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# The role of water in fault lubrication

## Abstract

The friction between two adjacent tectonic plates under shear loading may dictate seismic activities. To advance the understanding of mechanisms underlying fault strength, we investigate the frictional characteristics of calcite in an aqueous environment. By conducting single-asperity friction experiments using an atomic force microscope, here we show three pathways of energy dissipation with increasing contact stresses: viscous shear of a lubricious solution film at low normal stresses; shear-promoted thermally activated slip, similar to dry friction but influenced by the hydrated ions localized at the interface; and pressure-solution facilitated slip at sufficiently high stresses and slow sliding velocities, which leads to a prominent decrease in friction. It is also shown that the composition of the aqueous solution affects the frictional response. We use this nanoscale evidence to scrutinize the role of brines on fault behavior and argue that pressure solution provides a weakening mechanism of the fault strength at the level of single-asperity contacts.

## Introduction

When the static friction between two adjacent tectonic plates under shear loading is overcome, the two plates move relative to each other, which is known as a fault slip; under conditions of unstable slip, that is, when an intermittent motion (or stick–slip) occurs, an earthquake rupture may take place1,2. While earthquakes are one of the most notorious natural disasters, it remains elusive to accurately predict their occurrence and magnitude, which emphasizes the lack of understanding of the underlying mechanisms3. Along these lines, it has been proposed that water may play an important role in fault dynamics and earthquake nucleation4, which has motivated this work.

Rate and state friction (RSF) constitutive equations3,5 are commonly used to phenomenologically describe the frictional response of tectonic fault zones and earthquake nucleation. According to RSF equations, only velocity-weakening friction, i.e., a decreasing friction with an increase in slip rate, can yield an unstable fault slip. Regarding the effect of water, RSF equations predict that fluid overpressure decreases the critical fault stiffness, and therefore, it should promote aseismic creep. In contrast to these predictions, seismologic studies4 as well as laboratory experiments6 suggest that a high fluid pore pressure can trigger a dynamic slip instability, indicating that fault weakening—likely as a result of the reduced effective (compressive) stress, which brings the fault closer to failure—can overcome velocity-strengthening friction. Other mechanisms have been proposed to explain the effect of water on fault friction. For instance, water may promote fracture of asperities by chemically reacting with the strained apexes of microcracks (also called subcritical crack growth)7,8,9, thereby lowering the resistance to slip, i.e., friction. The weakening of wet fault gouges may also rely on intergranular lubrication facilitated by water adsorption to the grain surface10,11, like the oil lubricating an engine. Velocity-weakening friction and stick–slip were observed in experiments on water-saturated limestone gouge at high temperature (150 °C), whereas stable sliding was observed under dry and low temperature conditions10. Here, the authors could not unambiguously identify the underlying mechanisms, and proposed that crystal plasticity at asperity contacts coupled to granular flow and perhaps pressure solution—a process by which the contact stress leads to mineral dissolution, coupled to diffusion of the solutes and re-precipitation of the mineral on unstressed regions12,13—could play a role14. In fact, pressure solution has been concurrently observed in other experiments with water-saturated calcite-rich gouge that showed a reduction in friction compared to dry conditions10. Controversially, re-precipitation of calcite after its pressure-induced dissolution has been also related to overall gouge compaction15, and thereby, to an increase in the friction coefficient16. The complexity of the investigated systems has hindered discerning the contribution of these multiple mechanisms to the frictional response of fault materials in the presence of water, which calls for fundamental research on more simple systems.

Atomic force microscopy (AFM) is a powerful tool to investigate the fundamental mechanisms of frictional behavior at the nanoscale17,18. Although AFM experiments are highly idealized compared to the complexity in nature and in laboratory experiments with simulated fault gouge, they allow isolating and inspecting physical and chemical mechanisms. For instance, a previous AFM study demonstrated that the origin for the increase in the static friction of single-asperity silica–silica contacts of nanometer size is the formation of chemical bonds between the silica surfaces18,19,20. A precedent study of the frictional characteristics of calcite under dry conditions by AFM showed that both stick–slip behavior and kinetic friction depend on the crystallographic sliding direction21. While these recent works illustrate the first attempts to bridge the gap between nanoscale friction measurements and fault dynamics, many questions still remain open.

