Unit 2 Student Formula Sheet
This sheet is allowed on the Unit 2 test. It gives you the equations — it does NOT tell you which one to use or how to set up a problem. Knowing when to reach for each equation is the actual skill the test checks.
1. Systems and Center of Mass (Topic 2.1)
Variable definitions: $x_{cm}$ = center of mass position, $m_i$ = mass of object $i$, $x_i$ = position of object $i$.
$$x_{cm}=\dfrac{\sum m_ix_i}{\sum m_i}$$
Tips:
- This is a weighted average — heavier masses count more.
- Works the same way in 2D: find $x_{cm}$ and $y_{cm}$ separately.
2. Forces, Free-Body Diagrams, and Newton's Laws (Topics 2.2–2.5)
Variable definitions: $F$ = force (N), $m$ = mass (kg), $a$ = acceleration (m/s²), $g$ = $9.8\text{ m/s}^2$, $\theta$ = angle from horizontal.
| Equation | Use |
|---|---|
| $\sum F=ma$ | Newton's Second Law — net force causes acceleration |
| $F_{g,weight}=mg$ | Weight (force of gravity on an object near Earth's surface) |
| $F_{on,A,by,B}=-F_{on,B,by,A}$ | Newton's Third Law — equal, opposite, on two different objects |
| $F_{\parallel}=mg\sin\theta$ | Component of gravity along an incline |
| $F_{\perp}=mg\cos\theta$ | Component of gravity into an incline |
Tips:
- Equilibrium ($a=0$) means $\sum F=0$ — forces still exist, they just balance.
- On an incline, rotate your axes to run parallel/perpendicular to the surface.
- A connected system (two blocks, a block and a hanging mass) can be treated as one object to find acceleration; isolate a single object only to solve for internal tension.
- A hand pulling with force $F$ is NOT the same as hanging a weight of that same size $F$ — the hanging object adds its own mass to the system.
3. Gravitational Force (Topic 2.6)
Variable definitions: $F_g$ = gravitational force (N), $G=6.67\times10^{-11}\text{ N}\cdot\text{m}^2/\text{kg}^2$, $m_1,m_2$ = masses, $r$ = distance between centers.
$$F_g=\dfrac{Gm_1m_2}{r^2}$$
Tips:
- Inverse-square law: doubling $r$ divides the force by 4, not 2.
- $r$ is measured center-to-center, never surface-to-surface.
- With three or more objects, add forces as vectors — same-direction pulls add, opposite-direction pulls can cancel.
4. Friction (Topic 2.7)
Variable definitions: $F_{f,s}$ = static friction, $F_{f,k}$ = kinetic friction, $\mu_s$ = coefficient of static friction, $\mu_k$ = coefficient of kinetic friction, $F_N$ = normal force.
| Equation | Use |
|---|---|
| $F_{f,s}\le\mu_sF_N$ | Static friction — adjusts up to a maximum, object not sliding |
| $F_{f,k}=\mu_kF_N$ | Kinetic friction — fixed value, object already sliding |
Tips:
- "On the verge of slipping" is the one moment you can treat static friction as exactly $\mu_sF_N$.
- $\mu_s$ is always $\ge\mu_k$ for the same surfaces.
- Friction depends on normal force, not on surface area or speed.
5. Spring Forces (Topic 2.8)
Variable definitions: $F_s$ = spring force (N), $k$ = spring constant (N/m), $\Delta x$ = displacement from natural length.
$$F_s=k\Delta x$$
Tips:
- $k$ is a fixed property of the spring — find it once from any known trial, then reuse it.
- The spring force always points back toward the spring's natural (unstretched) length.
6. Circular Motion (Topic 2.9)
Variable definitions: $a_c$ = centripetal acceleration, $v$ = speed, $r$ = radius, $\omega$ = angular speed (rad/s), $F_c$ = net centripetal force.
| Equation | Use |
|---|---|
| $a_c=\dfrac{v^2}{r}$ | Centripetal acceleration from linear speed |
| $a_c=\omega^2r$ | Centripetal acceleration from angular speed |
| $F_c=\dfrac{mv^2}{r}=m\omega^2r$ | Net centripetal force |
| $v_{min}=\sqrt{gr}$ | Minimum speed at the top of a vertical circle |
Tips:
- Centripetal force is never a new, separate force — it's whatever real force (tension, friction, gravity, normal force) happens to point toward the center at that moment.
- At constant angular speed $\omega$, points farther from the center ($r$ larger) have MORE centripetal acceleration, not less.
- For a conical pendulum, the radius of the circle is $L\sin\theta$, not the full string length $L$.
- Vertical circle, top of the loop: $F_N+mg=\dfrac{mv^2}{r}$. Bottom of the loop: $F_N-mg=\dfrac{mv^2}{r}$.
Orbits (Kepler's Third Law):
$$v_{orbit}=\sqrt{\dfrac{GM}{r}} \qquad\qquad T=2\pi\sqrt{\dfrac{r^3}{GM}} \qquad\qquad \dfrac{T_A^2}{T_B^2}=\dfrac{r_A^3}{r_B^3}$$
- These come from setting gravitational force equal to the required centripetal force: $\dfrac{GMm}{r^2}=\dfrac{mv^2}{r}$ — mass of the orbiting object always cancels out.
- Tripling the orbital radius means the new orbital speed is the old speed divided by $\sqrt{3}$, not divided by 3.
- Kepler's Third Law compares two orbits around the SAME central body — $T^2\propto r^3$ for any pair of orbits sharing the same $M$.
This sheet is not collected and does not count toward your grade — it's here so you can spend your time thinking instead of memorizing.