Gravitational Potential Energy Change Research Articles

A low-cost method for quantifying bouncing ball dynamics with smartphone video

Abstract This study employed a low-cost method for quantifying bouncing ball dynamics with smartphone video. High-speed video analysis was used to investigate the bouncing behaviour of different balls, including table tennis, rubber, and tennis balls, on a table tennis surface. To examine the effect of surface material on bouncing dynamics, the table tennis ball was also tested on wood and tile surfaces. Key parameters analysed included bounce height decay, the coefficient of restitution (COR), and the bounce half-life. Additionally, we measured the total bouncing time and analysed the relationship between changes in gravitational potential energy, kinetic energy, and total mechanical energy (ME). Our results revealed a significant correlation between the COR and the decay rate of bounce height. Balls with higher COR values, indicative of lower energy loss per bounce, exhibited slower decay rates, resulting in longer bounce half-lives and extended total bounce times. For instance, the table tennis ball on a tile surface demonstrated a high COR of 0.93, a bounce half-life of 4.65 bounces, and a total bounce time of 8.22 ± 0.34 s. This suggests that material properties such as stiffness and elasticity of both the ball and the surface significantly influence the COR and decay rate, beyond just surface smoothness. Furthermore, our analysis of ME changes in the bouncing balls aligns with theoretical predictions, providing insights into the energy transformations occurring during bouncing. This improved understanding can also be a valuable tool for educators, helping teachers and students bridge the gap between theory and real-world observations, leading to a deeper grasp of the concepts.

The role of surface processes in basin inversion and breakup unconformity

In the context of continental extension, transient compressional episodes (stress inversion) and phases of uplift (depth inversion) are commonly recorded with no corresponding change in plate motion. Changes in gravitational potential energy during the rifting process have been invoked as a possible source of compressional stresses, but their magnitude, timing, and relationship with depth inversions remain unclear. Using high-resolution two-dimensional numerical experiments of the full rifting process, we track the dynamic interplay between the far-field tectonic forces, loading and unloading of the surface via surface processes, and gravitational body forces. Our results show that rift basins tend to localize compressive stresses; they record transient phases of compressional stresses as high as 30 MPa and experience a profound depth inversion, 2 km in magnitude, when sediment supply ceases, providing an additional driver for the breakup unconformity, a well-documented phase of regional uplift typically associated with continental breakup.

The role of surface processes in basin inversion and breakup unconformity

Abstract In the context of continental extension, transient compressional episodes (stress inversion) and phases of uplift (depth inversion) are commonly recorded with no corresponding change in plate motion. Changes in gravitational potential energy during the rifting process have been invoked as a possible source of compressional stresses, but their magnitude, timing, and relationship with depth inversions remain unclear. Using high-resolution two-dimensional numerical experiments of the full rifting process, we track the dynamic interplay between the far-field tectonic forces, loading and unloading of the surface via surface processes, and gravitational body forces. Our results show that rift basins tend to localize compressive stresses; they record transient phases of compressional stresses as high as 30 MPa and experience a profound depth inversion, 2 km in magnitude, when sediment supply ceases, providing an additional driver for the breakup unconformity, a well-documented phase of regional uplift typically associated with continental breakup.

Coplanar Electrowetting-Induced Droplet Detachment from Radially Symmetric Electrodes.

This work demonstrates electrowetting-induced droplet detachment in air from coplanar electrodes using a single voltage pulse. It also presents two models to predict when this detachment will occur. Previous works approximated the minimum energy for detachment based on (i) adhesion work at the solid-liquid interface and (ii) interfacial energy changes along all three interfaces in the system. This investigation updates those models to include changes in gravitational potential energy during detachment and provides validation by testing predicted detachment thresholds against experimental observations. Droplets of varying volume were ejected from electrowetting devices with (i) radially symmetric four-part coplanar electrodes and (ii) single electrodes with a ground wire inserted directly into the droplet. All experiments were performed in air. Incorporation of gravitational potential energy improves predictions for critical electrowetting number and captures the observed increase in applied voltage required with increased droplet volume. These new models will be of particular benefit in three-dimensional digital microfluidics applications that manipulate droplets in air.

Pendular energy transduction within the step during human walking on slopes at different speeds.