Recognizing that carbonate-dominated lithologies play an important role in mid- to shallow-crustal faulting16, the purpose of this article is thus to rationalize the frictional characteristics of calcite in the presence of an aqueous solution at nanoscale. An AFM was used to measure the friction force as a function of the applied load and sliding velocity. By sliding an oxidized silicon tip along the atomically flat calcite ($10 1 ̄ 4$) plane, the friction force was measured while isolated from other processes occurring concurrently in natural systems and macroscale experiments. An AFM tip with a silica surface was selected because its frictional characteristics are similar to those of silicate rocks18,22. The selected range of normal stresses is relevant to that along faults in the upper Earth crust, which is in the order of 100 MPa23, but higher in the multiple-asperity contacts that determine the fault strength. Scrutinizing the change in friction caused by varying stress, velocity, and brine concentration helps to understand the pathways of energy dissipation, i.e., the origin of friction. Based on the measured frictional behavior of calcite in aqueous solutions, three different mechanisms of energy dissipation are distinguished (Fig. 1): viscous dissipation at low stresses (abbreviated as LS, Hertzian contact stresses σn ≤ 200 MPa), shear-promoted thermally activated slip at intermediate stresses (IS, 200 < σn ≤ 400 MPa); and pressure-solution facilitated slip at higher stresses (HS, σn> 400 MPa). This work thus provides evidence for the presence of a lubricating film composed of ions and water that remains confined between calcite and silica surfaces even under high applied pressures. Our experimental study also demonstrates that pressure-induced dissolution of calcite at sufficiently high stresses and slow sliding velocities can cause a significant reduction of friction and facilitate slip, which is proposed to be a potential mechanism of fault-strength weakening at the level of single-asperity contacts.

## Results

### Viscous shear and hydration lubrication in the LS regime

Figure 1a–e show the friction force as a function of velocity at applied loads of 0.5, 1, and 2 nN (σn ≤ 200 MPa, Supplementary Table 1) and at calcium chloride concentrations of 0 mM (no CaCl2 added), 1, 10, 100 mM, and 1 M to simulate calcium concentrations in nature, from ground water24 to concentrated brines25. As shown here, friction increases with load but the change is very small in this narrow range of loads (0.5–2 nN). The velocity dependence of the friction force in the LS regime can be fit by a power law at all selected concentrations (see solid lines), which is reminiscent of the shear response of fluids as described by the Ostwald–de Waele equation26

$F L ~A η 0 ( v ∕ D ) n ,$
(1)

where FL is the friction force, A is the area, η0 is a constant, v/D is the shear strain rate, D is the fluid film thickness, and n an exponent. This relation suggests that friction in the LS regime is mainly due to a viscous force that results from shearing the confined aqueous solution. The lubrication mechanism provided by a nanoconfined solution has been previously referred to as “hydration lubrication”27.

The composition of the confined aqueous solution between calcite and a (micron-sized) silica tip was scrutinized in our previous AFM study28, by measuring the normal surface forces down to subnanometer calcite–silica separations. This work showed that a strong repulsion between the confining surfaces—also termed disjoining pressure—is originated by the colloidal forces between the two confining surfaces across the thin film of solution, and it prevents the solution to be squeezed-out under an applied pressure29,30,31. The confined solution has a thickness of a few nanometers and consists of water layers that are strongly adsorbed to calcite and located underneath a layer of partially dehydrated calcium ions, and fully hydrated calcium ions further away from the surface.

Assuming that the change in D is negligible across the investigated range of loads and velocities, the average exponent n for all selected concentrations is estimated to be 0.38±0.06. An exponent n smaller than 1 indicates that the confined fluid behaves as a non-Newtonian fluid with shear-thinning behavior, i.e., its viscosity decreases with increasing shear strain rates. A calculation of the viscosity is, however, not possible, because the absolute separation between the tip and the calcite surface (D) cannot be unambiguously determined. Nevertheless, a similar shear-thinning behavior has been observed for confined water between mica and an AFM tip at stresses ≥ 100 MPa32, while a constant viscosity was reported at much smaller normal stresses ( < 1 MPa)27. This suggests that the high normal stresses applied in the LS regime may be responsible for the observed non-Newtonian behavior of the confined aqueous solution.

### Shear-promoted thermally activated slip in the IS regime

Figure 1f–j show the friction force as a function of velocity in the IS regime (σn~ 200–400 MPa). The friction force scales with the logarithm of velocity, i.e., FL~ln(v), thereby indicating a mechanism of energy dissipation that differs from the viscous dissipation in the LS regime. This logarithmic relation is attributed to a shear-promoted thermally activated slip in the context of transition state theory33. This theory was applied by Eyring34 to describe liquid viscosity at the molecular level based on activated flow and is often applied to describe the origin of friction in thin-film lubrication35. That is, for slip to occur, the molecule, initially in an equilibrium position (an energy minimum) needs to jump over a “transition state” before reaching the adjacent energetic minimum, which requires an activation energy (∆E) (Fig. 2a). Although the thermal energy of the molecules might be sufficient to overcome this energy barrier, the shear force applied on the molecule reduces the energy barrier, and thereby, promotes the slip; when the molecule falls in the adjacent minimum, the applied work is irreversibly dissipated (lost). Although described at the molecular level, this theory has been often applied to explain friction in macroscale contacts, as well36,37.

