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 .