏$ − 4𝑎𝑐
𝑡)*+,-. =
𝑅 = 𝑥&'( =
𝐻=
𝑎𝑥 $ + 𝑏𝑥 + 𝑐 = 0 𝑥 =
𝑔
𝑔
2𝑔
2𝑎
Dynamics
Net Force
Weight
Acc. gravity on earth, 𝑔
Kinetic friction
Static friction
$
⃗
𝐹
=
𝑚𝑔
𝑔
=
9.8
m/s
𝐹
=
𝜇
𝐹
𝐹3 ≤ 𝜇3 𝐹2
∑𝐅/ = 𝑚𝐚-⃗
0
1
1 2
Uniform Circular motion & Gravitation
Frequency
Tangent Velocity & acceleration Centripetal acceleration
Centripetal force
$
1
2𝜋𝑟
𝑣.'4,54.
𝑓=
𝜔 = 2𝜋𝑓
𝑣.'4,54. =
𝑎.'4,54. = 0
𝑎6'7+'* = −
𝑣6'7+'* = 0 𝐹6'7+'* = 𝑚𝑎6'7+'*
𝑇
𝑇
𝑟
Gravitational Force
Universal G constant
Orbits
Kepler’s 3rd law
𝑚# ⋅ 𝑚$
𝑇:$ 𝑆:%
𝑀
9##
$
$
⁄
𝐹8 = 𝐺
𝐺
=
6.67
×
10
N
⋅
m
kg
=
𝑣
=
𝐺
𝑟$
𝑇;$ 𝑆;%
𝑟
Work and Energy
Work
Power – Intensity
∆𝐸
∆𝐸
= 𝐹𝑣 cos 𝜙
𝐼=
𝑊< = 𝐅⃗45. ⋅ 𝐝⃗ = 𝐹45. ∆𝑥 cos 𝜙 𝑃 =
∆𝑡
∆𝑡 ∆𝑎
Mechanical energy – Conservation of energy Conservative work
𝐸 = 𝐾𝐸 + 𝑈
(𝐾𝐸 + 𝑈)+4+.+'* + 𝑊 ∗ = (𝐾𝐸 + 𝑈))+4'*
𝑊>A4 = −Δ𝑈>A4
Escape speed
2𝐺𝑀
$
𝑣5=>
=
𝑅
Work – Energy Theo.; Kinetic Energy
#
𝐾𝐸? = $𝑚𝑣 $
𝑊45. = Δ𝐾𝐸
𝐺𝑀𝑚 G≈I!
𝐺𝑀𝑚
𝑈,6'C+.D = −
l⎯⎯n 𝑚𝑔𝑦 + const.
𝑟
𝑟$ ⟹
#
𝑈=F6+4, = $𝑘𝑥 $
= −𝑘𝑥
(,6'C+.D)
=
→ i𝐹
𝐹(=F6+4,)
Linear Momentum
Momentum
Impulse
Conservation of momentum (𝐅⃗45. = 0) Center of mass = (𝑥>& , 𝑦>& , 𝑦>& )
∆𝐩
-⃗
𝐩
-⃗ = 𝑚𝐯-⃗
∆𝑡
Coefficient of restitution
𝑣$ − 𝑣#
𝜖=
𝑢# − 𝑢$
𝑢 → initial ; 𝑣 → final
𝐉⃗ = ∆𝐩
-⃗ = 𝐅⃗∆𝑡
𝐩
-⃗+4+.+'* = 𝐩
-⃗)+4'*
Total elastic collision
Inelastic collision
𝐾𝐸+4+.+'* = 𝐾𝐸)+4'*
𝜖=1
1-D/2-body
𝐾𝐸# ≠ 𝐾𝐸$
𝜖=0
𝑣$ = 𝑣#
𝐅⃗ =
∑ 𝑚/ 𝑥/
∑ 𝑚/
∑ 𝑚/ 𝑦/
=
∑ 𝑚/
∑ 𝑚/ 𝑧/
=
∑ 𝑚/
𝑥>& =
𝑦>&
𝑧>&
Rotational Motion
Angular velocity
𝛥𝜃
𝜔
z=
= 2𝜋𝑓
𝛥𝑡
Angular acceleration
𝛥𝜔
𝛼} =
𝛥𝑡
∆𝜃 = 𝜃" + 𝜔" 𝑡 +
'
𝜔 = 𝜔" + 𝛼𝑡
Moment of Inertia
Parallel Axes
Angle–position
𝜔 −velocity
𝛼–acc.
Radial acc
∆𝑠 = 𝑟∆𝜃
𝑣.'4,54. = 𝑟𝜔
𝑎.'4,54. = 𝑟𝛼
𝑎6'7+'* = −𝑟𝜔$
𝐼45. = 𝐼# + 𝐼$ + ⋯
𝐼|| = 𝐼>& + 𝑀𝑑
$
Kinetic Eqs.
