Equation List
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Summary
| Metric | Count |
|---|---|
| Total Equations | 535 |
Display Equations ($$...$$) |
287 |
Inline Equations ($...$) |
248 |
Equations by Chapter
Other Content
96 equations (47 display, 49 inline)
| # | Type | Equation | Source |
|---|---|---|---|
| 1 | Display | \(\(m_1v_{1x,i} + m_2v_{2x,i} = m_1v_{1x,f} + m_2v_{2x,f}\)\) | Line 21 |
| 2 | Display | \(\(m_1v_{1y,i} + m_2v_{2y,i} = m_1v_{1y,f} + m_2v_{2y,f}\)\) | Line 24 |
| 3 | Inline | \(v_{0x} = v_0 \cos\theta\) | Line 19 |
| 4 | Inline | \(v_{0y} = v_0 \sin\theta\) | Line 20 |
| 5 | Inline | \(y_{max} = y_0 + \frac{v_{0y}^2}{2g}\) | Line 21 |
| 6 | Inline | \(R = \frac{v_0^2 \sin(2\theta)}{g}\) | Line 22 |
| 7 | Inline | \(F_x = F \cos\theta\) | Line 18 |
| 8 | Inline | \(F_y = F \sin\theta\) | Line 19 |
| 9 | Inline | \(N = mg - F_y\) | Line 20 |
| 10 | Inline | \(f_k = \mu_k N\) | Line 21 |
| 11 | Inline | \(a = (F_x - f_k) / m\) | Line 22 |
| 12 | Inline | \(T - m_1g = m_1a\) | Line 34 |
| 13 | Inline | \(m_2g - T = m_2a\) | Line 35 |
| 14 | Inline | \(a = \frac{(m_2 - m_1)g}{m_1 + m_2}\) | Line 36 |
| 15 | Inline | \(T = \frac{2m_1m_2g}{m_1 + m_2}\) | Line 37 |
| 16 | Inline | \(N \cos\theta = mg\) | Line 18 |
| 17 | Inline | \(N \sin\theta + f = \frac{mv^2}{r}\) | Line 19 |
| 18 | Inline | \(v_{ideal} = \sqrt{rg \tan\theta}\) | Line 20 |
| 19 | Inline | \(v_{ideal} = \sqrt{rg \tan\theta}\) | Line 18 |
| 20 | Inline | \(v_{max} = \sqrt{\frac{rg(\sin\theta + \mu\cos\theta)}{\cos\theta - \mu\sin\theta}}\) | Line 19 |
| 21 | Inline | \(\mu_{required} = \frac{\|f\|}{N}\) | Line 20 |
| 22 | Inline | \(a_c = \frac{v^2}{r} = \omega^2 r\) | Line 18 |
| 23 | Inline | \(F_c = ma_c = \frac{mv^2}{r}\) | Line 19 |
| 24 | Inline | \(\omega = \frac{v}{r}\) | Line 20 |
| 25 | Display | \(\(m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'\)\) | Line 54 |
| 26 | Display | \(\(\frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2 = \frac{1}{2}m_1 v_1'^2 + \frac{1}{2}m_2 v_2'^2\)\) | Line 58 |
| 27 | Display | \(\(F = k \frac{\|q_1 q_2\|}{r^2}\)\) | Line 13 |
| 28 | Inline | \((Coulomb's constant) -\) | Line 18 |
| 29 | Inline | \(are the charges (Coulombs) -\) | Line 19 |
| 30 | Display | \(\(\epsilon_{back} = k \cdot \omega\)\) | Line 114 |
| 31 | Display | \(\(\eta = \frac{P_{mechanical}}{P_{electrical}} = \frac{\tau \cdot \omega}{V \cdot I}\)\) | Line 140 |
| 32 | Display | \(\(F = BIL\)\) | Line 276 |
| 33 | Display | \(\(\tau = r \times F = r \cdot BIL\)\) | Line 279 |
| 34 | Display | \(\(\epsilon_{back} = k \cdot \omega\)\) | Line 282 |
| 35 | Display | \(\(I = \frac{V - \epsilon_{back}}{R}\)\) | Line 285 |
| 36 | Display | \(\(P_{in} = V \cdot I\)\) | Line 288 |
| 37 | Display | \(\(P_{out} = \tau \cdot \omega\)\) | Line 291 |
| 38 | Display | \(\(\eta = \frac{P_{out}}{P_{in}} \times 100\%\)\) | Line 294 |
| 39 | Display | \(\(\omega = \frac{V}{k} - \frac{R}{k^2} \cdot \tau\)\) | Line 300 |
| 40 | Display | \(\(\text{Efficiency} = \frac{E_{output}}{E_{input}} \times 100\%\)\) | Line 18 |
| 41 | Display | \(\(\text{Transit Depth} = \left(\frac{R_{\text{planet}}}{R_{\text{star}}}\right)^2\)\) | Line 63 |
| 42 | Display | \(\(J = F \cdot \Delta t = \Delta p\)\) | Line 19 |
| 43 | Display | \(\(F_1 \cdot \Delta t_1 = F_2 \cdot \Delta t_2\)\) | Line 22 |
| 44 | Inline | \(f_s^{max} = \mu_s N\) | Line 25 |
| 45 | Inline | \(f_k = \mu_k N\) | Line 26 |
| 46 | Display | \(\(\text{slope} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\)\) | Line 65 |
| 47 | Inline | \(t = \sqrt{\frac{2h}{g}}\) | Line 19 |
| 48 | Inline | \(R = v_x \cdot t\) | Line 20 |
| 49 | Inline | \(v_y = gt\) | Line 21 |
| 50 | Inline | \(t = \sqrt{2h/g}\) | Line 36 |
| 51 | Inline | \(R = v_x \cdot t\) | Line 37 |
| 52 | Inline | \(mg \sin\theta\) | Line 18 |
| 53 | Inline | \(mg \cos\theta\) | Line 19 |
| 54 | Inline | \(N = mg \cos\theta\) | Line 20 |
| 55 | Inline | \(f_s^{max} = \mu_s N\) | Line 21 |
| 56 | Inline | \(\theta_c = \arctan(\mu_s)\) | Line 22 |
| 57 | Display | \(\(\vec{p} = m\vec{v}\)\) | Line 18 |
| 58 | Display | \(\(\Delta v = v_e \times \ln\left(\frac{m_{initial}}{m_{final}}\right)\)\) | Line 18 |
| 59 | Inline | \(= change in velocity (m/s) -\) | Line 21 |
| 60 | Inline | \(= exhaust velocity (m/s) -\) | Line 22 |
| 61 | Inline | \(= initial mass (with fuel) -\) | Line 23 |
| 62 | Display | \(\(V = IR\)\) | Line 26 |
| 63 | Display | \(\(V = IR \quad \Rightarrow \quad I = \frac{V}{R} \quad \Rightarrow \quad R = \frac{V}{I}\)\) | Line 61 |
| 64 | Display | \(\(P = IV = I^2R = \frac{V^2}{R}\)\) | Line 88 |
| 65 | Display | \(\(PE_g = mgh\)\) | Line 19 |
| 66 | Display | \(\(PE_s = \frac{1}{2}kx^2\)\) | Line 22 |
| 67 | Inline | \(\vec{v}_{AB} = \vec{v}_A - \vec{v}_B\) | Line 18 |
| 68 | Inline | \(v_{ABx} = v_{Ax} - v_{Bx}\) | Line 20 |
| 69 | Inline | \(v_{ABy} = v_{Ay} - v_{By}\) | Line 21 |
| 70 | Inline | \(\|\vec{v}_{AB}\| = \sqrt{v_{ABx}^2 + v_{ABy}^2}\) | Line 22 |
| 71 | Inline | \(\theta = \tan^{-1}(v_{ABy}/v_{ABx})\) | Line 23 |
