Twenty minutes or less.

Week 5 — Dynamic Systems and Uniform Circular Motion. Pick a mode. Start a timer. That's it.

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The shortest path to walking in prepared.

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5-minute version

Two big set-ups. One sentence each.

Coupled bodies / inclines / friction — draw an FBD per body, write $\sum F = ma$ per axis, decide static vs kinetic friction.
Uniform circular motion — the net inward force equals $m v^{2} / r$ ; do not draw a separate $F_{c}$ arrow.

Open the cheatsheet quiz, do 3 easy questions, close it. You’re prepped.

Time	Action
0–5 min	Skim the cheatsheet tables (FBD workflow, circular motion set-ups).
5–10 min	Re-do one worked example from each topic by hand — covering pen: Example 1 (coupled blocks), Example 2 (incline), Example 3 (vertical circle).
10–15 min	Take the cheatsheet quiz. Don’t worry about the score.
15–20 min	Read the matching “common mistakes” section in the in-depth note.

Most students slip on three specific things in this week:

Adding a centripetal force arrow to the FBD of a circling body. It’s already there as the net of the real forces.
Skipping the sliding test on an incline and immediately using $μ_{k}$ without checking $μ_{s}$ vs $tan θ$ .
Wrong angle convention on the conical pendulum. In this course $θ$ is from the horizontal, so $r = L cos θ$ (not $L sin θ$ ).

F_{n e t} = m a F_{W} = m g F_{s} = k x F_{f r i c} = μ F_{N}

F_{N}^{incline} = m g cos θ a_{slide} = g sin θ - μ_{k} g cos θ μ_{s}^{req} = tan θ

a_{c} = \frac{v ^{2}}{r} F_{c} = \frac{m v ^{2}}{r} tan θ_{bank} = \frac{v ^{2}}{r g}

T_{top} = \frac{m v ^{2}}{r} - m g T_{bot} = \frac{m v ^{2}}{r} + m g

Task type	What the setup usually looks like
Two-body FBD	Two blocks on a surface joined by a rope/spring; find $a$ and the internal force
Incline + friction	Block on a ramp; decide slide/no-slide, then find $a$
Unbanked turn	Car on a flat curve; find required $μ$ from $v$ and $r$
Banked curve	No friction; find $θ$ for a given design speed
Conical pendulum	String at $θ$ from horizontal; find $v$
Vertical circle	Find tension at top vs bottom

Adding $F_{c}$ as a separate arrow on the FBD. It’s the net inward force.
Forgetting to convert km/h $\to$ m/s (divide by 3.6) before $a_{c} = v^{2} / r$ .
Using $μ_{s}$ when the object is already sliding (use $μ_{k}$ ); using $μ_{k}$ when checking whether it slides (compare $μ_{s}$ with $tan θ$ ).
Drawing the normal force not perpendicular to the surface.
Treating a negative answer for $μ$ as a real number — it just means a sign mistake on the FBD.
Forgetting the spring’s natural length: orbit radius $R = x_{0} + x$ , so spring force is $k (R - x_{0})$ .
Skipping units in the final answer.

Try these without notes. Six minutes total.

A 5 kg block sits on a 25° incline; $μ_{s} = 0.6$ . Does it slide?
A 1500 kg car takes an unbanked 200 m radius curve at 25 m/s. What is the minimum $μ$ needed?
A 0.4 kg ball on a 1.0 m string moves in a vertical circle at $v = 6$ m/s. What is the tension at the bottom?
Two blocks (1 kg and 2 kg) on a frictionless surface are pushed together by a 9 N force on the 2 kg block. What is the contact force between them?

Answers:

Q	Answer	Working hint
1	No, doesn’t slide	$tan 25° \approx 0.466 < 0.6 = μ_{s}$ , so static friction is sufficient
2	$μ \approx 0.32$	$μ = v^{2} / (r g) = 625/ (200 \times 9.8)$
3	$T = 18.32$ N	$T = m v^{2} / r + m g = 0.4 (36) /1 + 0.4 (9.8) = 14.4 + 3.92$
4	$F_{C} = 3$ N	System: $a = 9/3 = 3$ ; 1 kg FBD: $F_{C} = 1 \times 3 = 3$ N

Read the cheatsheet tables (FBD workflow and circular motion set-ups).
Re-derived $a_{c} = v^{2} / r$ direction (or convinced yourself it points inward).
Re-did one worked example by hand for each of: coupled blocks, incline, vertical circle.
Cheatsheet quiz attempted.
Mini self-test attempted.

That’s it. Close the laptop.