Physics 101 Cheatsheet

Cheatsheet Content

### Kinematics: Motion in 1D - **Velocity:** $v = v_0 + at$ - *Use when:* Velocity changes due to constant acceleration. - **Position:** $x = x_0 + v_0t + \frac{1}{2}at^2$ - *Use when:* Finding position after time $t$ with constant acceleration. - **Velocity without time:** $v^2 = v_0^2 + 2a(x - x_0)$ - *Use when:* Time is unknown but displacement is given. - **Average velocity:** $v_{avg} = \frac{x_f - x_i}{t_f - t_i}$ - *Use when:* Finding velocity over a time interval. #### Graphical Interpretations - Velocity = slope of position ($x$) vs. time ($t$) graph. - Acceleration = slope of velocity ($v$) vs. time ($t$) graph. - Area under acceleration ($a$) vs. time ($t$) graph = change in velocity. - Area under velocity ($v$) vs. time ($t$) graph = displacement. ### Projectile Motion (Motion in 2D) - Break into horizontal ($x$) and vertical ($y$) components. - **Horizontal motion:** No acceleration ($a_x = 0$, constant velocity). - $x = v_{0x} t$ - **Vertical motion:** $a_y = -g$ (gravity pulls downward). - $y = v_{0y} t - \frac{1}{2} g t^2$ - **Time of flight:** $t = \frac{2v_0 \sin\theta}{g}$ - *Use when:* Projectile starts and lands at the same height. - **Range:** $R = \frac{v_0^2 \sin(2\theta)}{g}$ - Maximum range at $\theta = 45^\circ$. - **Max height:** $h = \frac{v_0^2 \sin^2\theta}{2g}$ - Time to max height = $v_{0y} / g$. #### Problem-Solving Steps 1. Break initial velocity into components: $v_{0x} = v_0 \cos\theta$, $v_{0y} = v_0 \sin\theta$. 2. Use kinematic equations for $x$ and $y$ separately. 3. At maximum height, $v_y = 0$. 4. When solving for total time, analyze vertical motion. ### Vector Math - Vectors have both magnitude and direction. - **Components:** $A_x = A \cos\theta$, $A_y = A \sin\theta$. - **Magnitude:** $A = \sqrt{A_x^2 + A_y^2}$. - **Direction (angle $\theta$):** $\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$. - **Vector Addition:** Sum their x and y components separately. ### Newton’s Laws of Motion - **1st Law (Inertia):** An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction, unless acted upon by an unbalanced external force. - **2nd Law (Force & Acceleration):** $\sum \vec{F} = m\vec{a}$ - Force causes acceleration, proportional to mass. - **3rd Law (Action-Reaction):** For every action, there is an equal and opposite reaction. #### Common Forces - **Weight:** $F_g = mg$ (always downward). - **Normal force:** $F_N$ (force from a surface, perpendicular to the surface). - **Friction:** - **Static:** $f_s \le \mu_s N$ (prevents motion). - **Kinetic:** $f_k = \mu_k N$ (resists sliding). - **Tension ($T$):** Pulling force in ropes/strings. - **Inclined Plane:** - Force parallel to incline: $mg \sin\theta$. - Force perpendicular to incline: $mg \cos\theta$. ### Circular Motion - Acceleration and force always point toward the center of the circle. - **Centripetal acceleration:** $a_c = \frac{v^2}{r}$ - Required to keep an object moving in a circle. - **Centripetal force:** $F_c = \frac{mv^2}{r}$ ### Work and Energy - **Work done by a force:** $W = F D \cos\theta$ (where $\theta$ is angle between F and D) - **Hooke's Law (Spring Force):** $F_s = -kx$ - **Kinetic Energy (KE):** $KE = \frac{1}{2}mv^2$ - **Work-Energy Principle:** $W_{net} = \Delta KE$ - **Work on System:** $W_{on\ sys} = \Delta KE + \Delta PE$ - **Gravitational Potential Energy (PE_grav):** $PE_{grav} = mgh$ - **Elastic Potential Energy (PE_elastic):** $PE_{elastic} = \frac{1}{2}kx^2$ - **Conservation of Mechanical Energy (if no non-conservative forces):** - $KE_i + PE_i = KE_f + PE_f$ - **Power:** $P = \frac{W}{t}$ or $P = Fv$ ### Universal Gravitation and Orbits - **Newton's Law of Universal Gravitation:** $F_G = G \frac{m_1 m_2}{r^2}$ - $G = 6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$ ### Momentum and Collisions - **Momentum:** $\vec{p} = m\vec{v}$ - **Conservation of Momentum (1D):** $m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$ - Applies when no external net force acts on the system. - **Center of Mass:** - Position: $\vec{r}_{CM} = \frac{\sum m_i \vec{r}_i}{\sum m_i}$ - Velocity: $\vec{v}_{CM} = \frac{\sum m_i \vec{v}_i}{\sum m_i}$ ### Rotation and Torque - **Torque:** $\vec{\tau} = \vec{r} \times \vec{F}$ or $\tau = r F \sin\phi$ - $\phi$ is the angle between $\vec{r}$ and $\vec{F}$. - **Newton’s 2nd Law (Rotational):** $\sum \vec{\tau} = I \vec{\alpha}$ - $I$ = Moment of Inertia, $\vec{\alpha}$ = Angular Acceleration. - **Moment of Inertia ($I$):** - For a point mass: $I = mr^2$ - For collections of point masses: $I = \sum m_i r_i^2$ - **Angular Momentum:** $\vec{L} = I\vec{\omega}$ or $\vec{L} = \vec{r} \times \vec{p}$ - For a point mass: $L = m v_{tan} r = m \omega r^2$ - **Angular Velocity:** $\omega = \frac{\Delta\theta}{\Delta t}$ - Tangential speed $v = \omega R$ - **Rotational Kinetic Energy:** $K_{rot} = \frac{1}{2}I\omega^2$ ### Oscillations - **Hooke’s law for ideal spring:** $F_s = -kx$ - **Spring Potential Energy:** $U_s = \frac{1}{2}kx^2$ - **Period of oscillation:** - **Spring-mass system:** $T_{spring} = 2\pi \sqrt{\frac{m}{k}}$ - **Simple pendulum (small angles):** $T_{pendulum} = 2\pi \sqrt{\frac{L}{g}}$