Final Exam Chit Sheet : SHM, Waves and Fluids

Simple harmonic motion (mass-Spring system)
Hooke’s Law
𝐹 = −𝑘𝑥
Basic defns
𝑘
2𝜋
𝜔 = √ = 2𝜋𝑓 =
𝑚
𝑇
𝑇=
2𝜋
𝑚
= 2𝜋√
𝜔
𝑘
Position as a function of time
𝑥(𝑡) = 𝐴 𝑐𝑜𝑠𝜔𝑡
Velocity and acceleration
𝑣𝑚𝑎𝑥 = 𝑣(𝑚𝑖𝑑𝑑𝑙𝑒) = 𝜔𝐴
𝑎𝑚𝑎𝑥 = 𝑎(𝑒𝑑𝑔𝑒𝑠) = 𝜔2 𝐴
Energy
1 2
𝑘𝑥
2
1
𝐾. 𝐸 = 𝑚𝑣 2
2
𝑃𝐸 =
𝐸𝑡𝑜𝑡 = 𝑃. 𝐸 + 𝐾. 𝐸 =
1 2
𝑘𝐴
2
Simple Pendulum (small oscillations hence simple harmonic like spring)
𝑙
𝑇 = 2𝜋√
𝑔

Waves
Transverse Wave eqn
𝑦(𝑥, 𝑡) = 𝐴 sin(𝑘𝑥 − 𝜔𝑡)
Basic defns
2𝜋
, 𝜔 = 2𝜋𝑓
𝜆
𝜔
𝑣 = = 𝑓𝜆
𝑘
𝑘=
𝑣𝑚𝑎𝑥 = 𝜔𝐴
Max transverse speed
Speed of waves do not change unless they change medium

Fluid Statics
𝐹
𝐴
𝑚
𝜌=
𝑉
𝑃 = ℎ𝜌𝑔
𝑃=
𝑃𝑡𝑜𝑡 = ℎ𝜌𝑔 + 𝑃𝑎𝑡𝑚
𝑘𝑔
𝑘𝑔
𝑘𝑔
(𝜌𝑤 = 1000 𝑚3 , 𝜌𝑎𝑖𝑟 = 1.20 𝑚3 , 𝜌𝐻𝑔 = 13560 𝑚3 , 𝑃𝑎𝑡𝑚 = 101 𝑘𝑃𝑎)
𝐹𝑏 = 𝑤𝑡 𝑜𝑓 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑑 𝑤𝑎𝑡𝑒𝑟 = 𝜌𝑤 𝑉𝑖𝑛𝑠𝑖𝑑𝑒 𝑔
𝑊𝑎𝑝𝑝 = (𝜌𝑜𝑏𝑗 − 𝜌𝑤 )𝑉𝑖𝑛 𝑔
𝜌𝑜𝑏𝑗
𝜌𝑜𝑏𝑗
𝑉𝑜𝑢𝑡 = 𝑉𝑡𝑜𝑡 (1 −
) 𝑉𝑖𝑛 = 𝑉𝑡𝑜𝑡
𝜌𝜔
𝜌𝜔
𝜌𝑤
𝑎𝑑𝑜𝑤𝑛𝑤𝑎𝑟𝑑 = (1 −
)𝑔
𝜌𝑜𝑏𝑗
𝑚̇ = 𝜌𝑉̇ = 𝜌𝐴𝑣
Steady flow (𝑚̇ 1 = 𝑚̇ 2 ) leads to Continuity eqn
𝑣1 𝐴1 = 𝑣2 𝐴2
Bernoulli s eqn
1
1
𝑃1 + 𝜌𝑣12 + 𝜌𝑔𝑦1 = 𝑃2 + 𝜌𝑣22 + 𝜌𝑔𝑦2
2
2
For horizontal pipe: Narrow pipe, more pressure
For vertical pipe: More depth, more pressure