| 72 | Inline | \(\vec{v}_{result} = \vec{v}_{swimmer} + \vec{v}_{current}\) | Line 19 |
| 73 | Inline | \(t = \frac{d}{v_{swim}}\) | Line 20 |
| 74 | Inline | \(x_{drift} = v_{current} \times t\) | Line 21 |
| 75 | Inline | \(\theta = \arcsin(v_{current}/v_{swim})\) | Line 22 |
| 76 | Display | \(\(\vec{p}_{initial} = \vec{p}_{final}\)\) | Line 19 |
| 77 | Display | \(\(0 = \vec{p}_{rocket} + \vec{p}_{exhaust}\)\) | Line 20 |
| 78 | Display | \(\(\vec{p}_{rocket} = -\vec{p}_{exhaust}\)\) | Line 21 |
| 79 | Display | \(\(F_{thrust} = \frac{dm}{dt} \times v_{exhaust}\)\) | Line 24 |
| 80 | Inline | \(= mass flow rate of exhaust (kg/s) -\) | Line 27 |
| 81 | Display | \(\(KE_i + PE_i = KE_f + PE_f\)\) | Line 20 |
| 82 | Display | \(\(\frac{1}{2}mv_i^2 + mgh_i = \frac{1}{2}mv_f^2 + mgh_f\)\) | Line 22 |
| 83 | Display | \(\(R_{total} = R_1 + R_2 + R_3\)\) | Line 59 |
| 84 | Display | \(\(\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}\)\) | Line 71 |
| 85 | Display | \(\(R_{total} = R_1 + R_2 + R_3 + ...\)\) | Line 206 |
| 86 | Display | \(\(I_{total} = I_1 = I_2 = I_3\)\) | Line 207 |
| 87 | Display | \(\(V_{total} = V_1 + V_2 + V_3\)\) | Line 208 |
| 88 | Display | \(\(\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ...\)\) | Line 211 |
| 89 | Display | \(\(I_{total} = I_1 + I_2 + I_3\)\) | Line 212 |
| 90 | Display | \(\(V_{total} = V_1 = V_2 = V_3\)\) | Line 213 |
| 91 | Display | \(\(R_{total} = \frac{R_1 \times R_2}{R_1 + R_2}\)\) | Line 216 |
| 92 | Display | \(\(R_{total} = \frac{R}{n}\)\) | Line 219 |
| 93 | Display | \(\(T = 2\pi\sqrt{\frac{L}{g}}\)\) | Line 31 |
| 94 | Display | \(\(W = \int_{x_1}^{x_2} F(x) \, dx\)\) | Line 31 |
| 95 | Display | \(\(W_{net} = \Delta KE = KE_f - KE_i = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2\)\) | Line 20 |
| 96 | Display | \(\(W = F \cdot d \cdot \cos(\theta)\)\) | Line 26 |
Chapter 1: Scientific Foundations and Mathematical Tools
76 equations (10 display, 66 inline)
| # | Type | Equation | Source |
|---|---|---|---|
| 1 | Inline | \(\frac{100 \text{ cm}}{1 \text{ m}} = 1\) | Line 203 |
| 2 | Inline | \(\frac{1 \text{ m}}{100 \text{ cm}} = 1\) | Line 204 |
| 3 | Display | \(\(25 \text{ m} \times \frac{100 \text{ cm}}{1 \text{ m}} = 2{,}500 \text{ cm}\)\) | Line 208 |
| 4 | Display | $$60 \frac{\text{mi}}{\text{hr}} \times \frac{1.609 \text{ km}}{1 \text{ mi}} \times \frac{1{,}00... | Line 214 |
| 5 | Inline | \(is an integer **Examples:** -\) | Line 224 |
| 6 | Inline | \((move decimal left 3 places, positive exponent) -\) | Line 228 |
| 7 | Inline | \((move decimal right 4 places, negative exponent) -\) | Line 229 |
| 8 | Inline | \((move decimal left 7 places) **Why use scientific notation?** 1. **Clarity:**\) | Line 230 |
| 9 | Inline | $is much clearer than 602,000,000,000,000,000,000,000 2. Precision: Scientific notation makes... | Line 234 |
| 10 | Inline | \(- **Division:** Divide coefficients, subtract exponents:\) | Line 241 |
| 11 | Display | \(\(d = vt + \frac{1}{2}at^2\)\) | Line 277 |
| 12 | Inline | \(= distance,\) | Line 279 |
| 13 | Inline | \(= initial velocity,\) | Line 279 |
| 14 | Inline | \(= acceleration,\) | Line 279 |
| 15 | Inline | \(= time Let's check the dimensions (using brackets [ ] to denote "dimensions of"): -\) | Line 279 |
| 16 | Inline | \(✓ -\) | Line 284 |
| 17 | Inline | \(measured counterclockwise from the positive x-axis: - **x-component:**\) | Line 518 |
| 18 | Inline | \(- **y-component:**\) | Line 520 |
| 19 | Inline | \(**Example:** A velocity vector of 50 m/s at 30° above the horizontal has: -\) | Line 521 |
| 20 | Inline | \(m/s (horizontal) -\) | Line 525 |
| 21 | Inline | $m/s (vertical) Reverse process (finding magnitude and direction from components): If you k... | Line 526 |
| 22 | Inline | \(, you can find: - **Magnitude:**\) | Line 530 |
| 23 | Inline | \(- **Direction:**\) | Line 532 |
| 24 | Inline | \(: -\) | Line 539 |
| 25 | Inline | $Memory aid: SOH-CAH-TOA - Sine = Opposite/Hypotenuse - Cosine = Adjacen... | Line 543 |
| 26 | Display | \(\(c^2 = a^2 + b^2\)\) | Line 553 |
| 27 | Inline | $Important angles to memorize: | Angle | sin | cos | tan | |-------|-----|-----|---... | Line 559 |
| 28 | Inline | \(3. Add all y-components together:\) | Line 587 |
| 29 | Inline | \(4. Find magnitude:\) | Line 588 |
| 30 | Inline | \(5. Find direction:\) | Line 589 |
| 31 | Inline | \(**Example:** Add two vectors: -\) | Line 590 |
| 32 | Inline | \(m/s east (0°) -\) | Line 594 |
| 33 | Inline | \(m/s north (90°) Using components: -\) | Line 595 |
| 34 | Inline | \(m/s,\) | Line 599 |
| 35 | Inline | \(m/s -\) | Line 599 |
| 36 | Inline | \(m/s,\) | Line 600 |
| 37 | Inline | \(m/s -\) | Line 600 |
| 38 | Inline | \(m/s -\) | Line 601 |
| 39 | Inline | \(m/s -\) | Line 602 |
| 40 | Inline | \(m/s -\) | Line 603 |
| 41 | Inline | $north of east #### Diagram: Vector Addition Interactive MicroSim <iframe src="../../sims/vecto... | Line 604 |
| 42 | Inline | \(m/s east: -\) | Line 724 |
