_Omri

AFAIU the tension in the string matches the radial component of gravity ([math]gcos \theta[/math]) at every point during the swing for a net force of zero at the radial component, and they would both be zero at 90 degrees. At displacements above 90 degrees the string could become slack, but not less than that. Depends on the choice of the drag coefficient [math]C_d[/math] and the phone's surface area hitting the air [math]A[/math]. E.g. [math]C_d=0.35[/math] and [math]A=50 \text{ cm}^2=0.005 \text{ m}^2[/math] (half the phone HxW), [math]F_{drag} = \frac{1}{2}C_d\rho Av^2 = 0.0042 \text{N} [/math] Which is [math]100\frac{F_{drag_{max}}}{F_{t_{max}}}=100\frac{0.0042}{0.8428}=0.5\%[/math] of the maximum tangential force applied on the phone. More complete analysis throughout the whole movement is needed to confirm, but this would be consistent with the statement, 'drag is a non-negligible part of the discrepancy'.

I think I that still don't understand where you are going with this. If you are questioning drag, are you implying that the discrepancy should not be there, or do you mean that friction played a larger role, or that the experiment was flawed in some way? 30 degrees was measured from the vertical. Gravity keeps the string stretched.

Thank you! Yes, I wrote it, it seems plausible to me. I would expect the empirical observations to be higher than the theoretical one because of drag/friction but not by much since these are not big effects. Is there more justification needed? How could I quantify/estimate the effects of drag/friction? I was following this experiment, they used 30 degrees. https://www.sciencebuddies.org/science-fair-projects/project-ideas/Phys_p016/physics/pendulum-motion I tried plugging in different values into equation 3, it does make a difference relatively small, e.g. the values for 18 and 45 degrees are nearly identical. Generally speaking though, the bigger the angle the longer the period. It grows like [math]sin(\theta)E(\theta)[/math], we E is the complete elliptical integral of the first kind. I think this would shift the center of mass (essentially shortening the length) and depending on the thickness of the bar, also increase drag. Are there other methods you had in mind? Thanks so much for your feedback!

Introduction Hello folks, I've recently started teaching myself physics and one of the ways I do this was through hands-on experiments. I am looking for feedback on these, really any kind, for example Notes on experiment setup What would you have done differently? Mistakes, things not considered or mentioned How would you follow up on this experiment? Note: if the math on this page does not render properly for you, it is also available as a notion site here https://flawless-door-cdd.notion.site/Pendulum-Experiment-144c27137da88054b4eff55713e23c4e?pvs=74 Goal Measure the relationship between a pendulum length and its period Compare predictions based on Newton’s laws with empirical results Methods 3 separate trials for 3 different pendulum lengths (27.5, 37.5 and 50cm) were performed. The pendulum was constructed using of a piece of lightweight twine tied vertically to a ASUS Zenfone 10 (172.0g, 14.65cm height, 6.81cm width) on the lower end and a ruler on the top end. The ruler was weighed down using a textbook, allowing the twine and smartphone to hang off the edge of a desk. The smartphone was raised and released from a 30 degree angle and allowed to swing until rest. Data[math]^{[1]}[/math] was gathered using the phyphox app and the phone’s accelerometer. Predictions Length of Period Simplifying assumptions: Twine is massless Friction between twine and ruler, smartphone negligible Drag negligible Neglecting drag, we can approximate the maximum speed at the bottom of the pendulum using conservation of mechanical energy: [math]K_i+U_{Gi}=0+mg(L-Lcos\frac{\pi}{6})=\frac{1}{2}mv_{max}^2+0=K_f+U_{Gf}[/math] [math]\begin{equation} \tag{1} v_{max}=\sqrt{2gL(1-cos\frac{\pi}{6})} \end{equation}[/math] Similarly, we can use Newton’s 2nd law to get the same expression [math]F_{net_t}=-mgsin\theta =ma_t \rightarrow a_t=-gsin\theta[/math] [math]a_t=\alpha L=\frac{d \omega}{dt}L=\frac{\omega d \omega}{d\theta}L[/math] [math]-\frac{g}{L}sin\theta d\theta=\omega d \omega[/math] Integrating both sides, we get [math]\frac{g}{L}cos\theta=\frac{\omega^2}{2} + C[/math] For [math]\omega=0[/math], [math]\theta=\frac{\pi}{6}[/math], therefore [math]C=\frac{g}{L}cos\frac{\pi}{6}[/math] [math]\begin{equation} \tag{2} \omega(\theta)=\sqrt{2\frac{g}{L}(cos\theta-cos\frac{\pi}{6})}\end{equation}[/math] For [math]\omega_{max}[/math], [math]\theta=0[/math], therefore [math]v_{max}=\omega_{max}L=\sqrt{2gL(1-cos\frac{\pi}{6})}[/math] Using [math](2)[/math], we can find the period by integrating over the first quarter of the pendulum’s motion [math]\omega(\theta)=\frac{d\theta}{dt} \rightarrow \int_{0}^{\frac{T}{4}}dt=\int_{-\frac{\pi}{6}}^{0}\frac{1}{w(\theta)}[/math] [math]\begin{equation} \tag{3} T=4\int_{-\frac{\pi}{6}}^{0}\frac{1}{\sqrt{2\frac{g}{L}(cos\theta-cos\frac{\pi}{6})}}\end{equation}[/math] The right hand side can be computed numerically[math]^{[2]}[/math] for the different values of [math]L[/math], yielding the following predictions [math]T(L=0.5)=1.44\text{s}[/math] [math]T(L=0.37)=1.24\text{s}[/math] [math]T(L=0.27)=1.06\text{s}[/math] We also predict the results to be proportional to the square root of the length, i.e. [math]T \propto \sqrt{L}[/math]. Results Periods were calculated using the average difference between subsequent acceleration peaks during the first 10 seconds in each trial. Length (cm, measured to middle of phone) Avg. Trial 1 Period (s) Avg. Trial 2 Period (s) Avg. Trial 3 Period (s) Avg. of Trials (s) 27.5 1.09 1.09 1.09 1.09 37.5 1.26 1.26 1.26 1.26 50 1.46 1.48 1.47 1.47 Discussion The results of the experiment agree very closely with our predictions. There is a consistent discrepancy in the empirical data showing longer periods by 2.5-3.3 seconds, presumably owning to drag and friction. Raw data and results can be accessed here https://drive.google.com/drive/u/0/folders/1nR-IkVcfyhPUA8RYpGqp_Z4cSbe8epto https://www.integral-calculator.com/ was used for this task