________
Fluids:
Formula Sheet, Phys 101
________
𝑃 = 𝑃0 + 𝜌𝑔ℎ
𝐴1 𝑣1 = 𝐴2 𝑣2
1
𝑃 + 𝜌𝑣 2 + 𝜌𝑔ℎ = constant
2
𝐹𝐵 = 𝑚𝑓 𝑔 = 𝜌𝑓 𝑔𝑉
__
𝑃gauge = 𝑃 − 𝑃0
𝜌=
𝑚
𝑉
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Harmonic
Motion:
𝐹 = −𝑘𝑥
1
𝑃𝐸 = 𝑘𝑥 2
2
1
𝐾𝐸 = 𝑚𝑣 2
2
𝑥(𝑡 ) = 𝐴 cos(𝜔𝑡 + 𝜙0 )
𝑣(𝑡 ) = −𝐴𝜔 sin(𝜔𝑡 + 𝜙0 )
𝑎(𝑡 ) = −𝐴𝜔 2 cos(𝜔𝑡 + 𝜙0 ) = −𝜔 2 𝑥(𝑡 )
𝜔 = 2𝜋𝑓 =
2𝜋
𝑇
1
(𝑓 = )
𝑇
1
1
2
𝐸 = 𝑘𝐴2 = 𝑚𝑣max
2
2
𝐸 (𝑡 ) = 𝐸0 𝑒 −𝑏𝑡⁄𝑚 = 𝐸0 𝑒 −2𝑡/𝜏𝐴
𝜔=√
𝑥max (𝑡 ) =
𝑔
𝜔=√
𝐿
𝑘
(spring)
𝑚
𝐴𝑒 −𝑏𝑡⁄2𝑚
=
𝐴𝑒 −𝑡/𝜏𝐴
2
𝑘
𝑏2
√1 − ( 𝑇0 )
𝜔𝐷 = √ −
=
𝜔
0
𝑚 4𝑚2
2𝜋𝜏𝐴
(simple pendulum)
𝑥(𝑡 ) = 𝐴𝑒 −𝑏𝑡⁄2𝑚 cos(𝜔𝐷 𝑡 + 𝜙0 )
= 𝐴𝑒 −𝑡⁄𝜏𝐴 cos(𝜔𝐷 𝑡 + 𝜙0 )
, where 𝜔0 = √
𝑘
𝑚
and 𝑇0 = 2𝜋/𝜔0 for a
damped mass on a spring
________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ ______________________________
Travelling
Waves:
𝑘=
2𝜋
𝜆
General equation for a travelling
wave:
𝑣 = 𝑓𝜆 =
𝜔
𝑘
𝑇
𝑣=√
𝜇
𝐷(𝑥, 𝑡 ) = 𝐷𝑀 sin(𝑘𝑥 ∓ 𝜔𝑡 + 𝜙0 )
________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ ______________________________
Sound:
𝐼=
𝑃
𝐴
𝐼=
𝐼
𝛽 (dB) = 10 log10 ( )
𝐼0
𝑃
(spherical wave)
4𝜋𝑟 2
𝑣
1± 𝑜
𝑣 ± 𝑣𝑜
𝑣
𝑓 = 𝑓0 (
𝑣 ) = 𝑓0 ( 𝑣 ∓ 𝑣 )
𝑠
1∓ 𝑠
𝑣
𝑓𝑏𝑒𝑎𝑡 = |𝑓2 − 𝑓1 |
________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ ______________________________
Interference
&
diffraction:
𝑛=
𝑐
𝑣
2-slit interference,
with small angle approximation
(𝑚 = 0, ±1, ±2, … ):
𝜆𝑛 =
𝜆
𝑛
Bright fringes:
𝑦bright = 𝑚
𝜆𝐿
𝑑
Phase difference for interference:
Δ𝑥
Δ𝜙 = 2𝜋
+ Δ𝜙0
𝜆
Dark fringes:
1 𝜆𝐿
𝑦dark = (𝑚 + 2)
𝑑
𝜆
Single-slit diffraction minima with small angle approximation:
Circular aperture: 𝜃min = 1.22
𝐷
𝜆𝐿
𝑦dark = 𝑝 , where 𝑝 = ±1, ±2, …
𝑎
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Areas &
volumes:
𝐴(circle) = 𝜋𝑟 2
𝐴(sphere) = 4𝜋𝑟 2
4
𝑉(sphere) = 𝜋𝑟 3
3
𝑉(cylinder) = 𝜋𝑟 2 ℎ
________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ ______________________________
Constants:
𝑃0 = 𝑃atmosphere = 1.013 × 105 Pa;
𝜌fresh water = 1.00 × 103 kg⁄m3 ;
3
⁄
𝜌air = 1.29 kg m ;
𝜌mercury = 13.6 × 103 kg⁄m3 ;
−12
2
⁄
(
𝐼0 = 10
W m ;
𝑣sound in air) = 343 m⁄s
𝑔 = 9.81 m⁄s 2