| 43 | Inline | \(m/s,\) | Line 726 |
| 44 | Inline | \(m/s -\) | Line 726 |
| 45 | Inline | \(m/s,\) | Line 727 |
| 46 | Inline | \(m/s -\) | Line 727 |
| 47 | Inline | \(m/s -\) | Line 728 |
| 48 | Inline | \(m/s -\) | Line 729 |
| 49 | Inline | \(m/s -\) | Line 730 |
| 50 | Display | \(\(\vec{A} \cdot \vec{B} = AB\cos(\theta)\)\) | Line 741 |
| 51 | Display | \(\(\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z\)\) | Line 749 |
| 52 | Inline | \(), then\) | Line 755 |
| 53 | Inline | \(), then\) | Line 756 |
| 54 | Inline | \(**Physical applications:** 1. **Work** in physics is defined as\) | Line 756 |
| 55 | Inline | \(**Example:** Find the dot product of\) | Line 762 |
| 56 | Display | \(\(\vec{A} \cdot \vec{B} = (3)(1) + (4)(2) = 3 + 8 = 11\)\) | Line 766 |
| 57 | Display | \(\(\vec{A} \times \vec{B} = (AB\sin\theta)\hat{n}\)\) | Line 774 |
| 58 | Inline | \(is a unit vector perpendicular to both. **Magnitude:**\) | Line 776 |
| 59 | Inline | $Direction: Determined by the right-hand rule: 1. Point fingers of your right hand along... | Line 778 |
| 60 | Inline | \(**Component formula** (for 3D vectors): If\) | Line 784 |
| 61 | Display | \(\(\vec{A} \times \vec{B} = (A_y B_z - A_z B_y, A_z B_x - A_x B_z, A_x B_y - A_y B_x)\)\) | Line 790 |
| 62 | Inline | \((NOT commutative!) - If vectors are parallel (\) | Line 795 |
| 63 | Inline | \(), then\) | Line 796 |
| 64 | Inline | \(), then\) | Line 797 |
| 65 | Inline | \(**Physical applications:** 1. **Torque** is defined as\) | Line 797 |
| 66 | Inline | \((position vector cross force) 2. **Angular momentum** is\) | Line 801 |
| 67 | Inline | \(, meaning: - When\) | Line 812 |
| 68 | Inline | \((where\) | Line 816 |
| 69 | Inline | $. If you travel 60 miles in 1 hour, you'll travel 120 miles in 2 hours. **Inverse proportionali... | Line 818 |
| 70 | Inline | \(, meaning: - When\) | Line 820 |
| 71 | Inline | $. If you double the distance, force becomes one-fourth as strong. More complex proportions:... | Line 826 |
| 72 | Inline | \(- Slope\) | Line 854 |
| 73 | Inline | \(tells you starting value 2. **Quadratic:** Parabola, equation\) | Line 856 |
| 74 | Inline | \() / (change in\) | Line 871 |
| 75 | Inline | \() =\) | Line 871 |
| 76 | Display | \(\(m = \frac{y_2 - y_1}{x_2 - x_1}\)\) | Line 875 |
Chapter 2: Motion in One Dimension
51 equations (28 display, 23 inline)
| # | Type | Equation | Source |
|---|---|---|---|
| 1 | Inline | \(\Delta x\) | Line 58 |
| 2 | Display | \(\(\Delta x = x_{\text{final}} - x_{\text{initial}}\)\) | Line 60 |
| 3 | Display | \(\(\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}\)\) | Line 127 |
| 4 | Display | \(\(\text{Average velocity} = \frac{\text{Displacement}}{\text{Time interval}}\)\) | Line 133 |
| 5 | Display | $$a_{\text{avg}} = \frac{\Delta v}{\Delta t} = \frac{v_{\text{final}} - v_{\text{initial}}}{t_{\t... | Line 242 |
| 6 | Inline | $. The SI unit for acceleration is meters per second squared (m/s²), which can be read as "meter... | Line 244 |
| 7 | Inline | \(m/s -\) | Line 269 |
| 8 | Inline | \(m/s -\) | Line 270 |
| 9 | Display | \(\(x = x_0 + vt\)\) | Line 301 |
| 10 | Inline | $- Relates: velocities, acceleration, displacement - Missing: time - Use when: Time is unkn... | Line 842 |
| 11 | Inline | \(m/s,\) | Line 877 |
| 12 | Inline | \(m/s²,\) | Line 877 |
| 13 | Inline | \(For velocity, use equation 1:\) | Line 879 |
| 14 | Display | \(\(v = v_0 + at = 0 + (3.0)(5.0) = 15 \text{ m/s}\)\) | Line 883 |
| 15 | Display | $$x = x_0 + v_0 t + \frac{1}{2}at^2 = 0 + 0 + \frac{1}{2}(3.0)(5.0)^2 = \frac{1}{2}(3.0)(25) = 37... | Line 887 |
| 16 | Inline | \(m/s,\) | Line 893 |
| 17 | Inline | \(m/s,\) | Line 893 |
| 18 | Display | \(\(v^2 = v_0^2 + 2a(x - x_0)\)\) | Line 899 |
| 19 | Display | \(\((80)^2 = 0^2 + 2a(1200)\)\) | Line 900 |
| 20 | Display | \(\(6400 = 2400a\)\) | Line 901 |
| 21 | Display | \(\(a = \frac{6400}{2400} = 2.67 \text{ m/s}^2\)\) | Line 902 |
| 22 | Inline | \(m/s,\) | Line 908 |
| 23 | Inline | \(m/s (comes to rest),\) | Line 908 |
| 24 | Inline | \(m/s² Find:\) | Line 908 |
| 25 | Inline | \(For time, use equation 1:\) | Line 910 |
| 26 | Display | \(\(v = v_0 + at\)\) | Line 914 |
| 27 | Display | \(\(0 = 5.0 + (-0.50)t\)\) | Line 915 |
| 28 | Display | \(\(0.50t = 5.0\)\) | Line 916 |
| 29 | Display | \(\(t = 10 \text{ s}\)\) | Line 917 |
| 30 | Display | \(\(v^2 = v_0^2 + 2a(x - x_0)\)\) | Line 921 |
| 31 | Display | \(\(0^2 = (5.0)^2 + 2(-0.50)(x - x_0)\)\) | Line 922 |
| 32 | Display | \(\(0 = 25 - 1.0(x - x_0)\)\) | Line 923 |
| 33 | Display | \(\(x - x_0 = 25 \text{ m}\)\) | Line 924 |
| 34 | Display | \(\(g = 9.8 \text{ m/s}^2\)\) | Line 994 |
| 35 | Inline | \(m/s² (negative because it points down) - **Positive downward, negative upward:** With this choice,\) | Line 1000 |
| 36 | Inline | \(m/s (upward),\) | Line 1017 |
| 37 | Inline | \(m/s²,\) | Line 1017 |
| 38 | Display | \(\(v^2 = v_0^2 + 2a(x - x_0)\)\) | Line 1023 |
| 39 | Display | \(\(0^2 = (15)^2 + 2(-9.8)(x - 0)\)\) | Line 1024 |