When ascending (descending) a slope, positive (negative) work must be performed to overcome changes in gravitational potential energy at the center of body mass (COM). This modifies the pendulum-like behavior of walking. The aim of this study is to analyze how energy exchange and mechanical work done vary within a step across slopes and speeds. Ten subjects walked on an instrumented treadmill at different slopes (from -9° to 9°), and speeds (between 0.56 and 2.22 m s-1). From the ground reaction forces, we evaluated energy of the COM, recovery (i.e. the potential-kinetic energy transduction) and pendular energy savings (i.e. the theoretical reduction in work due to this recovered energy) throughout the step. When walking uphill as compared to level, pendular energy savings increase during the first part of stance (when the COM is lifted) and decreases during the second part. Conversely in downhill walking, pendular energy savings decrease during the first part of stance and increase during the second part (when the COM is lowered). In uphill and downhill walking, the main phase of external work occurs around double support. Uphill, the positive work phase is extended during the beginning of single support to raise the body. Downhill, the negative work phase starts before double support, slowing the downward velocity of the body. Changes of the pendulum-like behavior as a function of slope can be illustrated by tilting the 'classical compass model' backwards (uphill) or forwards (downhill).

CO‐SEISMIC GRAVITATIONAL POTENTIAL ENERGY CHANGE AND ITS TECTONIC IMPLICATIONS: A CASE STUDY IN TIBETAN PLATEAU AREA

AbstractEarthquakes not only release seismic‐wave energy, which decays eventually, but also give rise to permanent deformation in the Earth. This permanent deformation causes gravitational potential energy (GPE) change. Previous researches showed that the GPE change is a good indicator of extensional and compressional tectonics. In this paper, we first proposed an effective method to compute co‐seismic GPE change for a SNREI (spherically symmetric Non‐rotational Elastic Isotropic) Earth according to the point dislocation theory. This method is applicable to both shear and tensile earthquakes. We then applied this method to compute the contributions from earthquakes occurred over Tibetan plateau area from 1976 to 2013 to the crustal GPE change. The results show that earthquakes occurred in central and western part of Tibetan plateau caused the crustal GPE to decrease while those in eastern part caused it to increase, which correspond to extensional and compressional tectonic status, respectively, in this area. Furthermore, the impact of earthquakes occurred from 1999 to 2013 was larger than that of earthquakes occurred from 1976 to 1998.

Crustal Gravitational Potential Energy Change and Subduction Earthquakes

Crustal gravitational potential energy (GPE) change induced by earthquakes is an important subject in geophysics and seismology. For the past forty years the research on this subject stayed in the stage of qualitative estimate. In recent few years the 3D dynamic faulting theory provided a quantitative solution of this subject. The theory deduced a quantitative calculating formula for the crustal GPE change using the mathematic method of tensor analysis under the principal stresses system. This formula contains only the vertical principal stress, rupture area, slip, dip, and rake; it does not include the horizontal principal stresses. It is just involved in simple mathematical operations and does not hold complicated surface or volume integrals. Moreover, the hanging wall vertical moving (up or down) height has a very simple expression containing only slip, dip, and rake. The above results are significant to investigate crustal GPE change. Commonly, the vertical principal stress is related to the gravitational field, substituting the relationship between the vertical principal stress and gravitational force into the above formula yields an alternative formula of crustal GPE change. The alternative formula indicates that even with lack of in situ borehole measured stress data, scientists can still quantitatively calculate crustal GPE change. The 3D dynamic faulting theory can be used for research on continental fault earthquakes; it also can be applied to investigate subduction earthquakes between oceanic and continental plates. Subduction earthquakes hold three types: (a) crust only on the vertical up side of the rupture area; (b) crust and seawater both on the vertical up side of the rupture area; (c) crust only on the vertical up side of the partial rupture area, and crust and seawater both on the vertical up side of the remaining rupture area. For each type we provide its quantitative formula of the crustal GPE change. We also establish a simplified model (called CRW Model) as follows: for Type B and Type C subduction earthquakes, if the seawater average depth on the vertical up side of the rupture area is less than a tenth of the hypocenter depth, then take the approximation that the seawater above the continental plate is replaced by the upper crustal material of the continental plate. The formula of quantitative calculating the crustal GPE change is also provided for this model. Finally, for 16 September 2015 Mw 8.3 Illapel Chile earthquake, we apply CRW Model and obtain the following results: the crustal GPE change is equal to 1.8 × 1019 J, and the hanging wall vertical moving-up height is 1.9 m with respect to the footwall. We believe this paper might be the first report on the quantitative solution of the crustal GPE change for this subduction earthquake; our results and related method will be helpful in research into the earthquakes in Peru–Chile subduction zone and the Andean orogeny. In short, this study expounds a new method for quantitative determining the crustal GPE change caused by subduction earthquakes, which is different from other existing methods.