Based on our previous work28, we assume that hydrated calcium ions are not squeezed-out under high applied pressures, but instead they remain localized on oppositely charged surface sites and undergo similar thermally activated slip, as shown in Fig. 2a. Considering that the jump frequency of the hydrated ions f is increased by the applied lateral force according to f~ f*exp(−(∆E−FLΩ/A)/kBT), FLΩ/A involving the decrease of the energy barrier imposed by the lateral force, Ω the stress activated volume, and f* the characteristic frequency of water and ions, the following expression27 is obtained for the friction force FL:

$F L = Δ E λ + k B T λ ln v ∕ v 0 ,$
(2)

where kB being the Boltzmann constant, T is the absolute temperature, v = d·f is the sliding velocity, d being the jump amplitude, v0 = d·f* is the reference velocity, and λ = Ω/A is the stress-activation length. The solid lines in Fig. 1f–j show the fits of Eq. (2) to the experimental results. λ represents the characteristic length over which the confined ions and water can build up strain in response to shear, and it appears in Fig. 2a as the distance from the initial equilibrium position to the transition state. The calculated λ significantly decreases with increasing normal loads at all investigated concentrations (Fig. 2b); that is, the localized ions and water can build up strain over shorter distances, which suggests a decrease in mobility of the confined solution with increase in normal stress.

As shown in Fig. 1f–j, friction collectively decreases when the salt concentration increases from 0 to 10 mM, while it increases by further increasing the concentration to 1 M, thereby indicating the action of at least two phenomena with opposite effects. Furthermore, the non-monotonic change in friction with concentration coincides with the inverse trend of λ, first increasing and then decreasing with rising concentration. To better understand these results, reference measurements were made with mica in NaCl solutions (Supplementary Fig. 3). CaCl2 was not selected here because it had been previously shown that although calcium ions adsorb on mica38, the hydration force between mica surfaces in CaCl2 solutions does not reveal the presence of multiple layers of ions and water at concentrations below 100 mM30,39,40. In contrast, hydrated sodium ions accumulate on mica surfaces and form multiple layers at lower concentrations, similar to our observations for calcium ions on calcite28. Interestingly, we also observed opposite trends of friction and λ on mica in NaCl solutions. Supplementary Fig. 3 shows a decrease of friction and a concurrent elongation of λ by increasing the concentration from 0 to 1 M. Previous studies have proved that a higher amount of strongly hydrated sodium ions accumulate at the mica surface with increasing concentration41. Further, theory has showed that the adsorption of ions to surfaces leads to an electric field in the neighboring water molecules42, thereby creating boundary films with high fluidity43, while the fluidity of the hydration shell of the ions has been proposed to facilitate slip27. Accordingly, the observed increase of λ with concentration in Supplementary Fig. 3 reflects the enhanced fluidity or mobility of the confined solution, since more water is retained in the contact within the hydration shell of the adsorbed sodium ions, which decreases the resistance to sliding, i.e., friction.

In the experiments on calcite, the non-monotonic trend of friction (and of λ) with increasing concentration is intriguing, since our previous work also revealed an increasing amount of calcium ions populating the calcite–solution interface with increasing concentration28. This suggests that another mechanism is at play. Indeed, the increasing friction force at concentrations above 10 mM coincides with the charge reversal of the confined calcite surface28. We thus speculate that charge reversal of calcite (from negative to positive) magnifies the energy barrier at concentrations above 10 mM due to an additional electrostatic attraction to the naturally oxidized silicon tip, which is negatively charged. As a result, the increase in friction force at concentrations above 10 mM can be justified. Although the conditions for charge reversal are mineral-specific, this phenomenon is expected to be universal for mineral surfaces in contact with calcium-rich brines, and thus, to affect friction.