| 40 | Display | \(\(0 = 225 - 19.6x\)\) | Line 1025 |
| 41 | Display | \(\(x = \frac{225}{19.6} \approx 11.5 \text{ m}\)\) | Line 1026 |
| 42 | Inline | \((back to starting position):\) | Line 1028 |
| 43 | Display | \(\(x = x_0 + v_0 t + \frac{1}{2}at^2\)\) | Line 1030 |
| 44 | Display | \(\(0 = 0 + 15t + \frac{1}{2}(-9.8)t^2\)\) | Line 1031 |
| 45 | Display | \(\(0 = 15t - 4.9t^2\)\) | Line 1032 |
| 46 | Display | \(\(0 = t(15 - 4.9t)\)\) | Line 1033 |
| 47 | Inline | \((initial throw) or\) | Line 1035 |
| 48 | Inline | $s (return to hand) Free fall problems directly apply the kinematic equations with a = -9.8 m/s²... | Line 1035 |
| 49 | Inline | \(2.\) | Line 1146 |
| 50 | Inline | \(3.\) | Line 1147 |
| 51 | Inline | \(4.\) | Line 1148 |
Chapter 3: Motion in Two Dimensions
48 equations (11 display, 37 inline)
| # | Type | Equation | Source |
|---|---|---|---|
| 1 | Inline | \(x = x_0 + v_0t + \frac{1}{2}at^2\) | Line 37 |
| 2 | Inline | \(v = v_0 + at\) | Line 38 |
| 3 | Inline | \(v^2 = v_0^2 + 2a(x - x_0)\) | Line 39 |
| 4 | Inline | \(v_x\) | Line 47 |
| 5 | Inline | \(a_x\) | Line 47 |
| 6 | Inline | \(v_y\) | Line 48 |
| 7 | Inline | \(a_y\) | Line 48 |
| 8 | Inline | \(\vec{v}\) | Line 67 |
| 9 | Inline | \(v_x\) | Line 68 |
| 10 | Inline | \(v_y\) | Line 69 |
| 11 | Inline | \(\vec{v}\) | Line 71 |
| 12 | Inline | \(\vec{v}\) | Line 75 |
| 13 | Inline | \(v_x = v\cos\theta\) | Line 76 |
| 14 | Inline | \(v_y = v\sin\theta\) | Line 77 |
| 15 | Inline | \(\theta\) | Line 78 |
| 16 | Inline | \(v_{0x} = v_0\cos\theta\) | Line 90 |
| 17 | Inline | \(v_{0y} = v_0\sin\theta\) | Line 91 |
| 18 | Inline | \(v_0\) | Line 93 |
| 19 | Inline | \(\theta\) | Line 93 |
| 20 | Inline | \(g = 9.8 \text{ m/s}^2\) | Line 97 |
| 21 | Display | \(\(a_y = -g = -9.8 \text{ m/s}^2\)\) | Line 101 |
| 22 | Inline | \(, object starts from rest - **Thrown upward:**\) | Line 115 |
| 23 | Inline | \(, object rises then falls - **Thrown downward:**\) | Line 116 |
| 24 | Inline | \(, object speeds up throughout fall - **At maximum height:**\) | Line 117 |
| 25 | Inline | $- Independence: Horizontal and vertical motions occur simultaneously but don't affect each o... | Line 191 |
| 26 | Inline | \(\| \| Velocity \| Constant (\) | Line 199 |
| 27 | Inline | \() \| Changes (\) | Line 200 |
| 28 | Inline | \() \| \| Position equation \|\) | Line 200 |
| 29 | Display | \(\(x = v_0 t\)\) | Line 267 |
| 30 | Display | \(\(y = y_0 - \frac{1}{2}gt^2\)\) | Line 270 |
| 31 | Display | \(\(v_y = -gt\)\) | Line 271 |
| 32 | Inline | \(: **Initial velocity components:**\) | Line 357 |
| 33 | Display | \(\(v_{0x} = v_0\cos\theta\)\) | Line 360 |
| 34 | Display | \(\(v_{0y} = v_0\sin\theta\)\) | Line 361 |
| 35 | Display | \(\(x = (v_0\cos\theta)t\)\) | Line 364 |
| 36 | Display | \(\(y = y_0 + (v_0\sin\theta)t - \frac{1}{2}gt^2\)\) | Line 365 |
| 37 | Display | \(\(v_x = v_0\cos\theta\)\) | Line 368 |
| 38 | Display | \(\(v_y = v_0\sin\theta - gt\)\) | Line 369 |
| 39 | Inline | \((when\) | Line 375 |
| 40 | Inline | \() - **Maximum height:**\) | Line 375 |
| 41 | Inline | \(- **Total flight time:**\) | Line 376 |
| 42 | Inline | \((for level ground,\) | Line 377 |
| 43 | Inline | \() - **Horizontal range:**\) | Line 377 |
| 44 | Inline | $is maximum). However, complementary angles (like 30° and 60°) produce the same range—they just h... | Line 380 |
| 45 | Display | \(\(\vec{v}_{AB} = \vec{v}_A - \vec{v}_B\)\) | Line 498 |
| 46 | Inline | \(" for swimmer velocity (blue) - "\) | Line 539 |
| 47 | Inline | \(" for current velocity (red) - "\) | Line 540 |
| 48 | Inline | $" for resultant velocity (purple) - "Starting point (A)" at bottom - "Intended destinati... | Line 541 |
Chapter 4: Forces and Newton's Laws
16 equations (5 display, 11 inline)
| # | Type | Equation | Source |
|---|---|---|---|
| 1 | Display | \(\(\vec{F}_{\text{net}} = m\vec{a}\)\) | Line 233 |
| 2 | Inline | \(: -\) | Line 251 |
| 3 | Display | \(\(1 \text{ N} = 1 \text{ kg} \cdot \text{m/s}^2\)\) | Line 270 |
| 4 | Inline | \() pulls down - Normal force (\) | Line 491 |
| 5 | Inline | \() pushes up -\) | Line 492 |
| 6 | Inline | $- Velocity constant (no acceleration) ### Conditions for Equilibrium For an object to be in eq... | Line 504 |
| 7 | Display | \(\(\sum F_x = 0\)\) | Line 516 |
| 8 | Display | \(\(\sum F_y = 0\)\) | Line 517 |
| 9 | Display | \(\(W = mg\)\) | Line 600 |
| 10 | Inline | \(is weight (measured in newtons, N) -\) | Line 604 |
| 11 | Inline | \(is mass (measured in kilograms, kg) -\) | Line 605 |
| 12 | Inline | $is the acceleration due to gravity (9.8 m/s² on Earth's surface) ### Mass vs Weight: A Critical... | Line 606 |
| 13 | Inline | \(N - On the Moon:\) | Line 622 |
| 14 | Inline | \(N (Moon's gravity is weaker) - In deep space:\) | Line 623 |
| 15 | Inline | $N (no nearby gravitational source) But the mass stays 70 kg in all three locations! Mass is an ... | Line 624 |
| 16 | Inline | \((normal force decreases) **On an inclined plane:** - Normal force\) | Line 746 |