A climatic dataset of ocean vertical turbulent mixing coefficient based on real energy sources

Using data on wind stress, significant height of combined wind waves and swell, potential temperature, salinity and seawater velocity, as well as objectively-analyzed in situ temperature and salinity, we established a global ocean dataset of calculated wind- and tide-induced vertical turbulent mixing coefficients. We then examined energy conservation of ocean vertical mixing from the point of view of ocean wind energy inputs, gravitational potential energy change due to mixing (with and without artificially limiting themixing coefficient), and K-theory vertical turbulent parameterization schemes regardless of energy inputs. Our research showed that calculating the mixing coefficient with average data and artificial limiting the mixing coefficient can cause a remarkable lack of energy conservation, with energy losses of up to 90% and changes in the energy oscillation period. The data also show that wind can introduce a huge amount of energy into the upper layers of the Southern Ocean, and that tidesdo so in regions around underwater mountains. We argue that it is necessary to take wind and tidal energy inputs into account forlong-term ocean climate numerical simulations. We believe that using this ocean vertical turbulent mixing coefficient climatic dataset is a fast and efficient method to maintain the ocean energy balance in ocean modeling research.

Energetics of lateral eddy diffusion/advection: Part II. Numerical diffusion/diffusivity and gravitational potential energy change due to isopycnal diffusion

Study of oceanic circulation and climate requires models which can simulate tracer eddy diffusion and advection accurately. It is shown that the traditional Eulerian coordinates can introduce large artificial horizontal diffusivity/viscosity due to the incorrect alignment of the axis. Therefore, such models can smear sharp fronts and introduce other numerical artifacts. For simulation with relatively low resolution, large lateral diffusion was explicitly used in models; therefore, such numerical diffusion may not be a problem. However, with the increase of horizontal resolution, the artificial diffusivity/viscosity associated with horizontal advection in the commonly used Eulerian coordinates may become one of the most challenging obstacles for modeling the ocean circulation accurately. Isopycnal eddy diffusion (mixing) has been widely used in numerical models. The common wisdom is that mixing along isopycnal is energy free. However, a careful examination reveals that this is not the case. In fact, eddy diffusion can be conceptually separated into two steps: stirring and subscale diffusion. Due to the thermobaric effect, stirring, or exchanging water masses, along isopycnal surface is associated with the change of GPE in the mean state. This is a new type of instability, called the thermobaric instability. In addition, due to cabbeling subscale diffusion of water parcels always leads to the release of GPE. The release of GPE due to isopycnal stirring and subscale diffusion may lead to the thermobaric instability.

Normal and shear stresses acting on arbitrarily oriented faults, earthquake energy, crustal GPE change, and the coefficient of friction

As usual, earthquake energy is defined as the total energy released from an earthquake, which is partitioned into radiated energy, friction energy, and rupture energy regardless of crustal gravitational potential energy (GPE) change. We analyze the energy and stress parameters in earthquake energy budget. For arbitrarily oriented faults, we deduce the formulas for calculating the normal and shear stresses acting on the fault under principal stresses. We show that shear stress is composed of horizontal and vertical shear stresses. Then, we provide the expressions for computing crustal GPE change and the coefficient of friction. The GPE change should be considered, except strike-slip faulting, when investigating earthquakes. Also, for various faulting types, we show that the ratio of differential stresses is related to the fault orientation and the relative magnitudes of stresses. Finally, “12 May, 2008, Wenchuan, Sichuan, China, MW 7.9 Earthquake” is cited to analyze and calculate various energy/stress parameters and the coefficient of friction. Our result of GPE change coincides with the post-event field observations.

