Hydrostatics has a long history in the field of fluid mechanics (just shout “Eureka!” to remind you of this—consider jumping out of a bathtub for dramatic effect).^{1,2} However, it is often overshadowed by topics involving fluids in motion in courses that cover fluid mechanics principles. That is understandable given the need to learn about the fundamental principles that underlie fluids in motion as our modern world is dependent on moving and controlling fluids from micro to macro scales. But hydrostatics problems, in particular those that require students to investigate forces on submerged surfaces and, by extension, buoyancy, are excellent tools to reinforce concepts of statics. And statics is the gateway to dynamics. Therefore, a solid foundation in hydrostatics, fostered by exposure to interesting examples, can pay dividends as students progress and attempt to understand fluids in motion.

What can often make a course problem stand out is if (1) it is readily understandable by a wide audience of students (i.e., it is relatable to their own experience); (2) it can be easily visualized; (3) the result can help strengthen or correct intuition; (4) instead of a single result, there is a trend that can be explored; and (5) it can be connected to a data set, demonstration, or experiment to promote the connection between the theoretical analysis and a practical and measurable result.

In contrast to hydrostatic questions involving dams and gates, the “tethered buoy” problem is a straightforward hydrostatics problem that addresses many of the characteristics described above. It is a conceptually accessible problem, as it takes something all of us are familiar with—a floating body—and reasonably extends it in complexity. Many students might already be familiar with the basic idea, e.g., sitting on the end of a floating surfboard, trying the “dead man” float in a pool, or watching competitors in birling (i.e., log rolling). Thus, the problem is understandable, easily visualized, and intuitive. The analysis required to build a theoretical model is rather simple, yet it requires crucial problem-solving elements. Furthermore, the result provides the opportunity to explore trends. Last, as we will show, the problem is amendable to experiments at the benchtop scale. And, from these experiments, a range of conditions can be tested, and interesting data can be collected and compared to a model to have students practice moving beyond the “single-number answer.”

We now present a complete description of the theoretical and experimental analysis of a tethered buoy suitable for an undergraduate college or high-school physics, engineering, or STEM audience.

## Theoretical model

We start with the typical textbook problem for a tethered buoy.^{3,4} In the problem, students are shown a scenario similar to that presented in Fig. 1.

A buoy formed from a rigid and homogenous material with density *ρ*_{b}, diameter *D*, and length *L* is tethered at its base and partially submerged by an amount *d* = *d*_{1} – *d*_{2} in a liquid with density *ρ*_{l}. Without the tether, the buoy would lay horizontally, as the vertical orientation is unstable—an interesting problem on its own.^{4} But, with the tether, the buoy takes on an angle *θ* with respect to the horizontal gas–liquid interface. We consider the gas to have a density *ρ*_{g} ≪ *ρ*_{l}. The goal of this problem is to find *θ* = *f*(*ρ*_{l}, *ρ*_{b}, *L*, *D*, *g*, *d*).^{5}

*F*

_{T}, the buoy weight

*F*

_{W}, and the buoyant force

*F*

_{B}. The point of connection of the tether is assumed to support no moment. As the buoy is static, we can write

*θ*, we need only one equation, and Eq. (2) is sufficient, as it eliminates the need to solve for the tension in the tether.

^{6}In writing Eq. (2), we are locating the line of action of the buoyant force through the center of mass of a cylinder with length

*d*/(2 sin

*θ*) and neglecting the influence of the small submerged volume near the gas–liquid interface. The buoyant force is related to the density and displaced liquid volume:

*θ*yields

*not*a function of the constant cross section of the buoy (i.e., not a function of shape or size). Equation (5) only applies for

*d*≤

*L*(i.e., up to

*θ*= 90°). For

*d*/

*L*> 1, the tethered buoy is stable in its vertical orientation, and

*θ*= 90° no matter the value of the depth.