Chapter 5: Applications of Newton's Laws
64 equations (58 display, 6 inline)
| # | Type | Equation | Source |
|---|---|---|---|
| 1 | Display | \(\(f_s^{max} = \mu_s N\)\) | Line 108 |
| 2 | Inline | $. #### Diagram: Interactive Static Friction MicroSim <iframe src="../../sims/static-friction/m... | Line 110 |
| 3 | Display | \(\(f_k = \mu_k N\)\) | Line 179 |
| 4 | Inline | $is the normal force. Notice that this equation doesn't depend on the object's velocity—an object... | Line 181 |
| 5 | Display | \(\(N = mg = (25 \text{ kg})(9.8 \text{ m/s}^2) = 245 \text{ N}\)\) | Line 273 |
| 6 | Display | \(\(f_s^{max} = \mu_s N = (0.5)(245 \text{ N}) = 122.5 \text{ N}\)\) | Line 277 |
| 7 | Display | \(\(f_k = \mu_k N = (0.3)(245 \text{ N}) = 73.5 \text{ N}\)\) | Line 288 |
| 8 | Display | \(\(\sum F_x = F_{applied} - f_k = ma\)\) | Line 292 |
| 9 | Display | \(\(150 \text{ N} - 73.5 \text{ N} = (25 \text{ kg})a\)\) | Line 294 |
| 10 | Display | \(\(a = \frac{76.5 \text{ N}}{25 \text{ kg}} = 3.06 \text{ m/s}^2\)\) | Line 296 |
| 11 | Display | \(\(\sum F_y = T - mg = 0\)\) | Line 371 |
| 12 | Display | \(\(T = mg = (15 \text{ kg})(9.8 \text{ m/s}^2) = 147 \text{ N}\)\) | Line 373 |
| 13 | Display | \(\(\sum F_x = T - f_k = ma\)\) | Line 392 |
| 14 | Display | \(\(T - 300 \text{ N} = (1500 \text{ kg})(1.2 \text{ m/s}^2)\)\) | Line 394 |
| 15 | Display | \(\(T = 1800 \text{ N} + 300 \text{ N} = 2100 \text{ N}\)\) | Line 396 |
| 16 | Inline | \(N - Vertical:\) | Line 410 |
| 17 | Display | \(\(N + F_y = mg\)\) | Line 415 |
| 18 | Display | \(\(N = mg - F_y = (20)(9.8) - 25 = 171 \text{ N}\)\) | Line 417 |
| 19 | Display | \(\(f_k = \mu_k N = (0.25)(171) = 42.75 \text{ N}\)\) | Line 421 |
| 20 | Display | \(\(\sum F_x = F_x - f_k = ma\)\) | Line 425 |
| 21 | Display | \(\(43.3 - 42.75 = 20a\)\) | Line 427 |
| 22 | Display | \(\(a = 0.028 \text{ m/s}^2\)\) | Line 429 |
| 23 | Inline | \((parallel to incline),\) | Line 512 |
| 24 | Display | \(\(\sum F_y = N - mg \cos \theta = 0\)\) | Line 570 |
| 25 | Display | \(\(N = mg \cos \theta\)\) | Line 572 |
| 26 | Display | \(\(\sum F_x = f_s - mg \sin \theta = 0\)\) | Line 576 |
| 27 | Display | \(\(f_s = mg \sin \theta\)\) | Line 578 |
| 28 | Display | \(\(mg \sin \theta_{critical} = \mu_s (mg \cos \theta_{critical})\)\) | Line 586 |
| 29 | Display | \(\(\tan \theta_{critical} = \mu_s\)\) | Line 588 |
| 30 | Display | \(\(\theta_{critical} = \arctan(\mu_s)\)\) | Line 590 |
| 31 | Display | \(\(N = mg \cos \theta = (5)(9.8) \cos 35° = 40.1 \text{ N}\)\) | Line 605 |
| 32 | Display | \(\(f_k = \mu_k N = (0.25)(40.1) = 10.0 \text{ N}\)\) | Line 609 |
| 33 | Display | \(\(\sum F_x = mg \sin \theta - f_k = ma\)\) | Line 613 |
| 34 | Display | \(\((5)(9.8) \sin 35° - 10.0 = 5a\)\) | Line 615 |
| 35 | Display | \(\(28.1 - 10.0 = 5a\)\) | Line 617 |
| 36 | Display | \(\(a = 3.62 \text{ m/s}^2\)\) | Line 619 |
| 37 | Display | \(\(T - m_1 g = m_1 a\)\) | Line 779 |
| 38 | Display | \(\(m_2 g - T = m_2 a\)\) | Line 782 |
| 39 | Display | \(\(m_2 g - m_1 g = m_1 a + m_2 a\)\) | Line 785 |
| 40 | Display | \(\((m_2 - m_1)g = (m_1 + m_2)a\)\) | Line 787 |
| 41 | Display | $$a = \frac{(m_2 - m_1)g}{m_1 + m_2} = \frac{(7-5)(9.8)}{5+7} = \frac{19.6}{12} = 1.63 \text{ m/s... | Line 789 |
| 42 | Display | \(\(T = m_1(g + a) = 5(9.8 + 1.63) = 57.2 \text{ N}\)\) | Line 792 |
| 43 | Display | \(\(T = m_2(g - a) = 7(9.8 - 1.63) = 57.2 \text{ N}\)\) | Line 795 |
| 44 | Display | \(\(F = \frac{mg}{2} = \frac{(120)(9.8)}{2} = 588 \text{ N}\)\) | Line 894 |
| 45 | Display | \(\(d_{rope} = 2 \times d_{load} = 2 \times 3 = 6 \text{ m}\)\) | Line 897 |
| 46 | Inline | \(J - Work output:\) | Line 902 |
| 47 | Display | \(\(a_c = \frac{v^2}{r}\)\) | Line 984 |
| 48 | Display | \(\(a_c = \omega^2 r\)\) | Line 990 |
| 49 | Display | \(\(a_c = \frac{v^2}{r} = \frac{(20)^2}{50} = \frac{400}{50} = 8.0 \text{ m/s}^2\)\) | Line 999 |
| 50 | Display | \(\(F_c = ma_c = m\frac{v^2}{r}\)\) | Line 1007 |
| 51 | Display | \(\(F_c^{max} = f_s^{max} = \mu_s N = \mu_s mg\)\) | Line 1089 |
| 52 | Display | \(\(\mu_s mg = m\frac{v^2}{r}\)\) | Line 1093 |
| 53 | Display | \(\(\mu_s g = \frac{v^2}{r}\)\) | Line 1097 |
| 54 | Display | \(\(v = \sqrt{\mu_s gr} = \sqrt{(0.85)(9.8)(80)} = \sqrt{666.4} = 25.8 \text{ m/s}\)\) | Line 1101 |
| 55 | Inline | $km/h The maximum safe speed is about 93 km/h (58 mph). In wet conditions, with μs ≈ 0.5, this d... | Line 1103 |
| 56 | Display | \(\(v_{ideal} = \sqrt{rg \tan \theta}\)\) | Line 1190 |
| 57 | Display | \(\(v_{ideal} = \sqrt{rg \tan \theta} = \sqrt{(120)(9.8) \tan 15°}\)\) | Line 1259 |
| 58 | Display | \(\(v_{ideal} = \sqrt{(1176)(0.268)} = \sqrt{315.2} = 17.8 \text{ m/s}\)\) | Line 1261 |
| 59 | Display | \(\(N \cos 15° = mg\)\) | Line 1274 |
| 60 | Display | \(\(N = \frac{mg}{\cos 15°} = \frac{(1200)(9.8)}{0.966} = 12,180 \text{ N}\)\) | Line 1276 |
| 61 | Display | \(\(F_c = \frac{mv^2}{r} = \frac{(1200)(25)^2}{120} = 6250 \text{ N}\)\) | Line 1280 |
| 62 | Display | \(\(N \sin 15° = (12,180)(0.259) = 3155 \text{ N}\)\) | Line 1284 |
