Formula Sheet – PHY 161 – Exam B Chapter 1: Introduction 4 Sphere (radius 𝑅): Area 𝐴 = 4𝜋𝑅 2 , Volume 𝑉 = 𝜋 𝑅 3 . 3 Cylinder (radius 𝑅, height ℎ): Area 𝐴 = 2𝜋𝑅 2 + 2𝜋𝑅ℎ, Volume 𝑉 = 𝜋𝑅 2 ℎ. Density (mass 𝑀, volume 𝑉): 𝜌 = 𝑀/𝑉. Chapter 2: One-Dimensional Kinematics Definitions Change (∆) is final value minus initial value: ∆𝑥 = 𝑥 − 𝑥0 , ∆𝑡 = 𝑡 − 𝑡0 , ∆𝑣 = 𝑣 − 𝑣0 . Instantaneous quantities: 𝑣 = 𝑑𝑥 𝑑𝑡 Average quantities: 𝑣̅ = 𝑣𝑎𝑣𝑔 = , 𝑎= ∆𝑥 ∆𝑡 𝑑𝑣 𝑑𝑡 = , 𝑎𝑎𝑣𝑔 = 𝑑2 𝑥 𝑑𝑡 2 ∆𝑣 ∆𝑡 . . Speed (𝑠 always positive or zero): 𝑠 = |𝑣| , 𝑠𝑎𝑣𝑔 = Acceleration due to gravity: 𝑎 = −𝑔 , 𝑔 = 9.8 Kinematic equations of motion If we let 𝑎 = constant and 𝑡0 = 0, then 1 𝑥 = 𝑥0 + 𝑣0 𝑡 + 𝑎𝑡 2 , 𝑣 = 𝑣0 + 𝑎𝑡, 2 𝑣 2 = 𝑣02 + 2𝑎∆𝑥 , 𝑣𝑎𝑣𝑔 = 𝑣0 +𝑣 2 . 𝑚 𝑠2 𝑡𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡 . , “g-factor”= 𝑎/𝑔. Chapter 3: Vectors Components: 𝑉𝑥 = 𝑉 cos 𝜃, 𝑉𝑦 = 𝑉 sin 𝜃, 𝑉 = √𝑉𝑥2 + 𝑉𝑦2 , tan 𝜃 = 𝑉𝑦 /𝑉𝑥 . ⃗⃗, if and only if 𝐶𝑥 = 𝐴𝑥 + 𝐵𝑥 and 𝐶𝑦 = 𝐴𝑦 + 𝐵𝑦 . Addition: 𝐶⃗ = 𝐴⃗ + 𝐵 ⃗⃗ = 𝐴𝑥 𝐵𝑥 + 𝐴𝑦 𝐵𝑦 + 𝐴𝑧 𝐵𝑧 = 𝐴𝐵 cos 𝜃. Scalar (dot) product: 𝐴⃗ ∙ 𝐵 Vector (cross) product: ⃗⃗ = (𝐴𝑦 𝐵𝑧 − 𝐴𝑧 𝐵𝑦 )𝑖̂ + (𝐴𝑧 𝐵𝑥 − 𝐴𝑥 𝐵𝑧 )𝑗̂ + (𝐴𝑥 𝐵𝑦 − 𝐴𝑦 𝐵𝑥 )𝑘̂. 𝐴⃗ × 𝐵 Chapter 4: Two-Dimensional Kinematics ⃗⃗𝐴𝐶 = 𝑉 ⃗⃗𝐴𝐵 + 𝑉 ⃗⃗𝐵𝐶 . Relative motion: 𝑉 Uniform circular motion (speed 𝑣, radius 𝑅, period 𝑇): 2 𝑎 = 𝑣 ⁄𝑅 , 𝑣 = 2𝜋𝑅⁄𝑇 . Chapter 5: Newton’s Laws, Part 1 Here 𝐹⃗ refers to the total force acting on an object, equal to the vector sum of all individual forces acting on the object: 𝐹⃗ = ⃗⃗⃗⃗ 𝐹1 + ⃗⃗⃗⃗⃗ 𝐹2 + ⃗⃗⃗⃗⃗ 𝐹3 + ⋯ First Law: 𝐹⃗ = 0 if and only if 𝑎⃗ = 0, where 𝑎⃗ is the object's acceleration. Second Law: 𝐹⃗ = 𝑚𝑎⃗, where 𝑚 is the object's (inertial) mass, a constant. Third Law: If two objects are interacting, then 𝐹⃗21 = −𝐹⃗12 . Weight: 𝑤 = 𝑚𝑔, where 𝑔 = 9.8𝑚/𝑠 2 .