Earthquake-induced crustal gravitational potential energy change in the Philippine area

The crustal gravitational potential energy change (ΔGPE) caused by earthquakes in the Philippine area from January 1976 to November 2011 was estimated in this study. The active convergence between the Philippine Sea Plate and the Sundaland–Eurasian margin is reflected by the greatest gains in GPE along the Philippine, Negros and Cotabato trenches, whereas the Manila Trench is covered by a GPE loss pattern. Although the Philippine Mobile Belt (PMB) itself is actually affected by the ongoing collision and subduction processes, almost the entire Philippine Fault Zone is dominated by GPE loss, revealing a slightly extensional environment along the fault. The time evolution of the cumulated ΔGPE for different segments along the Philippine archipelago shows distinct patterns. Due to the numerous large underthrusting events that have occurred along the Philippine Trench, the cumulated ΔGPE is regularly increasing in its most southern segment. However, in the middle segments, where the Palawan Block enters into collision with the PMB, the increase in cumulated ΔGPE is relatively small. In the most northern segment, where the North Luzon is located, a decrease of cumulated ΔGPE demonstrates that the seismic characteristic of the Manila Trench is dissimilar from other subduction systems in the world. We suggest that the collision of both the Palawan Block and the Benham Rise with the PMB promotes the rotation of the PMB and facilitates the northward escape of the northeastern Luzon, resulting in a decrease of cumulated ΔGPE in the northern Philippines.

Pliocene sinistral slip across the Adobe Hills, eastern California–western Nevada: Kinematics of fault slip transfer across the Mina deflection

The Adobe Hills region (California and Nevada, USA) is a faulted volcanic field located within the western Mina deflection, a right-stepping zone of faults that connects the northern Eastern California shear zone (ECSZ) to the south with the Walker Lane belt (WLB) to the north. New detailed geologic mapping, structural studies, and 40 Ar/ 39 Ar geochronology in the Adobe Hills allow us to calculate fault slip rates and test predictions for the kinematics of fault slip transfer into the Mina deflection. The Adobe Hills are dominated by Pliocene tuffaceous sandstone, basaltic lavas that yield 40 Ar/ 39 Ar ages between 3.13 ± 0.02 and 3.43 ± 0.01 Ma, and basaltic cinder cones. These Pliocene units unconformably overlie Middle Miocene latite ignimbrite that yields an 40 Ar/ 39 Ar age of 11.17 ± 0.04 Ma, and Quaternary tuffaceous sands, alluvium, and lacustrine deposits cap the sequence. Northwest-striking normal faults, west-northwest–striking dextral faults, and northeast-striking sinistral faults cut all units; the northeast-striking sinistral faults are the youngest and most well developed fault set. We calculate ∼0.1 mm/yr of approximately east-west horizontal extension and northwest dextral shear since the Pliocene. The prominent northeast-striking sinistral faults offset basalt ridgelines, normal fault–hanging-wall intersections, a channelized basalt flow, a basalt flow edge, and a basalt flow contact a net minimum of 921 ± 184 to 1318 ± 264 m across the Adobe Hills. These measured sinistral offsets yield a minimum Pliocene sinistral fault slip rate of 0.2–0.5 mm/yr; our preferred minimum slip rate is 0.4–0.5 mm/yr. The geometry and orientation of the prominent sinistral faults are consistent with simple shear/couple clockwise block rotation within a broad dextral shear zone. Vertical axis block rotation data are needed to test this interpretation. We propose that a set of faults subparallel to Sierra Nevada–North America motion and associated releasing steps, located west of the White Mountains fault zone and east of the Long Valley Caldera, transfer a portion of dextral Owens Valley fault slip northwestward onto the sinistral faults in the Adobe Hills. Dextral slip distributed across faults between the White Mountains fault zone and the Sierra Nevada and east of the Fish Lake Valley fault zone may account for the apparent discrepancy between summed long-term geologic slip rates and present-day geodetic rates across the northern ECSZ. Fault slip in the Adobe Hills is part of a regional pattern of initiation and renewal of dextral, sinistral, and normal fault slip during the Pliocene that extends from lat ∼40°N to ∼36°N within the ECSZ-WLB and along the western margin of the Basin and Range Province. This regional deformation episode may be related to changes in gravitational potential energy.