Torque
#
𝛼𝑡 $
$
𝜏 = ±𝑟𝐹 sin𝜙
𝜏 = 𝑟𝐹J or 𝜏 = 𝑟J 𝐹
⇒ 𝜔$ = 𝜔"$ + 2𝛼∆𝜃
Torque & L
Rigid rotator about a static point O or the c.m.
Δ𝐿-⃗
† 𝐿+4+.+'* = 𝐿)+4'* ‡if ˆ 𝜏 = 0‰
Δ𝑡
𝐿 = 𝑟𝑝 sin 𝜙 = 𝑟𝑝J = 𝑟J 𝑝
𝑊 (I) = 𝜏∆𝜃 cos 𝜙
-𝝉⃗ =
(?)
𝑊A4 M
𝑊=
⃗ = 𝐼𝛚
𝐋
---⃗
+
(I)
𝑊'NAO. M
(?)
Momenta of Inertia examples
𝑠>& = 𝑅𝛥𝜃'NAO. >& 𝑣>& = 𝑅𝜔'NAO. >&
𝑎>& = 𝑅𝛼'NAO. >&
𝐼=F-565 = P𝑚𝑅$
$
(I)
𝐾𝐸 = 𝐾𝐸A) M + 𝐾𝐸'NAO. M (𝐾𝐸(?) = "#𝑀𝑣 $ )
-⃗ = 𝐼M 𝛂
𝝉
-⃗
Rolling without slipping
𝑃 = 𝜏𝜔 cos 𝜙
#
𝐾𝐸 (I) = $𝐼𝜔$
#
𝐼>54.56 6A7 = #$𝑚ℓ$
𝐼>D*+4756 = 𝑚𝑅$
#
𝐼>D*+4756 = $𝑚𝑅$
Equation Sheet – PHYS–2350
Fluids
Density & Specific gravity Pressure
𝑀
𝜌)*O+7
𝐹J
𝜌=
𝑠8 =
𝑃
=
𝑉
𝜌Q#M
𝐴
Archimedes
Density of Water
Absolute pressure
Atmospheric P
Vibrations
Max. velocity
Continuity
Acceleration
Vol. flow rate
∆𝑉
𝑄=
= 𝐴𝑣
∆𝑡
Bernoulli’s equation
#
#
𝑃# + $𝜌𝑣#$ + 𝜌𝑔𝑦# = 𝑃$ + $𝜌𝑣$$ + 𝜌𝑔𝑦$
Simple pendulum
$
𝑣&'( = 𝜔𝐴
Pressure in fluids
𝑃('N=A*O.5) = 𝑃'.& + ∆𝑃 1.013 × 10P Pa Δ𝑃)*O+7 = 𝜌)*O+7 𝑔ℎ
𝜌Q#M = 10% kg/m% 𝜌# 𝐴# 𝑣# = 𝜌$ 𝐴$ 𝑣$
𝐵 = 𝜌)*O+7 𝑔 𝑉7+=F*'>57
2
Physical pendulum
𝑎=𝜔 𝑥
𝑎&'( = 𝜔$ 𝐴
𝜔=–
𝑔
→ 𝑇 = 2𝜋—ℓ/𝑔
ℓ
𝑚
𝜔 = —𝑘/𝑚 → 𝑇 = 2𝜋–
𝑘
𝑥 = 𝐴 sin(𝜔𝑡 + 𝛿)
𝑣 = 𝐴𝜔 cos(𝜔𝑡 + 𝛿)
𝜔 = —𝑚𝑔𝑑/𝐼M → 𝑇 = 2𝜋—𝐼M /𝑚𝑔𝑑
Mass-spring
#
#
#
𝐸 = $𝑚𝑣 $ + $𝑘𝑥 $ = $𝑘𝐴$
𝑣 = ±𝑣&'( —1 − (𝑥/𝐴)$
Waves
Wave velocity
𝜔
𝑣 = 𝜆𝑓 =
𝑘
𝑘 = 2𝜋/𝜆
String
𝑣 = —𝑇/𝜇 (𝜇 = 𝑚/𝐿)
𝑣
𝑓R = 𝑛 ‡ ‰ , 𝑛 = 1, 2, 3, … standing waves
2𝐿
Sound
Decibel Intensity
Velocity of sound
S
𝛽 = 10log ‡S ‰ 𝐼" = 109#$ W/m$
𝑣 = 331¢
$
Beats
𝑓N5'. = |𝑓# − 𝑓$ |
Z
$
$
T
𝑇
[m/s] 𝑓R = 𝑛 ‡$U‰ , 𝑛 = 1, 2, 3, … open
273 K
9$%
Ideal Gas
