[Experiment] Period of a Pendulum

[_Omri]

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.

1. Raw data and results can be accessed here https://drive.google.com/drive/u/0/folders/1nR-IkVcfyhPUA8RYpGqp_Z4cSbe8epto
2. https://www.integral-calculator.com/ was used for this task

Edited Monday at 12:14 PM by _Omri
Math formatting again

[studiot]

Just now, _Omri said:

Hello folks, I've recently started teaching myself physics and one of the ways I do this was through hands-on experiments.

Good on you. +1

 

Just now, _Omri said:

I am looking for feedback on these, really any kind,

 It was noted in the experimental link that

Quote

Discussion

 

The results of the experiment agree very closely with our predictions. There is a consistent discrepency in the empirical data showing longer periods by 2.5-3.3 seconds, presumably owning to drag and friction.

Do you agree with this ?

If so or not why ?

 

Whay was the lift angle chosen to be 30 degrees ?

What difference does it make if any ?

 

The pendulum is one of those experiments that can offer much more insight than just verifying the period formula.

Why was twine used as the suspension method and would other methods perform differently ?
Note pendulums clocks often have rigid bar support arms.

Other aspects of this experiment are being discussed here.  Scroll down the page to the sketch.

 

[_Omri]

22 hours ago, studiot said:

Good on you. +1

Thank you!

 

22 hours ago, studiot said:

Do you agree with this ?

If so or not why ?

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?

22 hours ago, studiot said:

Whay was the lift angle chosen to be 30 degrees ?

What difference does it make if any ?

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.

22 hours ago, studiot said:

Why was twine used as the suspension method and would other methods perform differently ?
Note pendulums clocks often have rigid bar support arms.

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!

[studiot]

Perhaps you misunderstood my comments.

I was questioning the importance of the drag effect.

Further I am having trouble understanding how a displacement of 30^o could keep the swing arm properly constant.
Wouldn't the string be slack ?

And you should make clear whether the 30^o was measured from the horizontal or vertical.

[_Omri]

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.

Edited Tuesday at 12:51 PM by _Omri
grammar

[studiot]

Just now, _Omri said:

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.

Look at your formula estimate for maximum speed.

I make it less than 2m/s or walking pace ie less than 4  miles per hour.

How much air resistance do you anticipate at this speed ?

 

Look at this google page in response to a question "how far should a pendulum swing ?

 

 

Gravity only acts vertically.

The string tension has to have both vertical and horizontal components.

What hppens if you increase the angular displacement to 90^o ?

 

Basically I am encouraging you to think about the mechanics of the setup.

[_Omri]

3 hours ago, studiot said:

Gravity only acts vertically.

The string tension has to have both vertical and horizontal components.

What hppens if you increase the angular displacement to 90^o ?

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.

 

3 hours ago, studiot said:

Look at your formula estimate for maximum speed.

I make it less than 2m/s or walking pace ie less than 4  miles per hour.

How much air resistance do you anticipate at this speed ?

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'.

Edited Tuesday at 09:26 PM by _Omri
Math formatting