Thermodynamics/Dynamics Coupling in Weakly Compressible Turbulent Stratified Fluids

In traditional and geophysical fluid dynamics, it is common to describe stratified turbulent fluid flows with low Mach number and small relative density variations by means of the incompressible Boussinesq approximation. Although such an approximation is often interpreted as decoupling the thermodynamics from the dynamics, this paper reviews recent results and derive new ones that show that the reality is actually more subtle and complex when diabatic effects and a nonlinear equation of state are retained. Such an analysis reveals indeed: (1) that the compressible work of expansion/contraction remains of comparable importance as the mechanical energy conversions in contrast to what is usually assumed; (2) in a Boussinesq fluid, compressible effects occur in the guise of changes in gravitational potential energy due to density changes. This makes it possible to construct a fully consistent description of the thermodynamics of incompressible fluids for an arbitrary nonlinear equation of state; (3) rigorous methods based on using the available potential energy and potential enthalpy budgets can be used to quantify the work of expansion/contraction in steady and transient flows, which reveals that is predominantly controlled by molecular diffusive effects, and act as a significant sink of kinetic energy.

Sandbox Experimental Study on the Influence of Rock Strength and Gravity on Formation of Thrusts

ABSTRACT A sandbox experiment model was designed to simulate how differences in rock strength and gravity between two blocks can influence the formation characteristics of thrusts. In the experiment the compression was from one direction with basement shortening and the initial surfaces of the model were oblique. The results show that if the initial surface was horizontal or the slope angle was smaller than 7°, the compression induced two groups of thrusts with opposite dip orientations. If the slope angle of the initial surface was greater than 7°, the compression induced only one group of thrusts with a dip orientation contrary to the original compression direction. This result is similar to the actual section of a collision zone between two continental blocks. By applying stress analysis, rock strength is shown to be an important factor in deformation. As other boundary conditions are changeless, it is the change of gravitational potential energy that leads to different deformation styles.

Earthquake-induced gravitational potential energy change in the active Taiwan orogenic belt

SUMMARY The Philippine Sea Plate is converging against the Eurasian Plate near Taiwan at a velocity of 7‐8 cm yr −1 ; this has caused the Taiwan orogenesis and induced abundant earthquakes. In this study we examine the corresponding change of gravitational potential energy (�GPE) using 757 earthquakes from the earthquake catalogue of the Broadband Array in Taiwan for Seismology (BATS) from 1995 July to 2003 December. Our results show that the variation of

Giant Galápagos tortoises walk without inverted pendulum mechanical-energy exchange

Animals must perform mechanical work during walking, but most conserve substantial mechanical energy via an inverted-pendulum-like mechanism of energy recovery in which fluctuations of kinetic energy (KE) and gravitational potential energy (GPE) are of similar magnitude and 180 degrees out of phase. The greatest energy recovery typically occurs at intermediate speeds. Tortoises are known for their slow speeds, which we anticipated would lead to small fluctuations in KE. To have an effective exchange of mechanical energy using the inverted-pendulum mechanism, tortoises would need to walk with only small changes in GPE corresponding to vertical center-of-mass (COM) fluctuations of < 0.5 mm. Thus, we hypothesized that giant Galapagos tortoises would not conserve substantial mechanical energy using the inverted-pendulum mechanism. We studied five adult giant Galapagos tortoises Geochelone elephantopus (mean mass=142 kg; range= 103-196 kg). Walking speed was extremely slow (0.16+/-0.052 m s(-1); mean +/- 1 s.d.). The fluctuations in kinetic energy (8.1+/-3.98 J stride(-1)) were only one-third as large as the fluctuations in gravitational potential energy (22.7+/-8.04 J stride(-1)). In addition, these energies fluctuated nearly randomly and were only sporadically out of phase. Because of the dissimilar amplitudes and inconsistent phase relationships of these energies, tortoises conserved little mechanical energy during steady walking, recovering only 29.8+/-3.77% of the mechanical energy (range=13-52%). Thus, giant Galapagos tortoises do not utilize effectively an inverted-pendulum mechanism of energy conservation. Nonetheless, the mass-specific external mechanical work required per distance (0.41+/-0.092 J kg(-1) m(-1)) was not different from most other legged animals. Other turtle species use less than half as much metabolic energy to walk as other terrestrial animals of similar mass. It is not yet known if Galapagos tortoises are economical walkers. Nevertheless, contrary to biomechanical convention, poor inverted-pendulum mechanics during walking do not necessarily correspond to high mechanical work and may not result in a high metabolic cost.