Equation (5) provides a model that predicts that the angle of the buoy, *θ*, is a function of the depth *d*, the buoy length *L*, and the density ratio *ρ*_{b}/*ρ*_{l} (essentially the specific gravity of the buoy if *ρ*_{l} = *ρ*_{H2O}).^{7} The model demonstrates that the angle is not dependent on the cross-sectional area, nor the cross-sectional shape, so long as the area and shape are constant along the length of the buoy. And, instead of immediately jumping to a single value of *θ* (which is often what textbooks encourage in order to get a “correct answer”), Eq. (5) yields a symbolic solution that allows us to explore the influence of each variable. This exploration is best done through plots.

The trends in Fig. 3 highlight the influence of buoy length *L* on angle *θ*. To observe only the influence of length, the density ratio is held fixed at *ρ*_{b}/*ρ*_{l} = 0.50 (we would see similar trends for any chosen ratio). Within this plot, we can see that as length increases, the angle decreases for any submerged depth. And, as would be expected, the submerged depth at which the buoy becomes vertical increases. This is because the vertical inclination of the buoy is achieved when the lines of action of both the weight and the buoyant force coincide, and the line of action of the buoy weight always increases with length. Note that the *d*/*L* to achieve *θ* = 90° always corresponds to $\rho b/\rho 1$ (0.707 for a density ratio of 0.5). We can also observe the increase in the slope of the curves as *θ* = 90° is approached, meaning that we can expect that measurements in this region will be difficult to resolve, as small changes in *d* can lead to large changes in *θ*.

The influence of buoy density ratio *ρ*_{b}/*ρ*_{l} is seen in Fig. 4, where we have chosen to show five cases: 0.01, 0.25, 0.50, 0.75, and 0.99—the three middle values corresponding to density ratios achieved quite easily in experiments. What we observe in Fig. 4 is the same increase in angle with submerged depth that was found in Fig. 3 (not surprising since both *L* and *ρ*_{b}/*ρ*_{l} are in the denominator). It is intuitive that as the density ratio increases—*ρ*_{b}→*ρ*_{l}—the angle decreases for any fixed submerged depth. In other words, a smaller angle at fixed submerged depth implies larger displaced volume (hence larger magnitude of *F*_{B}), which is needed to balance the moment of the larger weight *F*_{W} of the denser buoy.

We can also see in Fig. 4 that as *ρ*_{b}/*ρ*_{l}→0, *θ*→90° for *d* ∼ 0; i.e., the buoy becomes vertical almost as soon as it is submerged. Again, this should be intuitive, as with a very small weight, very little displaced volume is needed to cause a moment from the buoyant force to set the buoy upright. For any of us who has ever tried to pull a “pool noodle” underwater by its end, we can attest to this behavior!

The final way to view the trends of Eq. (5) is in a dimensionless form. This is captured in Fig. 5. The dimensionless trend has a form similar to the plots in Figs 3 and 4. As with _{,} Fig. 3, a vertical orientation is achieved when $d/L=\rho b/\rho 1$, or as shown in Fig. 5 when $(1/\rho b/\rho 1)(d/L)=1$. Both Eq. (5) and Fig. 5 indicate that any combination of buoy density ratio or length should collapse into a single trend when plotted in dimensionless form.

## The experiment setup

The model for our tethered buoy guides the choice of our experiments. To assess the validity of the model, we needed to manufacture buoys with various length *L*, density *ρ*_{b}, and diameter^{8} (or side length) *D*, and to create experimental conditions where we can vary submerged depth *d* and measure the resulting angle *θ*. We provide brief descriptions of the buoys, tank setup, and image analysis technique here. Considerable detail, for the benefit of instructors, is given in the Appendix.^{9}

### Buoys, tank, and image analysis

In order that experiments be easily adaptable to the classroom, buoys needed to be small in scale and made of readily available materials. To meet these requirements, we settled on buoys made from wood dowels, wood strips, and plastic drinking straws. The hollow straws allowed for a filler to be inserted to adjust the average density. Figure 6 shows several buoys. Overall, 19 different buoys were constructed, providing a range of length, density, and diameter. A summary of this information can be found in Table I.

