Absolutely! I’ll provide a **step-by-step guide** for a sample project based on your assignment. Let's use a **simple pendulum** (like the one you used in class) as the vibrating object, but you can easily adapt these steps to another object (like a ruler clamped to a desk, a car antenna, etc.). --- ## **Step-by-Step Solution** ### **1. Select a Vibrating Object** - **Object:** Simple Pendulum (a small weight on a string) - **Reason:** Easily accessible, frequency < 60 Hz, simple to model. --- ### **2. Model the Object (1 DOF)** A simple pendulum is well-approximated as a single degree-of-freedom (1 DOF) system. **Equation of motion:** \[ mL^2 \ddot{\theta} + b\dot{\theta} + mgL\theta = 0 \] where: - \( m \) = mass - \( L \) = length - \( b \) = damping coefficient - \( \theta \) = angle from vertical For small angles, \(\sin\theta \approx \theta\). --- ### **3. Measurement from Video** - **Record** a video of the pendulum swinging (use a phone camera). - **Extract Frequency:** - Use a stopwatch or video timer to measure the time for several oscillations (say, 10 swings). - **Period \(T\):** Average time per swing = total time / number of swings. - **Frequency \(f\):** \( f = 1/T \) - **Estimate Mass:** - Use a small object (known mass, e.g., a 50 g weight or similar). - **Estimate Length:** - Measure from pivot to center of mass of the bob. - **Damping:** - Observe amplitude decrease per cycle. - **Logarithmic decrement:** \[ \delta = \ln \left( \frac{A_1}{A_2} \right) \] where \(A_1, A_2\) are the amplitudes of two successive peaks. --- ### **4. Calculate Model Parameters** - **Natural frequency (undamped):** \[ f_n = \frac{1}{2\pi} \sqrt{\frac{g}{L}} \] (for small angles) - **Damping Ratio:** \[ \zeta = \frac{\delta}{\sqrt{4\pi^2 + \delta^2}} \] - **Damped frequency:** \[ f_d = f_n \sqrt{1-\zeta^2} \] --- ### **5. Analyze Response to Input** Let's analyze response to **initial conditions** (release from angle \(\theta_0\), zero velocity): - **Solution:** \[ \theta(t) = \theta_0 e^{-\zeta \omega_n t} \cos(\omega_d t) \] where \(\omega_n = 2\pi f_n\), \(\omega_d = \omega_n \sqrt{1-\zeta^2}\). - **Compare model prediction to video data**: Plot measured vs. predicted amplitude over time. --- ### **6. MATLAB/Hand Calculation** You can simulate the above equation in MATLAB or Excel, or just plot by hand. --- ## **Summary Example (for your report)** ### **Description of Project** We recorded the vibration of a simple pendulum made from a 50g weight and a string of length 0.5m. A video was taken of the pendulum swinging after being released from a small angle. Using video timing, we measured the period of oscillation and estimated mass and length. ### **Model and Parameters** The pendulum is modeled as a 1 DOF system. Using the measured length (\(L = 0.5\) m), the theoretical natural frequency is: \[ f_n = \frac{1}{2\pi} \sqrt{\frac{9.81}{0.5}} \approx 0.7 \text{ Hz} \] Video measurement: 10 swings in 14 seconds; \(T = 1.4\) s; \(f = 0.71\) Hz (matches theory). Measured amplitude dropped from 10 cm to 7 cm in 5 cycles: \[ \delta = \ln(10/7) = 0.357 \] \[ \zeta = \frac{0.357}{\sqrt{4\pi^2 + 0.357^2}} \approx 0.057 \] ### **Analysis** The response to initial displacement matches the model prediction closely for frequency and damping. Simulated and measured amplitude decay are similar. Slight discrepancies could be due to air resistance or pivot friction. ### **PowerPoint Slides (Suggested Layout)** 1. **Title Slide**: Vibration of a Simple Pendulum 2. **Object/Measurement**: Photo of setup, video screenshot, measured values 3. **Model**: Equation, 1DOF diagram, parameter table 4. **Analysis**: Plots of measured vs. model response, discussion of accuracy --- ## **Final Answer (Summary Table)** | Parameter | Measured Value | Theory/Model | |----------------------|---------------|--------------| | Length | 0.5 m | 0.5 m | | Mass | 0.05 kg | 0.05 kg | | Period (T) | 1.4 s | 1.43 s | | Frequency (f) | 0.71 Hz | 0.7 Hz | | Damping Ratio (\(\zeta\)) | 0.057 | -- | **Conclusion:** The simple pendulum is accurately modeled as a 1DOF system. The measured vibration frequency and damping match theoretical predictions closely. --- ### **What to Submit** - **3-page written summary**: As above (object, model, measurement, analysis, conclusion) - **PowerPoint**: 4 slides as suggested above --- ### **Tips for Other Objects** - For a ruler or beam, use a cantilever beam formula for frequency. - For a mass-spring system, use \( f_n = \frac{1}{2\pi}\sqrt{k/m} \). - You can estimate mass with a scale, length with a ruler, frequency from video. --- **If you want an example using a specific object (like a phone spring, a car antenna, a ruler, etc.), let me know and I'll tailor the steps!**