| 63 | Display | \(\(f = F_c - N \sin \theta = 6250 - 3155 = 3095 \text{ N}\)\) | Line 1288 |
| 64 | Display | \(\(\mu = \frac{f}{N} = \frac{3095}{12,180} = 0.25\)\) | Line 1292 |
Chapter 6: Work, Energy, and Power
71 equations (33 display, 38 inline)
| # | Type | Equation | Source |
|---|---|---|---|
| 1 | Display | \(\(W = F \cdot d \cdot \cos(\theta)\)\) | Line 61 |
| 2 | Inline | \(= work (measured in joules, J) -\) | Line 65 |
| 3 | Inline | \(= magnitude of the applied force (newtons, N) -\) | Line 66 |
| 4 | Inline | \(= displacement of the object (meters, m) -\) | Line 67 |
| 5 | Inline | \(), work is maximum:\) | Line 72 |
| 6 | Inline | \(), no work is done:\) | Line 73 |
| 7 | Inline | \(), work is negative:\) | Line 74 |
| 8 | Display | \(\(W = (50 \text{ N})(3 \text{ m})\cos(0°) = 150 \text{ J}\)\) | Line 78 |
| 9 | Display | \(\(W = \int_{x_1}^{x_2} F(x) \, dx\)\) | Line 166 |
| 10 | Display | \(\(W = \frac{1}{2}kx^2\)\) | Line 172 |
| 11 | Display | \(\(KE = \frac{1}{2}mv^2\)\) | Line 251 |
| 12 | Inline | \(= kinetic energy (joules, J) -\) | Line 255 |
| 13 | Inline | \(= mass (kilograms, kg) -\) | Line 256 |
| 14 | Display | \(\(KE = \frac{1}{2}(1200 \text{ kg})(25 \text{ m/s})^2 = 375,000 \text{ J}\)\) | Line 269 |
| 15 | Display | \(\(W_{net} = \Delta KE = KE_f - KE_i\)\) | Line 281 |
| 16 | Display | \(\(W_{net} = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2\)\) | Line 285 |
| 17 | Inline | \(m/s² 2. Use kinematic equation:\) | Line 301 |
| 18 | Inline | \(3. Solve for\) | Line 302 |
| 19 | Inline | \(m Using work-energy theorem: 1. Initial KE:\) | Line 303 |
| 20 | Inline | \(J 2. Final KE: 0 J (block stops) 3. Work by friction:\) | Line 306 |
| 21 | Inline | \(4. Apply theorem:\) | Line 308 |
| 22 | Display | \(\(PE_g = mgh\)\) | Line 409 |
| 23 | Inline | \(= gravitational potential energy (joules, J) -\) | Line 413 |
| 24 | Inline | \(= mass (kg) -\) | Line 414 |
| 25 | Inline | \(= gravitational acceleration (9.8 m/s²) -\) | Line 415 |
| 26 | Display | \(\(PE_g = (10 \text{ kg})(9.8 \text{ m/s}^2)(2 \text{ m}) = 196 \text{ J}\)\) | Line 427 |
| 27 | Display | \(\(PE_s = \frac{1}{2}kx^2\)\) | Line 435 |
| 28 | Inline | \(= elastic potential energy (J) -\) | Line 439 |
| 29 | Inline | \(= spring constant (N/m), measuring the spring's stiffness -\) | Line 440 |
| 30 | Inline | \(N/m by 0.3 meters stores:\) | Line 452 |
| 31 | Display | \(\(PE_s = \frac{1}{2}(200 \text{ N/m})(0.3 \text{ m})^2 = 9 \text{ J}\)\) | Line 454 |
| 32 | Display | \(\(E_{total} = E_{kinetic} + E_{potential} + E_{thermal} + E_{chemical} + ... = \text{constant}\)\) | Line 583 |
| 33 | Display | \(\(E_{mechanical} = KE + PE = \text{constant}\)\) | Line 587 |
| 34 | Display | \(\(KE_i + PE_i = KE_f + PE_f\)\) | Line 591 |
| 35 | Display | \(\(KE_i + PE_i = KE_f + PE_f + E_{thermal}\)\) | Line 614 |
| 36 | Display | \(\(W_{non-conservative} = \Delta KE + \Delta PE\)\) | Line 618 |
| 37 | Display | \(\(v = \sqrt{2gh} = \sqrt{2(9.8)(5)} = 9.9 \text{ m/s}\)\) | Line 626 |
| 38 | Display | \(\(mgh - 20 = \frac{1}{2}mv^2\)\) | Line 630 |
| 39 | Display | \(\((2)(9.8)(5) - 20 = \frac{1}{2}(2)v^2\)\) | Line 631 |
| 40 | Display | \(\(v = 8.1 \text{ m/s}\)\) | Line 632 |
| 41 | Inline | $- Maximum speed: Occurs where PE is minimum - Range of motion: Between turning points wh... | Line 775 |
| 42 | Display | \(\(P = \frac{W}{t} = \frac{\Delta E}{t}\)\) | Line 905 |
| 43 | Inline | \(= power (watts, W) -\) | Line 909 |
| 44 | Inline | \(= work done (joules, J) -\) | Line 910 |
| 45 | Inline | \(= energy transferred (J) -\) | Line 911 |
| 46 | Display | \(\(W = Pt\)\) | Line 918 |
| 47 | Display | \(\(P = F \cdot v = Fv\cos(\theta)\)\) | Line 939 |
| 48 | Inline | \(= force magnitude (N) -\) | Line 943 |
| 49 | Inline | \(= velocity magnitude (m/s) -\) | Line 944 |
| 50 | Inline | \():\) | Line 947 |
| 51 | Display | \(\(P = Fv\)\) | Line 949 |
| 52 | Display | \(\(\text{Efficiency} = \frac{E_{output}}{E_{input}} = \frac{P_{output}}{P_{input}}\)\) | Line 974 |
| 53 | Display | \(\(\text{Efficiency} = \frac{E_{output}}{E_{input}} \times 100\%\)\) | Line 978 |
| 54 | Display | \(\(\text{Efficiency} = \frac{1600}{2000} = 0.80 = 80\%\)\) | Line 1002 |
| 55 | Display | \(\(MA = \frac{F_{output}}{F_{input}}\)\) | Line 1133 |
| 56 | Display | \(\(W_{input} = W_{output}\)\) | Line 1139 |
| 57 | Display | \(\(F_{input} \cdot d_{input} = F_{output} \cdot d_{output}\)\) | Line 1141 |
| 58 | Display | \(\(MA = \frac{F_{output}}{F_{input}} = \frac{d_{input}}{d_{output}}\)\) | Line 1145 |
| 59 | Inline | $(distances from fulcrum) Pulley System: - Single fixed pulley: MA = 1 (changes direction on... | Line 1154 |
| 60 | Display | \(\(\text{Efficiency} = \frac{AMA}{IMA} \times 100\%\)\) | Line 1176 |
| 61 | Inline | $| Crowbar, scissors, seesaw, wheelbarrow | Lifting heavy objects, cutting, balancing | | **P... | Line 1187 |
| 62 | Inline | $(length ÷ width) | Axe, knife, chisel, doorstop | Splitting, cutting, holding objects in place... | Line 1190 |