Universal gas constant
𝑃𝑉 = 𝑛𝑅𝑇
𝑃𝑉 = 𝑁𝑘𝑇
𝑛
𝜌
𝑃𝑉 = =
𝑉 ℳ
Thermodynamics
1st Law
𝑅 = 8.314 J/mol ⋅ K
Avogadro’s Number
𝑁: = 6.022 × 10$%
𝑃 = 2𝜋 $ (𝑓𝐴)$ 𝜇𝑣
𝐼 = 2𝜋 $ (𝑓𝐴)$ 𝜌𝑣
T
𝑓R = 𝑚 VU , 𝑚 = 1, 3, 5, …closed
Expansion
Number of moles
𝑚 (g)
𝑁
𝑣6&= = —}}}
𝑣 $ = —3𝑘𝑇/ℳ Δ𝐿 = 𝛼𝐿" Δ𝑇 linear
𝑛=
=
(g/mol)
Δ𝑉 = 𝛽𝑉" Δ𝑇 volumetric
ℳ
𝑁:
J/K ℳ = 𝑚𝑁:
Heat flow
∆𝑇
∆𝑠
= ℎ𝐴(𝑇\ − 𝑇=O6) )
𝑄̇>A47 = 𝜅𝐴
𝑄̇6'7
Work (area under 𝑃𝑉 curve)
Δ𝑈 = 𝑄 − 𝑊
%
𝑈 = $𝑛𝑅𝑇
𝑊∆_`" = 𝑃𝛥𝑉
𝑊 ∆?`" = 𝑛𝑅𝑇 ln(𝑉)+4 /𝑉+4+ )
Ideal gas
Processes (apply to 1st law)
𝛥𝑈 = 𝑛𝐶b ∆𝑇
#
𝐶b = $𝑁75, A) )6557A& 𝑅
Isothermal:
Isobaric:
Isochoric:
Adiabatic:
Cyclic:
e&
Power & Intensity
X
𝑣=AO47 ± 𝑣AN=56C56
𝑓W = 𝑓 ¨
© obs ln source
𝑣=AO47
rms speed
𝑇[ = 𝑇Y + 273.15 𝑘; = 𝑅/𝑁: = 1.38 × 10
𝐶c = 𝐶b + 𝑅
𝑃𝑉 d = constant
e
𝛾 = % adiabats
Boundary conditions
𝜆# 𝜆$
=
𝑣# 𝑣$
𝑓# = 𝑓$
Pipe standing waves
Doppler Effect
X
𝑣=AO47
𝑓W = 𝑓 ¨
© source ln obs
𝑣=AO47 ± 𝑣=AO6>5
Temperature & Kinetic Theory
Scales
Average kinetic energy
P
# }}}
$ = %𝑘 𝑇
𝑇Y = (𝑇< − 32) }}}}
𝐾𝐸 = 𝑚𝑣
;
𝑎 = −𝐴𝜔$ sin(𝜔𝑡 + 𝛿)
𝜔𝑥"
tan(𝛿) =
𝑣"
+75'* ,'=
By system 𝑊 > 0
On system 𝑊 < 0
Carnot engine
V
V
𝑄̇6'7 = 𝑒𝜎𝐴³𝑇=O66
− 𝑇NA7D
´
9]
$ V
𝜎 = 5.67 × 10 W/m K
Heat
Specific Heat
𝑄∆a`" = 𝑛𝐶a ∆𝑇
𝑄∆_`" = 𝑛𝐶_ ∆𝑇
𝑄 = 𝑚𝑐∆𝑇
𝑄 = 𝑚𝐿 Latent heat
∑R 𝑄R = 0 isolated system
Heat engine
Entropy (2nd Law)
∆𝑇 = 0 → ∆𝑈 = 0
𝑊 = 𝑄f − 𝑄U
𝛿𝑄 𝑄
𝑄
𝑇
∆𝑃 = 0 → 𝑊 = 𝑃∆𝑉 − U = U
=
𝑊
𝑄U Δ𝑆gO'=+ =.'.+> = ˆ
𝑇
𝑄f 𝑇f
𝑇
𝑒
=
=
1
−
∆𝑉 = 0 → 𝑊 = 0
𝑄f
𝑄f Δ𝑆
𝑇U
≥
0
+=A*'.57 =D=.5&
𝑒Y'64A. = 1 −
𝑄
𝑄 = 0 → ∆𝑈 =– 𝑊
𝑇f 𝜅 = − U
Δ𝑆 = 𝑆U − 𝑆f engine
∆𝑈 = 0 → 𝑄 = 𝑊>D>*5
𝑊