Phys M20A—Concept Summary via Equations

Ch 2: Motion in 1D

𝑣 =𝑑𝑥

𝑑𝑡𝑎 =

𝑑𝑣

𝑑𝑡=

𝑑2𝑥

𝑑𝑡2

𝑥(𝑡) = 𝑥𝑖 + 𝑣𝑖(𝑡 − 𝑡𝑖) +1

2𝑎(𝑡 − 𝑡𝑖)

2

𝑣(𝑡) = 𝑣𝑖 + 𝑎 (𝑡 − 𝑡𝑖)

𝑣2 = 𝑣𝑖2 + 2𝑎 𝑥𝑓 − 𝑥𝑖

Ch4: Motion in 2D

𝑥(𝑡) = 𝑥𝑖 + 𝑣𝑖𝑥 𝑡 − 𝑡𝑖 +1

2𝑎𝑥 𝑡 − 𝑡𝑖

2

𝑣𝑥(𝑡) = 𝑣𝑖𝑥 + 𝑎𝑥(𝑡 − 𝑡𝑖)

𝑣𝑥2 = 𝑣𝑖𝑥

2 + 2𝑎𝑥(𝑥 − 𝑥𝑖)

𝑦(𝑡) = 𝑦𝑖 + 𝑣𝑖𝑦 𝑡 − 𝑡𝑖 +1

2𝑎𝑦 𝑡 − 𝑡𝑖

2

𝑣𝑦(𝑡) = 𝑣𝑖𝑦 + 𝑎𝑦(𝑡 − 𝑡𝑖)

𝑣𝑦2 = 𝑣𝑖𝑦

2 + 2𝑎𝑦(𝑥 − 𝑥𝑖)

𝑎𝑐 =𝑣2

𝑟

Ch4: Projectile MotionNOTE: NOT on equation sheet!

𝑥 = 𝑥𝑖 + 𝑣𝑖𝑥∆𝑡𝑣𝑥 = 𝑣𝑖𝑥

𝑦 = 𝑦𝑖 + 𝑣𝑖𝑦∆𝑡 −1

2𝑔∆𝑡2

𝑣𝑦 = 𝑣𝑖𝑦 − 𝑔∆𝑡

𝑣𝑦2 = 𝑣𝑖𝑦

2 − 2𝑔(𝑦 − 𝑦𝑖)

𝑎𝑥 = 0 𝑎𝑦 = −𝑔

Ch5: Newton’s Laws 𝐹 = 𝑚 𝑎

1. Draw free-body (force) diagrams for each object – External forces acting on object

2. Find components of forces from diagrams3. Add forces minding signs—”LHS” 4. Determine the state of the object: accelerating?

– Write down as “RHS”

5. Make sure # eqns = # unknows– If not then may need another eqn (from another object,

etc.)

6. Solve for unknowns

Ch5: Newton’s Laws: Friction

𝑓𝑘 = 𝜇𝑘𝑛 (kinetic friction)

𝑓𝑠 ≤ 𝜇𝑠𝑛 (static friction)

Ch 6: Circular Motion & Applications

𝑎𝑐 =𝑣2

𝑟

𝑅 = −𝑏𝑣

𝑣𝑇 =𝑚𝑔

𝑏

𝑣 𝑡 = 𝑣𝑇 1 − 𝑒−𝑏𝑡/𝑚

Ch 7: Energy of a System𝑊 = 𝐹 ⋅ 𝑑 𝑟

𝑊 = 𝐹 ∆ 𝑟 cos 𝜃

𝐾 =1

2𝑚𝑣2

𝑈𝑔 = 𝑚𝑔𝑦 (gravity)

𝑈𝑠 =1

2𝑘𝑥2 (spring)

𝑊𝑛𝑒𝑡 = ∆𝐾𝑊𝐶 = −∆𝑈

𝐹𝑥 = −𝑑𝑈

𝑑𝑥 𝐹 = −𝛻𝑈

Ch 8: Conservation of Energy𝑊𝑁𝐶 = ∆𝐾 + ∆𝑈

𝑃 =𝑑𝑊

𝑑𝑡= 𝐹 ⋅ 𝑣

Ch 9: Momentum

𝑝 = 𝑚 𝑣

𝐹 =𝑑 𝑝

𝑑𝑡

𝐼 = 𝐹∆𝑡

𝑝𝑖 = 𝑝𝑓

𝑣𝐴𝑖 − 𝑣𝐵𝑖 = −(𝑣𝐴𝑓 − 𝑣𝐵𝑓) (elastic collisions)

𝑥𝐶𝑀 = 𝑚𝑖𝑥𝑖 𝑚𝑖

𝑥𝐶𝑀 = 𝑥𝑑𝑚

𝑑𝑚

Ch 10: Rotational Motion-Kinematics

𝜔 =𝑑𝜃

𝑑𝑡𝛼 =

𝑑𝜔

𝑑𝑡

𝜃 = 𝜃0 + 𝜔0Δ𝑡 +1

2𝛼Δ𝑡2

𝜔 = 𝜔0 + 𝛼Δ𝑡

𝜔2 = 𝜔02 + 2𝛼Δ𝜃

𝑠 = 𝑟𝜃𝑣 = 𝑟𝜔𝑎𝑡 = 𝑟𝛼

𝑎𝑟 =𝑣2

𝑟= 𝑟𝜔2

Ch 10: Rotational Motion-Dynamics

𝜏 = 𝑟 𝐹 sin𝜑

𝜏 = 𝐼𝛼

𝐾 =1

2𝐼𝜔2 (rotational)

𝐼 = 𝑟𝑖2𝑚𝑖 𝐼 = 𝑟2𝑑𝑚

𝑊 = 𝜏𝑑𝜃

𝑃 = 𝜏𝜔

Ch 11: Angular Momentum

𝜏 = 𝑟 × 𝐹

𝐿 = 𝑟 × 𝑝

𝐿𝑧 = 𝐼𝜔

𝜏 =𝑑𝐿

𝑑𝑡

Ch 12: Equilibrium

𝐹 = 0

𝜏 = 0

Ch 13: Universal Gravitation

𝐹𝑔 =𝐺𝑀𝑚

𝑟2

𝑔 = 𝐹𝑔

𝑚

𝑈 = −𝐺𝑀𝑚

𝑟

Ch 14: Fluids

𝐵 = 𝜌𝑓𝑙𝑢𝑖𝑑𝑉𝑑𝑖𝑠𝑝𝑔

𝑝 =𝐹

𝐴𝐴1𝑣1 = 𝐴2𝑣2

𝑝1 +1

2𝜌𝑣1

2 + 𝜌𝑔𝑦1 = 𝑝2 +1

2𝜌𝑣2

2 + 𝜌𝑔𝑦2

Ch 15: Oscillatory Motion

𝑑2𝑥

𝑑𝑡2= −𝜔2𝑥

𝑥 𝑡 = 𝐴 cos 𝜔𝑡 + 𝜙

𝜔 =𝑘

𝑚

𝜔 =𝑔

ℓ