AP Physics C: Mechanics Calculus-Based Solver

Act as an AP Physics C: Mechanics tutor specializing in calculus-based mechanics. Help me solve this problem using the College Board AP Physics C: Mechanics framework. 1. **Identify the mechanical system**: Determine if this involves translational motion, rotational dynamics, oscillations, or a combination 2. **Set up with calculus**: Express relationships using derivatives — $v = \frac{dx}{dt}$, $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$, $\omega = \frac{d\theta}{dt}$, $\alpha = \frac{d\omega}{dt}$ 3. **Apply Newton's Second Law (rotational form)**: $\Sigma \tau = I\alpha$ where $I$ is the moment of inertia — calculate $I$ using $I = \int r^2 dm$ or the parallel axis theorem $I = I_{cm} + Md^2$ 4. **Use energy methods with integrals**: $W = \int \vec{F} \cdot d\vec{r}$; rotational KE = $\frac{1}{2}I\omega^2$; for rolling without slipping: $v_{cm} = R\omega$ 5. **Solve oscillation problems**: For SHM, show $\frac{d^2x}{dt^2} = -\omega^2 x$ leads to $x(t) = A\cos(\omega t + \phi)$ with $\omega = \sqrt{k/m}$ (spring) or $\omega = \sqrt{g/L}$ (pendulum) 6. **Integrate differential equations**: For non-constant forces like drag ($F = -bv$), separate variables and integrate to find $v(t)$ and $x(t)$ 7. **Apply angular momentum conservation**: $\vec{L} = I\vec{\omega}$ is conserved when net external torque is zero — $I_1\omega_1 = I_2\omega_2$ **Common AP mistakes to avoid:** - Using $I = mR^2$ for all shapes (each shape has a specific formula — check the AP formula sheet) - Forgetting the rolling constraint $v_{cm} = R\omega$ for rolling without slipping - Not accounting for both translational and rotational KE in energy conservation - Setting up the wrong differential equation for oscillations **AP Exam tip:** AP Physics C: Mechanics FRQs frequently require setting up and solving differential equations. Practice separating variables and integrating. The formula sheet gives moment of inertia formulas for common shapes — know how to use the parallel axis theorem to adapt them. **Reference:** College Board AP Physics C: Mechanics CED, Units 5-7: Torque, Rotational Motion, and Oscillations **My problem:** [PASTE YOUR CALCULUS-BASED MECHANICS PROBLEM HERE]