Experiment Category . | L (mm)
. | D (mm)
. | ρ_{b} (g/cm^{3})
. |
---|---|---|---|

L variation (6 buoys) | 50–300 ± 1 | 7.7 ± 0.05 (7.7–7.8) | 0.45 ± 0.02 (0.42–0.47) |

D variation (7 buoys) | 200 ± 1 | 6.3–18.9 (±1%) | 0.47 ± 0.04 (0.40–0.51) |

ρ_{b} variation (8 buoys) | 200 ± 1 | 6.5–9.4 (±1%) | 0.11–0.73 (±3%) |

Experiment Category . | L (mm)
. | D (mm)
. | ρ_{b} (g/cm^{3})
. |
---|---|---|---|

L variation (6 buoys) | 50–300 ± 1 | 7.7 ± 0.05 (7.7–7.8) | 0.45 ± 0.02 (0.42–0.47) |

D variation (7 buoys) | 200 ± 1 | 6.3–18.9 (±1%) | 0.47 ± 0.04 (0.40–0.51) |

ρ_{b} variation (8 buoys) | 200 ± 1 | 6.5–9.4 (±1%) | 0.11–0.73 (±3%) |

An acrylic tank ∼46 cm × 46 cm × 46 cm was available for use, which provided sufficient room for the buoys (and set the longest buoy length tested, i.e., 300 mm) in addition to excellent optical access. Based on the number of buoys tested, it was decided to maintain the liquid level in the tank (*d*_{1} in Fig. 1) and to vary the depth of the bottom end of the buoy (*d*_{2} in Fig. 1) using a simple adjustable tether made from monofilament fishing line guided through a submerged eye bolt and wrapped around a wooden dowel. A diagram of the tank can be found in Fig. 7. A panel of images showing the process of submerging a buoy and a superposition of multiple images more completely showing the path taken by the buoy as it is submerged is captured in Fig. 8.

The two pieces of information collected from photographs were the submerged depth *d* and the buoy angle *θ*. A partially automated approach to image analysis was implemented. A MATLAB script was written that, for each image, prompted the user to identify the base of the buoy and any other single point along the buoy axis. The buoy base, referenced to the free surface location, yielded *d*, and *θ* was computed using the horizontal and vertical distances between the two selected points. This process allowed for the rapid extraction of data from photographs (see Fig. 9). Because of the fixed location of the camera, camera leveling, and the user input of points to evaluate, there is uncertainty in values of *d* and *θ*. We estimate these to be ±2.5 mm and ±2°, respectively.

## Results and discussion

We can assess the validity of Eq. (5) by comparing the theoretical predictions to data collected from experiments. Specifically, we can look at trends associated with variations in buoy length, density, and diameter.

To begin, let us look at the results shown in Fig. 10, which were obtained from experiments in which buoy length was varied, but buoy diameter and density were fixed.

In these experiments, the same dowel material was used, but there were small variations in buoy density and diameter (e.g., ±4.4% and ±2%, respectively). These variations do not play a role in the trends observed. Inspection of Fig. 10 leads us to conclude that the predicted trend in *θ* vs. *d* associated with changes in buoy length is correct. In other words, we can clearly see the qualitative agreement between theory and experiment, and this agreement spans a factor of six change in length.

Although qualitative agreement is excellent, a quantitative comparison shows that there can be large discrepancies. This is suggested in Fig. 10 for the *L* = 50 mm case, and is more pronounced for angles nearing *θ* = 90°. When we compute the difference in angle, Δ*θ* = *θ*_{theory} – *θ*, and plot Δ*θ* vs. measured angle *θ*, we can see the trend more clearly (see Fig. 11).

Referring to Fig. 11, differences in angle tend to be small for *θ* ≈ 0° owing to the small angles, remain small until ∼60°, and increase dramatically as *θ*→90°, but then decrease as the buoys become vertical. This increase is expected, as the steepness of the trend increases in this region, and so small differences in *d* lead to large changes in *θ*. In particular, the data for the shortest buoy exhibits larger Δ*θ* over a wide range of angles. One explanation for this may be that for small *L*/*D* aspect ratios, the approximation for the line action of the buoyant force breaks down. We did not test other short buoys to explore this hypothesis.

