Formula Sheet – PHY 161 – Exam D
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 .
Chapter 6: Newton’s Laws, Part 2
APPLICATIONS of Newton's Laws
Static friction: 𝑓𝑠 ≤ 𝑓𝑠,𝑚𝑎𝑥 = 𝜇𝑠 𝑁, where 𝜇𝑠 is a constant and 𝑁 is the normal
force.
Kinetic (or sliding) friction: 𝑓𝑘 = 𝜇𝑘 𝑁, where 𝜇𝑘 is a constant.
Centripetal force: This is required for circular motion. 𝐹 = 𝑚𝑣 2 /𝑅, where 𝑣 is
the speed and 𝑅 is the radius of curvature.
Drag force (air/fluid resistance): This can be parameterized as
𝐷 = 12𝐶𝜌𝐴𝑣 2 ,
where 𝜌is the fluid density, 𝐴 is the object’s cross-sectional area, 𝑣 is the
object’s speed relative to the fluid, and 𝐶 is parameter of order 1 called the drag
coefficient.
Chapter 7: Work and Energy
Kinetic Energy: 𝐾 = 12𝑚𝑣 2
Definition of work done by arbitrary force: 𝑑𝑊𝐹 = 𝐹⃗ ∙ 𝑑𝑥⃗
Work done by a constant force: 𝑊𝐹 = 𝐹⃗ ∙ ∆𝑥⃗
Work done by gravity: 𝑊𝑔 = −𝑚𝑔∆𝑦 , where ∆𝑦 is the change in elevation.
Work done by a spring: 𝐹𝑘 = −𝑘𝑥, 𝑊𝑘 = 12𝑘𝑥02 − 12𝑘𝑥 2
Work-Energy Theorem: 𝑊𝑎𝑙𝑙 𝑓𝑜𝑟𝑐𝑒𝑠 = ∆𝐾
Power: 𝑃𝑎𝑣𝑔 =
𝑊
∆𝑡
, 𝑃=
𝑑𝑊
𝑑𝑡
= 𝐹⃗ ∙ 𝑣⃗
Chapter 8: Conservation of Energy
Conservative force: 𝑊 (𝐴 → 𝐵) = 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑜𝑓 𝑝𝑎𝑡ℎ
Potential energy difference: only defined for conservative force,
∆𝑈(𝐴 → 𝐵) = −𝑊 (𝐴 → 𝐵)
Potential energy: where 𝑥0 is the reference point and we choose 𝑈0 at will,
𝑈(𝑥0 ) = 𝑈0 ,
𝑈(𝑥 ) = 𝑈(𝑥0 ) + ∆𝑈(𝑥0 → 𝑥 )
Force associated with potential energy: 𝐹 = −
𝑑𝑈
𝑑𝑥
Potential energy for specific forces
For gravity: ∆𝑈𝑔 = 𝑚𝑔∆𝑦, where 𝑦 is the elevation, or 𝑈𝑔 = 𝑚𝑔ℎ,
choosing the equilibrium point 𝑦0 = 0 as the reference point, 𝑈0 = 0, and ℎ = 𝑦.
For a spring-mass system: ∆𝑈𝑘 = 12𝑘𝑥 2 − 12𝑘𝑥02 → 𝑈𝑘 = 12𝑘𝑥 2 ,
choosing the equilibrium point 𝑥0 = 0 as the reference point, and 𝑈0 = 0.
Mechanical energy: 𝐸𝑚𝑒𝑐ℎ = 𝐾 + 𝑈, the sum of kinetic and potential energy.
Work-Energy Theorem (version 2): 𝑊𝑛𝑜𝑛−𝑐𝑜𝑛𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑣𝑒 𝑓𝑜𝑟𝑐𝑒𝑠 = ∆𝐸𝑚𝑒𝑐ℎ
Work-Energy Theorem (version 3):
𝑊𝑒𝑥𝑡𝑒𝑟𝑛𝑎𝑙 𝑓𝑜𝑟𝑐𝑒𝑠 = ∆𝐸 = ∆𝐸𝑚𝑒𝑐ℎ + ∆𝐸𝑡ℎ𝑒𝑟𝑚𝑎𝑙 + ∆𝐸𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙
Work-Energy Theorem (isolated system): 𝑊𝑒𝑥𝑡𝑒𝑟𝑛𝑎𝑙 𝑓𝑜𝑟𝑐𝑒𝑠 = 0
Chapter 9: Linear Momentum and Collisions
Systems of Particles
Center of Mass: In the following the particles are numbered, 𝑖 = 1,2,3, … , 𝑁.
The system total mass is 𝑀 = ∑𝑁
𝑖=1 𝑚𝑖 .
1
Position: 𝑅⃗⃗𝐶𝑀 = ( ) ∑𝑁
𝑖=1 𝑚𝑖 𝑟⃗𝑖 , and its components
1
𝑀
𝑋𝐶𝑀 = ( ) ∑𝑁
𝑖=1 𝑚𝑖 𝑥𝑖 ,
𝑀
1
𝑌𝐶𝑀 = ( ) ∑𝑁
𝑖=1 𝑚𝑖 𝑦𝑖 .
𝑀
⃗⃗𝐶𝑀 = 𝑑 𝑅⃗⃗𝐶𝑀 , and 𝐴⃗𝐶𝑀 = 𝑑 𝑉
⃗⃗ .
Velocity and acceleration: 𝑉
𝑑𝑡
𝑑𝑡 𝐶𝑀
𝑑𝑝⃗
Momentum, single particle: 𝑝⃗ = 𝑚𝑣⃗, 𝐹⃗ = 𝑚𝑎⃗ = .
Total Momentum: 𝑃⃗⃗𝑡𝑜𝑡𝑎𝑙 =
Impulse: 𝐽⃗ = ∆𝑝⃗ = 𝐹𝑎𝑣𝑔 ∆𝑡.
∑𝑁
𝑖=1 𝑚𝑖
𝑑𝑡
⃗⃗𝐶𝑀 , 𝐹⃗𝑡𝑜𝑡𝑎𝑙 = 𝑀𝐴⃗𝐶𝑀 =
𝑣⃗𝑖 = 𝑀𝑉
𝑑
⃗⃗
𝑃
.
𝑑𝑡 𝑡𝑜𝑡𝑎𝑙
Collisions
Completely inelastic (sticking) collisions: When two particles with masses
(𝑚1 , 𝑚2 ) and initial velocities (𝑣1 , 𝑣2 ) collide and stick together, they share a
final velocity 𝑉. Conservation of momentum gives
𝑚1 𝑣1 + 𝑚2 𝑣2 = (𝑚1 + 𝑚2 )𝑉.
Elastic collisions: When two particles with masses (𝑚1 , 𝑚2 ) and initial velocities
(𝑣1 , 𝑣2 ) undergo an elastic collision their final velocities are (𝑣′1 , 𝑣′2 ):
𝑣1′ = [
𝑚1 − 𝑚2
2𝑚2
] 𝑣1 + [
]𝑣 ,
𝑚1 + 𝑚2
𝑚1 + 𝑚2 2
2𝑚1
𝑚2 − 𝑚1
𝑣2′ = [
] 𝑣1 + [
]𝑣 .
𝑚1 + 𝑚2
𝑚1 + 𝑚2 2