| 63 | Inline | $(circumference ÷ pitch) | Bolts, jar lids, vise, cork screw | Fastening, lifting, pressing | ... | Line 1191 |
| 64 | Inline | \(- Variable force:\) | Line 1369 |
| 65 | Inline | \(**Energy:** - Kinetic:\) | Line 1370 |
| 66 | Inline | \(- Elastic potential:\) | Line 1374 |
| 67 | Inline | \(**Conservation:** - Mechanical energy (conservative forces only):\) | Line 1375 |
| 68 | Inline | \(- Work-energy theorem:\) | Line 1379 |
| 69 | Inline | \(**Power:** - Average power:\) | Line 1380 |
| 70 | Inline | \(**Efficiency:** -\) | Line 1384 |
| 71 | Inline | \(**Mechanical Advantage:** -\) | Line 1387 |
Chapter 7: Momentum and Collisions
29 equations (20 display, 9 inline)
| # | Type | Equation | Source |
|---|---|---|---|
| 1 | Display | \(\(\vec{p} = m\vec{v}\)\) | Line 45 |
| 2 | Inline | \(is momentum (measured in kg·m/s) -\) | Line 49 |
| 3 | Inline | \(is mass (measured in kg) -\) | Line 50 |
| 4 | Inline | $is velocity (measured in m/s) Notice that momentum is a vector quantity—it has both magnitu... | Line 51 |
| 5 | Display | \(\(\vec{J} = \vec{F}\Delta t\)\) | Line 136 |
| 6 | Inline | \(is impulse (measured in N·s or kg·m/s) -\) | Line 140 |
| 7 | Display | \(\(\vec{J} = \Delta\vec{p} = \vec{p}_f - \vec{p}_i\)\) | Line 161 |
| 8 | Display | \(\(\vec{F}\Delta t = m\vec{v}_f - m\vec{v}_i\)\) | Line 165 |
| 9 | Display | \(\(\vec{p}_{total,i} = \vec{p}_{total,f}\)\) | Line 244 |
| 10 | Display | \(\(m_1\vec{v}_{1i} + m_2\vec{v}_{2i} = m_1\vec{v}_{1f} + m_2\vec{v}_{2f}\)\) | Line 248 |
| 11 | Display | \(\(m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}\)\) | Line 352 |
| 12 | Display | $$\frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}... | Line 355 |
| 13 | Display | \(\(v_{1f} = \frac{m_1 - m_2}{m_1 + m_2}v_{1i} + \frac{2m_2}{m_1 + m_2}v_{2i}\)\) | Line 359 |
| 14 | Display | \(\(v_{2f} = \frac{2m_1}{m_1 + m_2}v_{1i} + \frac{m_2 - m_1}{m_1 + m_2}v_{2i}\)\) | Line 361 |
| 15 | Inline | $)**: The objects exchange velocities. If object 2 is initially at rest, object 1 stops and objec... | Line 367 |
| 16 | Display | \(\(m_1v_{1i} + m_2v_{2i} = (m_1 + m_2)v_f\)\) | Line 398 |
| 17 | Display | \(\(\Delta KE = KE_i - KE_f\)\) | Line 415 |
| 18 | Display | \(\(m_1v_{1ix} + m_2v_{2ix} = m_1v_{1fx} + m_2v_{2fx}\)\) | Line 505 |
| 19 | Display | \(\(m_1v_{1iy} + m_2v_{2iy} = m_1v_{1fy} + m_2v_{2fy}\)\) | Line 508 |
| 20 | Inline | \(- Direction:\) | Line 519 |
| 21 | Inline | $### Glancing Collisions When objects don't collide head-on, they undergo *glancing collisions... | Line 520 |
| 22 | Display | $$x_{cm} = \frac{m_1x_1 + m_2x_2 + ... + m_nx_n}{m_1 + m_2 + ... + m_n} = \frac{\sum m_ix_i}{\sum... | Line 608 |
| 23 | Display | \(\(x_{cm} = \frac{\sum m_ix_i}{M_{total}}, \quad y_{cm} = \frac{\sum m_iy_i}{M_{total}}\)\) | Line 611 |
| 24 | Display | \(\(\vec{v}_{cm} = \frac{\sum m_i\vec{v}_i}{M_{total}}\)\) | Line 620 |
| 25 | Display | \(\(\vec{p}_{total} = M_{total}\vec{v}_{cm}\)\) | Line 624 |
| 26 | Display | \(\(0 = (M - \Delta m)v - \Delta m \cdot v_e\)\) | Line 658 |
| 27 | Display | \(\(\Delta v = v_e \ln\left(\frac{M_i}{M_f}\right)\)\) | Line 666 |
| 28 | Inline | $must be large, meaning most of the rocket is fuel - Achieving high speeds requires multistage ro... | Line 679 |
| 29 | Inline | $). This vector quantity plays a central role in analyzing collisions and interactions. **Key Co... | Line 882 |
Chapter 8: Rotational Motion and Angular Momentum
29 equations (29 display, 0 inline)
| # | Type | Equation | Source |
|---|---|---|---|
| 1 | Display | \(\(s = r\theta\)\) | Line 50 |
| 2 | Display | \(\(\omega = \frac{\Delta\theta}{\Delta t}\)\) | Line 127 |
| 3 | Display | \(\(v = r\omega\)\) | Line 133 |
| 4 | Display | \(\(\alpha = \frac{\Delta\omega}{\Delta t}\)\) | Line 195 |
| 5 | Display | \(\(a_t = r\alpha\)\) | Line 201 |
| 6 | Display | \(\(\omega = \omega_0 + \alpha t\)\) | Line 228 |
| 7 | Display | \(\(\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2\)\) | Line 230 |
| 8 | Display | \(\(\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)\)\) | Line 232 |
| 9 | Display | \(\(\theta = \theta_0 + \frac{1}{2}(\omega_0 + \omega)t\)\) | Line 234 |
| 10 | Display | \(\(\tau = rF\sin\theta\)\) | Line 327 |
| 11 | Display | \(\(\tau = rF_\perp\)\) | Line 337 |
| 12 | Display | \(\(\sum \tau = \tau_1 + \tau_2 + \tau_3 + ...\)\) | Line 401 |
| 13 | Display | \(\(\sum \tau = 0\)\) | Line 405 |
| 14 | Display | \(\(I = mr^2\)\) | Line 428 |
| 15 | Display | \(\(I = \sum m_i r_i^2\)\) | Line 432 |
| 16 | Display | \(\(\sum \tau = I\alpha\)\) | Line 523 |
| 17 | Display | \(\(KE_{rot} = \frac{1}{2}I\omega^2\)\) | Line 558 |
| 18 | Display | \(\(KE_{total} = KE_{trans} + KE_{rot} = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2\)\) | Line 564 |
| 19 | Display | \(\(E_{initial} = E_{final}\)\) | Line 630 |
| 20 | Display | \(\(PE_i + KE_{trans,i} + KE_{rot,i} = PE_f + KE_{trans,f} + KE_{rot,f}\)\) | Line 631 |
| 21 | Display | \(\(L = I\omega\)\) | Line 639 |