A simpler explanation lies with the uncertainty in the measurement of *d*. We mentioned before that a conservative estimate of the uncertainty is ±2.5 mm, and in Fig. 11, it appears that our data nearly always underpredicts the theory. Thus, if there were a bias in the measurement of *d*, this would create pronounced differences between theory and experiment for the *L* = 50 mm case in particular. Applying a shift in the *data* for this buoy shows that the average Δ*θ* drops from 6.1% to 1.4%, and a visual of this can be seen in the inset of Fig. 11.

Using buoys constructed of different materials, we can investigate the influence of buoy density *ρ*_{b}. This is shown in Fig. 12. Here we see the behavior of eight different buoys with density ranging from *ρ*_{b} = 0.11 g/cm^{3} (hollow drinking straw) to *ρ*_{b} = 0.73 g/cm^{3} (oak). This is a nearly factor of 7 change in density. It is apparent that the experiments prove that the influence of buoy density is similar to that predicted by Eq. (5).

*y*=

*x*

^{a}, where

*a*is predicted to be –1/2. We can extract the experimental value of

*a*from a curve fit to the data. To do this, we use values of

*θ*and

*d*for angles between 10° and 80° to compute sin

*θ*(

*L*/

*d*) for each density ratio. We take the average for a data set and plot vs.

*ρ*

_{b}/

*ρ*

_{l}. The result is shown in Fig. 13, where log–log axes have been used to make the power-law trend appear linear. The slope of the data yields a power-law exponent of

*a*= 0.48 (in agreement with the theoretical prediction of 0.50). A similar approach can be taken with the data from Fig. 10 to assess the trend with changing

*L*.

Despite the quantitative agreement in overall trend, the data in Fig. 12 shows that the buoy with *ρ*_{b} = 0.27 g/cm^{3} (balsa wood) exhibits a large difference from theory (as high as ∼20° around *d* = 100 mm). Even when considering the uncertainty in the density (only ±2.1%) and uncertainty in *d*, and after replicating the results by a second experiment, we are confident that the discrepancy between theory and experiment is not the result of a simple measurement error. What is interesting about the buoy that yields this data is that it is the lowest-density wood buoy (buoys with lower densities are made with plastic straws). The wood buoy surfaces, compared to the plastic straw buoys, may be sufficiently hydrophilic to produce a capillary force at the air–water interface. This capillary force can result in a moment that adds to the moment produced by the buoy weight. Such a moment would reduce the inclination angle for any submerged depth and become less significant as the buoy weight increases (e.g., for larger buoy densities). Crude estimates based on this hypothesis confirm the trend but still do not fully explain the magnitude of the discrepancy between theory and experiment. A more detailed theoretical analysis, and supporting experiments with low-density hydrophilic buoys, might shed more light on these types of buoy conditions. However, despite this unknown, this single data set should not prevent us from concluding that the influence of *ρ*_{b}/*ρ*_{l} on *θ* vs. *d* is as predicted by Eq. (5).

What remains is to consider whether buoy diameter plays any important role in the inclination angle, since the derivation of Eq. (5) showed that diameter does *not*. Because this might be counter to our intuition, we can assess the validity of the lack of influence of *D* by considering the results of experiments performed with buoys of various diameter. These results are provided in Fig. 14, where buoys of different diameter but made from the same dowel material (poplar) were used. Buoys ranged from 6.3 mm to 18.9 mm in diameter. The collapse of the data onto a single trend, considering the small spread in buoy density, is evidence that Eq. (5) is correct—buoy diameter *does not* play a role.