It is important to discuss the influence of the AFM tip on these results. First, analogous results were obtained with a silicon nitride AFM tip (Supplementary Fig. 4a), and hence, the bulk material of the tip does not affect the mechanism of energy dissipation. Second, the friction force measured with a naturally oxidized silicon tip sliding on a silica surface decreases with velocity (Supplementary Fig. 5a), which has been attributed to the formation of hydrogen and/or siloxane bonds between the silica surfaces19,44. Therefore, the identified shear-promoted thermally activated slip in Fig. 1f–j mainly reflects the frictional behavior of calcite.

### Pressure-solution facilitated slip in the HS regime

At the sliding conditions at which the friction coefficient decreased, AFM imaging of the calcite surface showed the track of the tip (Supplementary Fig. 7), while re-precipitation sometimes occurred close to the track (see the bright spot in Supplementary Fig. 7c). Images taken after similar friction measurements in ethanol, where calcite is insoluble, do not show any change of the surface, which excludes brittle fracture and plastic deformation of calcite (Supplementary Fig. 8). The tracks are similar to the previously reported wear of a single-crystal calcite created by an AFM tip along the edges of pre-existing etch pits in aqueous solution46. Hence, the remarkable drop in friction with increase in load is attributed to the activation of the pressure solution of calcite at sufficiently high normal stresses and slow sliding velocities. As described earlier, the dehydration of calcite cannot happen at the highest applied pressure in this work, and hence, strongly adsorbed water and ions are still retained between the two confining surfaces, thereby providing a path for the pressure-induced dissolution of calcite. We hypothesize that if the calcite under the AFM tip could be dissolved and re-hydrated before the tip slides away, it would facilitate slip in a similar fashion as in the LS regime, that is, by hydration lubrication, but at much higher contact stresses.

The decrease of the friction coefficient shown in Fig. 3 is reflected in Fig. 1k–o as a crossover of the velocity-dependent friction force when the load is decreased (see arrows). Such a crossover was not observed in reference tests in the absence of the aqueous solution, as illustrated in Fig. 4a. Therefore, the crossover is considered to be a qualitative footprint for pressure-induced dissolution of calcite. Although one cannot rule out the possibility that pressure solution also takes place in the IS regime, both the relation FL~ln(v) and the increasing FL with load in this regime indicate that the influence of pressure solution, if any, is small. By the same token, pressure solution, if any, is small at 10 nN in Fig. 1k–o, and hence, this load is taken as reference to quantify the crossover velocity, which is shown in Fig. 2c. The general trend that higher loads lead to higher crossover velocities is consistent with the proposed mechanism: pressure solution is enhanced with increasing normal stresses, requiring less time for dissolution, and therefore, taking place at higher sliding velocities compared to lower stresses. Furthermore, the increasing crossover velocity with increasing concentration indicates the more prominent pressure solution under these conditions. This is most likely originated by the increasing reactivity of calcite with increase in calcium concentration (and the concurrent decrease in pH, Supplementary Table 1), as inferred from the more dramatic reconstruction of the unconfined calcite surface (Supplementary Fig. 9).

The proposed weakening mechanism is also consistent with the estimated kinetics of pressure-induced calcite dissolution. Taking a normal stress σn ~ 540 MPa, the chemical potential ∆µ that drives the dissolution of the stressed calcite is given by ∆μ/kBT = Vmσn/kBT = ln(ap/a0)~7.87, Vm being the molecular volume of calcite, and a0 and ap the activity of the solution in equilibrium with stress-free and stressed calcite, respectively47. The increase in chemical potential induced by σn leads to a sudden increase of the local solubility as well as a dramatic undersaturation of the interfacial solution. The reaction-limited dissolution rate under such conditions is proportional to the solution activity48. Thus, the pressure-induced dissolution rate is estimated as rp~r0ap/a0, being r0 the dissolution rate of stress-free calcite. Let us now take r0~1.8×10−10 mol cm2 s−1 as the reaction-limited dissolution rate of calcite in water49, equivalent to ~1 formula unit (CaCO3) nm−2 s−1. At the assumed contact stress, the rate will increase to rp~e7.87 formula units nm−2 s−1. Given the unit cell of calcite c~0.5 nm, the time required for a formula unit to be dissolved is ∆t = 1/(c2rp)~1.5 ms. Taking d = 6.8 nm as the estimated diameter of the Hertzian contact area, it requires a velocity slower than vc = d/∆t = 4.43 µm s−1 for a unit cell to be pressed by the AFM tip longer than ∆t. Theoretically, sliding events at either slower velocities (to allow longer tip–substrate contact time), or higher stresses (to enhance rp further) should let calcite dissolution happen while the tip slides. In the range of stresses in the HS regime (~400–700 MPa) the calculated vc lies between 0.7 and 94 µm s–1, which agrees well with the range of velocities in our study. We thus conclude that pressure solution of calcite may cause a decrease in friction by promoting hydration lubrication at high normal stresses and sufficiently slow sliding velocities.