| 22 | Display | \(\(\sum \tau = \frac{\Delta L}{\Delta t}\)\) | Line 645 |
| 23 | Display | \(\(L = mvr\)\) | Line 651 |
| 24 | Display | \(\(L_{initial} = L_{final}\)\) | Line 714 |
| 25 | Display | \(\(I_i\omega_i = I_f\omega_f\)\) | Line 715 |
| 26 | Display | \(\(v_{cm} = r\omega\)\) | Line 814 |
| 27 | Display | \(\(mgh = \frac{1}{2}mv_{cm}^2 + \frac{1}{2}I\omega^2\)\) | Line 904 |
| 28 | Display | \(\(v_{cm} = \sqrt{\frac{2gh}{1 + I/mr^2}}\)\) | Line 908 |
| 29 | Display | \(\(a_{cm} = \frac{g\sin\theta}{1 + I/mr^2}\)\) | Line 914 |
Chapter 10: Waves and Sound
20 equations (12 display, 8 inline)
| # | Type | Equation | Source |
|---|---|---|---|
| 1 | Display | \(\(T = \frac{1}{f}\)\) | Line 243 |
| 2 | Display | \(\(v = f\lambda\)\) | Line 253 |
| 3 | Display | \(\(y_{total} = y_1 + y_2\)\) | Line 339 |
| 4 | Display | \(\(f' = f \left(\frac{v}{v \pm v_s}\right)\)\) | Line 640 |
| 5 | Display | \(\(f' = f \left(\frac{v \pm v_o}{v}\right)\)\) | Line 650 |
| 6 | Display | \(\(v = 331 + 0.6T\)\) | Line 777 |
| 7 | Display | \(\(I = \frac{P}{4\pi r^2}\)\) | Line 796 |
| 8 | Display | \(\(\beta = 10 \log_{10}\left(\frac{I}{I_0}\right)\)\) | Line 804 |
| 9 | Display | \(\(f_{beat} = \|f_1 - f_2\|\)\) | Line 1023 |
| 10 | Display | \(\(f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}\)\) | Line 1061 |
| 11 | Display | \(\(f_n = \frac{nv}{2L}\)\) | Line 1066 |
| 12 | Display | \(\(f_n = \frac{nv}{4L}\)\) | Line 1069 |
| 13 | Inline | \(\| v = wave speed, f = frequency, λ = wavelength \| \| Period-frequency \|\) | Line 1228 |
| 14 | Inline | \(\| T = period, f = frequency \| \| Doppler (moving source) \|\) | Line 1229 |
| 15 | Inline | \(\| f' = observed frequency, v = wave speed, v_s = source speed \| \| Doppler (moving observer) \|\) | Line 1230 |
| 16 | Inline | \(\| v_o = observer speed \| \| Sound level \|\) | Line 1231 |
| 17 | Inline | \(\| β = decibels, I = intensity, I₀ = 10⁻¹² W/m² \| \| Beat frequency \|\) | Line 1232 |
| 18 | Inline | \(\| f₁, f₂ = frequencies of interfering waves \| \| String harmonics \|\) | Line 1233 |
| 19 | Inline | \(\| n = harmonic number, L = length, T = tension, μ = mass/length \| \| Open pipe harmonics \|\) | Line 1234 |
| 20 | Inline | \(\| n = 1, 2, 3... (all harmonics) \| \| Closed pipe harmonics \|\) | Line 1235 |
Chapter 12: Electric Charge and Electric Fields
18 equations (18 display, 0 inline)
| # | Type | Equation | Source |
|---|---|---|---|
| 1 | Display | \(\(F = k \frac{\|q_1 q_2\|}{r^2}\)\) | Line 338 |
| 2 | Display | \(\(F_g = G \frac{m_1 m_2}{r^2}\)\) | Line 349 |
| 3 | Display | \(\(\vec{F}_{total} = \vec{F}_{12} + \vec{F}_{13}\)\) | Line 426 |
| 4 | Display | \(\(\vec{E} = \frac{\vec{F}}{q_0}\)\) | Line 442 |
| 5 | Display | \(\(E = k \frac{\|Q\|}{r^2}\)\) | Line 448 |
| 6 | Display | \(\(F = qE\)\) | Line 545 |
| 7 | Display | \(\(U = k \frac{q_1 q_2}{r}\)\) | Line 580 |
| 8 | Display | \(\(W = -\Delta U = -(U_f - U_i)\)\) | Line 593 |
| 9 | Display | \(\(\Delta KE = W = -\Delta U\)\) | Line 599 |
| 10 | Display | \(\(V = \frac{U}{q}\)\) | Line 661 |
| 11 | Display | \(\(V = k \frac{Q}{r}\)\) | Line 667 |
| 12 | Display | \(\(\Delta V = V_B - V_A\)\) | Line 682 |
| 13 | Display | \(\(W = q \Delta V\)\) | Line 686 |
| 14 | Display | \(\(E = -\frac{\Delta V}{\Delta d}\)\) | Line 746 |
| 15 | Display | \(\(E = \frac{V}{d}\)\) | Line 750 |
| 16 | Display | \(\(E = \frac{100 \text{ V}}{0.1 \text{ m}} = 1000 \text{ V/m} = 1000 \text{ N/C}\)\) | Line 762 |
| 17 | Display | \(\(W_{net} = \Delta KE\)\) | Line 793 |
| 18 | Display | \(\(\Delta KE + \Delta PE = 0\)\) | Line 797 |
Chapter 13: Electric Circuits
17 equations (16 display, 1 inline)
| # | Type | Equation | Source |
|---|---|---|---|
| 1 | Display | \(\(I = \frac{\Delta Q}{\Delta t}\)\) | Line 56 |
| 2 | Display | \(\(R = \rho \frac{L}{A}\)\) | Line 148 |
| 3 | Display | \(\(V = IR\)\) | Line 156 |
| 4 | Display | \(\(C = \frac{Q}{V}\)\) | Line 265 |
| 5 | Display | \(\(C = \epsilon_0 \epsilon_r \frac{A}{d}\)\) | Line 281 |
| 6 | Display | \(\(U = \frac{1}{2}CV^2 = \frac{1}{2}QV = \frac{Q^2}{2C}\)\) | Line 362 |
| 7 | Display | \(\(V = L\frac{dI}{dt}\)\) | Line 382 |
| 8 | Display | \(\(L = \mu_0 \mu_r \frac{N^2 A}{l}\)\) | Line 397 |
| 9 | Display | \(\(V(t) = V_0 \sin(2\pi ft)\)\) | Line 485 |
| 10 | Display | \(\(\text{Capacity (Ah)} = \text{Current (A)} \times \text{Time (h)}\)\) | Line 588 |
| 11 | Display | \(\(E = \text{Capacity} \times \text{Voltage}\)\) | Line 594 |
| 12 | Display | \(\(P = IV\)\) | Line 669 |
| 13 | Display | \(\(P = I^2R = \frac{V^2}{R}\)\) | Line 673 |
| 14 | Display | \(\(\text{Energy (kWh)} = \text{Power (kW)} \times \text{Time (h)}\)\) | Line 691 |
| 15 | Display | $$## Series and Parallel Circuits ### Series Circuits In a series circuit, components are c... | Line 695 |
| 16 | Inline | $\text{Cost} = 1.5 \text{ kW} \times 0.25 \text{ h} \times $ | Line 695 |
| 17 | Display | \(\(**Power**: Mechanical power output:\)\) | Line 1082 |
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