Perhaps the best way to validate Eq. (5) is to plot the measurements from all 19 buoys together in dimensionless form. Figure 5 suggested that, in this way, the behavior of any buoy would follow a single trend, and we can see in Fig. 15 that this is what we observe in practice. The data representing buoys of various size, shape/length, and material collapse to form a single identifiable trend and are consistent with the prediction of Eq. (5) (shown as the solid line partially obscured by the data). The two data sets that diverge slightly from the pack are from the buoys already discussed (i.e., the *L* = 50 mm from Figs. 10 and 11 and the *ρ*_{b}/*ρ*_{l} = 0.27 buoy of Fig. 12).

## Conclusions

Analysis of a “tethered buoy,” including both theoretical work to yield a model for inclination angle and experiments to extract data for assessing the validity of several aspects of the model, have been described in detail for the benefit of physics and engineering faculty who are interested in having their students pursue a hydrostatics problem beyond the typical textbook “single-value answer.” Such a seemingly simple problem, and equally straightforward experiments, can yield a plethora of interesting results.^{9} In addition, there are multiple ways to consider the analysis of the data that incorporate both visuals (plots and qualitative comparisons) and quantitative techniques. We encourage both faculty and students to explore similar problems, using a theory–experiment approach like this, to engage students in deeper thinking about engineering and physics problems.

## References and endnotes

*Fundamentals of Fluid Mechanics*, 6th ed. (

*Statics*, 2nd ed. (

*Fluid Mechanics*, 8th ed. (

*θ*=

*f*(

*ρ*

_{b}/

*ρ*

_{l},

*d*/

*L*,

*D*/

*L*). However, these techniques are often encountered after fluid statics is taught in undergraduate fluid mechanics classes and rarely in basic physics. Therefore, we do not include the details of the approach here.

*θ*, we do not need to know the magnitude of the tension in the tether, it is important to point out that the tension provided by the tether is a necessary force to establish equilibrium for the inclined buoy. Why? Because a buoy of the design used in the experiments, with an equal distribution of mass and therefore a centrally located center of mass, is not stable in any orientation except horizontal. Achieving an equilibrium with an orientation at any angle other than

*θ*= 0° requires the application of a positive tension

*F*

_{T}. Thus, investigating the tension in the tether for these buoys provides an opportunity to further expand on the theory of hydrostatics and floating bodies including their stability. Such an investigation can also point to some interesting differences between the values of tension vs. submerged depth computed for the inclined buoys and values for buoys constrained to always be at

*θ*= 90°. For example, for inclined buoys, substitution of Eqs. (3)–(5) into Eq. (1) suggests that

*F*

_{T}is a positive constant value and independent of submerged depth

*d*for 0° <

*θ*≤ 90°. Once at

*θ*= 90°, the dimensionless submerged depth will be

*d*/

*L*= (

*ρ*

_{b}/

*ρ*

_{l})

^{1/2}. Beyond this, an increase in the tension force will further submerge the buoy until it is fully submerged (beyond which no further increase in tension is required to submerge further). Remember that Eq. (5) is based on summing moments. This is in contrast to a buoy constrained, because of the unstable nature of the buoy, to

*θ*= 90°. In this case, no analysis of moments is needed, and Eq. (1) predicts an upward force (i.e., a negative tension—not possible from a cable) required to keep the buoy at values

*d*/

*L*< (

*ρ*

_{b}/

*ρ*

_{l})

^{1/2}. Once the depth

*d*/

*L*= (

*ρ*

_{b}/

*ρ*

_{l})

^{1/2}is reached, the buoy floats, and the tension is computed to be

*F*

_{T}= 0. With an increase in tension from the tether, the buoy will submerge further. And so it may appear that two vertical buoys, submerged such that

*d*/

*L*> (

*ρ*

_{b}/

*ρ*

_{l})

^{1/2}, require two different values of tension based on whether they are allowed to incline or constrained vertically. Such a study of these cases, once highlighted by considering the tension in the tether, would make for an additional theoretical

*and*experimental analysis for a student physics/engineering audience.

*TPT Online*at https://doi.org/10.1119/5.0171072, in the “Supplementary Material” section.