Bernoulli’s equation

In a forthcoming article we will look at some examples of the application of Bernoulli’s equation. From this article I hope the reader has developed a feel for some aspects of fluid motion: the concept of a fluid particle, the two types of fluid acceleration and how motion in one part of the fluid causes motion in other parts of the fluid. Bernoulli’s equation can be viewed in two ways. One as Newton’s second law applied to a line of fluid particles in a stream-tube. The second as a statement of energy conservation: the change in gravitational potential energy plus the change in kinetic energy is equal to the work done by the pressure forces.

Analysis of microvascular responses in the finger to changes in arm position during cold water stimulation.

We evaluated the extent to which the myogenic autoregulatory response was affected by sympathetic vasoconstriction in humans. The stimulus used to produce sympathetic vasoconstriction was the cold water immersion of a contralateral finger. The stimulus used to elicit myogenic response was gravitational potential energy change (GPEC), that is, arm position change, i.e., the raising and lowering of the upper extremity 40 cm above and below the heart level. The response was observed in terms of the changes in amplitude on a differential digital photoplethysmogram (delta DPG) during this maneuver with and without cold stimulation. The subjects were seven healthy males. During non-immersion, the delta DPG amplitude increased to 55.7 +/- 7.6% (SE) at the elevated position and decreased to -56.3 +/- 4.0% at the lowered position. Mean finger arterial blood pressure (MBP) decreased by 34.0 +/- 1.8% at the elevated position and increased by 39.9 +/- 3.3% at the lowered position. During immersion, the delta DPG amplitude was significantly (p < 0.05) decreased by 25-40, 33-57, and 33-38% at the heart, elevated, and lowered position, respectively. MBP increased by 5-9%, but heart rate was unchanged throughout immersion. These results indicate that the myogenic response of arterioles is not entirely abolished by the present sympathetic vasoconstrictor effect. Thus interactions between these two effects may maintain both systemic and local circulatory homeostasis.

Application of energy concepts to groundwater flow: Adaptive modeling of a leaky aquifer

The net work and energy flux at the boundaries of an aquifer change its internal energy and overcome its resistance to flow. In saturated porous media, the change in internal (strain) energy is stored in the elastic soil matrix and in pore water compression. In unsaturated media, an additional term accounts for changes in gravitational potential energy. The energy approach complements conventional insight by allowing spatially distributed processes to be integrated into energy and work terms which characterize a system's response to a set of excitations. Specifically, a technique is developed in this paper to interpret the dynamic behavior of a one‐dimensional leaky aquifer in terms of its composite energy functions. In particular, the work interaction at the leaky boundary is used as an index of the significance of the leakage: when the work parameter indicates a relatively small leakage, the flow components of the multiaquifer can be isolated and modeled separately with a controllable loss of accuracy.

Double-diffusive convection with a non-linear equation of state: Part I. The accurate conservation of properties in a two-layer system

If the equation of state is nonlinear, a given flux of heat across a double diffusive interface causes different buoyancy fluxes in the upper and lower layers. This results in different convective activity in the two layers and can lead to preferential entrainment across the interface in one direction (i.e. a migration of the interface). In this paper we derive the conservation equations for properties (e.g. heat and a solute) across a double diffusive interface between two well-mixed layers. A nondimensional measure of the entrainment across an interface and the most suitable choice for the buoyancy flux ratio are presented. Some surprising facts emerge. First, even for a linear equation of state and in the absence of direct entrainment across the interface, the flux of water across a finger interface is shown to be important. Second, for the heat-solute system, the heat balance equations for each well-mixed layer contain terms proportional to the heat of solution of the solute and the partial specific enthalpy of pure water in a seawater solution. Third, the rate of change of gravitational potential energy of the two-layer system is shown to have several extra terms in addition to the two commonly quoted major terms.