Week 5 Cheatsheet — Newton's Laws, Friction, and Circular Motion

How this week breaks down

Two big applications of $\sum F=ma$. Skim this once, then revise from the in-depth note.

Topic	What you do
Coupled-body dynamics	One FBD per body, $\sum F_x=m{a}_x$ per body, solve simultaneous equations
Inclined plane + friction	Tilt axes along/perp to slope; weight splits into $mg\sin \theta$ (along) and $mg\cos \theta$ (perp)
Uniform circular motion	Net inward force $=m{v}^2/r$. Centripetal is not new — it’s the net

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1 — FBD workflow (slide 13)

1. Identify the body in question (one body at a time).
2. Define coordinate axes (choose them to make the algebra easy — along the slope, or toward the circle centre).
3. Model the object as a particle at the origin.
4. Represent every force as a labelled vector arrow from the origin.

Then write $\sum F_x=m{a}_x$ and $\sum F_y=m{a}_y$ from the diagram.

Common forces — quick reference

Force	Symbol	Magnitude	Direction
Weight	$F_W$	$mg$	Vertically down
Normal	$F_N$	from $\sum F_{\perp }=0$	Perpendicular to surface
Tension	$F_T$ or $T$	from equation	Along string, away from object
Spring	$F_s$	$kx$	Toward equilibrium (restoring)
Kinetic friction	$F_{fric}$	${\mu }_k{F}_N$	Opposite to motion
Static friction	$F_{fric}$	$\le {\mu }_s{F}_N$	Opposite to tendency of motion

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2 — Newton’s laws (slide 14)

$$\text{1st: }{F}_{net}=0\text{2nd: }{\vec{F}}_{net}=m\vec{a}\text{3rd: }{\vec{F}}_{A\to B}=-{\vec{F}}_{B\to A}$$

Component form (per axis):

$$\sum F_x=m{a}_x,\sum F_y=m{a}_y.$$

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3 — Friction (slide 12)

${\vec{F}}_{fric}=F_N\mu$

Type	When it applies	Formula
Static	Object at rest (or on the verge)	$F_{fric}\le {\mu }_s{F}_N$
Kinetic	Object already sliding	$F_{fric}={\mu }_k{F}_N$

Sliding test on an incline. For no motion: ${\mu }_s^{\text{req}}=\tan \theta$. If actual ${\mu }_s<\tan \theta$ the body slides; switch to ${\mu }_k$ for acceleration.

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4 — Inclined plane recipe (Example 2, slide 17)

Choose $x$ along the slope (down-slope positive), $y$ perpendicular.

Axis	Equation
$\sum F_y=0$	$F_N=mg\cos \theta$
$\sum F_x=ma$	$ma=mg\sin \theta -\mu F_N$

Combine:

$$a=g\sin \theta -{\mu }_{k}g\cos \theta .$$

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5 — Uniform circular motion (slides 19–24)

Quantity	Formula	Notes
Centripetal acceleration	$a_c=v^2/r$	Toward circle centre
Centripetal force (net)	$F_c=m{v}^2/r$	Sum of real forces resolved inward; not a new force

Standard set-ups

Set-up	$\sum F_y$	$\sum F_x$ (toward centre)
Conical pendulum, string at $\theta$ from horizontal, $r=L\cos \theta$	$T\sin \theta -mg=0$	$T\cos \theta =m{v}^2/r$
Banked curve, no friction, bank $\theta$	$F_N\cos \theta -mg=0$	$F_N\sin \theta =m{v}^2/r$
Vertical circle, top	n/a	$T+mg=m{v}^2/r$
Vertical circle, bottom	n/a	$T-mg=m{v}^2/r$
Unbanked curve, friction only	$F_N=mg$	$\mu F_N=m{v}^2/r$

The banked-curve design equation:

$$\tan \theta =\frac{v^2}{rg}.$$

Mass cancels — useful sanity check.

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6 — Coupled bodies (Example 1, slide 16)

Two equivalent methods.

Method	When to use
Separate FBDs, equate accelerations	Always works; needed if forces between the bodies are unknown
Treat both bodies as one system	Faster when only the common acceleration is needed

Recipe (two-body system)

1. FBD body A → $\sum F=m_{A}a$.
2. FBD body B → $\sum F=m_{B}a$ (note: same $a$, opposite signs on the internal force).
3. Add the equations to eliminate the internal force, solve for $a$.
4. Plug $a$ back into either equation to recover the internal force (tension / spring stretch).

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Common mistakes

1. Adding centripetal force as an extra arrow on the FBD. It is not a new force — it is the net of the real forces resolved toward the centre.
2. Confusing constant speed with constant velocity. Uniform circular motion has constant speed but changing direction, so there is acceleration and a non-zero net force.
3. Wrong angle convention on the conical pendulum. In this course $\theta$ is from the horizontal, so $r=L\cos \theta$ (not $L\sin \theta$).
4. Skipping the sliding test on an incline. Compare ${\mu }_s$ to $\tan \theta$. If the object doesn’t slide, $a=0$ and $F_{fric}$ takes whatever value balances $mg\sin \theta$.
5. Negative coefficient of friction in algebra. Always positive by definition; a negative sign means you drew $F_{fric}$ in the wrong direction on the FBD.
6. Forgetting unit conversion. km/h $\to$ m/s by dividing by 3.6 before plugging into $a_c=v^2/r$.
7. Spring + circular motion confusion. Orbit radius $R$ includes the natural length: extension is $R-x_0$, so spring force = $k(R-x_0)$.
8. Setting the centre as negative direction. Always pick “toward centre” as positive when writing the radial equation — sign errors get expensive otherwise.

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Key formulas

$${\vec{F}}_{net}=m\vec{a}{F}_W=mg{F}_s=kx{F}_{fric}=\mu F_N$$ $$F_N^{\text{incline}}=mg\cos \theta a_{\text{slide}}=g\sin \theta -{\mu }_{k}g\cos \theta$$ $$a_c=\frac{v^2}{r}{F}_c=\frac{m{v}^2}{r}\tan {\theta }_{\text{bank}}=\frac{v^2}{rg}$$ $$\text{Conical pendulum:}T\sin \theta =mg,T\cos \theta =\frac{m{v}^2}{r},r=L\cos \theta$$ $$\text{Vertical circle:}{T}_{\text{top}}=\frac{m{v}^2}{r}-mg,T_{\text{bot}}=\frac{m{v}^2}{r}+mg$$

For the why and a full exam-style worked example, see the in-depth note.