In summary, we have resolved three different mechanisms responsible for the frictional strength of this nanoscale asperity. Figure 5a shows the friction force divided by load as a function of the slip rate in the three regimes; note that in the HS regime this value significantly differs from the friction coefficient, which becomes negative under certain conditions (Fig. 3a). Dry friction is also displayed for comparison. Representative measurements for the lateral force as a function of the sliding distance (during trace and retrace of the tip) at three selected sliding velocities are shown in Fig. 5b–d. Hydration lubrication occurs at normal stresses σn ≤ 200 MPa, when the confined solution lubricates the nanoscale contact and friction is small and dictated by the viscous shear force of the fluid film. Here, the corrugation of the lateral force (Fig. 5b–d, blue) is indistinguishable from the noise (~0.02 nN), supporting smooth sliding, and thus, viscous energy dissipation. At higher normal stresses (σn~200–400 MPa, green), velocity-strengthening friction is originated by the shear-promoted thermally activated slip of the surface-localized hydrated ions, and it is strongly dependent on the solution composition. Here, the variation of the lateral force is much higher than the noise, revealing a periodic atomic-scale stick–slip with a slip length of ~0.5 nm, close to the unit cell of calcite, throughout all sliding velocities. Such discontinuous sliding is characteristic of the shear-promoted thermally activated slip in nanoscale experiments as a result of the jumps from the transition state to the adjacent energy minimum17,50. Intermittent slip prevails in the HS regime (red), although it becomes more irregular, which is likely associated with the concurrent pressure solution of calcite. The pressure-induced dissolution re-activates hydration lubrication at higher stresses (σn > 400 MPa), thereby causing a significant weakening of the frictional strength of this single asperity. Here, the red dashed line shows the calculated friction force assuming that the friction force at the highest velocities is still given by Eq. (2) and extrapolating to lower velocities; the area between calculated and measured friction force gives the decrease in dissipated energy per unit time as a result of pressure solution (see arrows in Fig. 5a). This is most prominent at slow sliding velocities (0.2 µm s–1, Fig. 5b), where the lateral force in the HS regime is of similar magnitude as in the LS and IS regimes but at much higher normal stresses. The lateral force increases when sliding occurs at 1 µm s−1, while the reproducible drops pointed by the arrows in Fig. 5c reflect the sequential weakening of the frictional strength upon sliding. The characteristic drops vanish with a further increase in the sliding velocity to 5 µm s–1 (Fig. 5d), indicating that pressure-solution facilitated slip is not noticeably active anymore. This mechanism is more pronounced with increasing calcium concentration, likely owing to the enhanced calcite surface reactivity; in nature, it will depend on the specific composition of the brine.

## Discussion

Implications of our work regarding the frictional strength of calcite-rich faults are discussed next. Our experiments have shown that the presence of the aqueous solution leads to a significant weakening of the frictional strength of the single-asperity contact compared to dry conditions, as the friction coefficient drops from ~0.05 to lower than ~0.005 at slow sliding velocities (Fig. 3a). Previous studies of calcite-rich fault gouge also reported a decrease in the friction coefficient from ~0.7 to ~0.6 under saturated conditions; however, much less pronounced10. We also note the significantly lower friction coefficients reported in our work compared to macroscale friction coefficients in fault gouge and rocks. There are multiple reasons for such discrepancy, as demonstrated previously for other substrates like mica27,51, quartz43,52, and graphite53. In the case of fault gouge and rock friction, multi-asperity interlocking during macroscopic sliding of such rough surfaces as well as asperity fracture can cause a significant increase in friction that is excluded in atomically smooth contacts, like in our nanoscale experiments. The simple geometry of our study enables us to evaluate the direct effect of hydration lubrication and pressure solution on the frictional characteristics of calcite at the level of a single asperity.

The strong affinity of the aqueous solution to the calcite surface results in the presence of a thin fluid film between the confining surfaces at high normal stresses. The confined fluid film is thus responsible for hydration lubrication under shear loading, serving as an excellent lubricant. In the case of the calcite–silica interface, a high disjoining pressure was measured (up to ~200 MPa), which makes hydration lubrication effective in this range of normal stresses28. While higher stresses bring the friction coefficient closer to that of the dry contact (insets of Fig. 3), we propose that hydration lubrication may be also active at much higher stresses (> 400 MPa), if pressure-induced dissolution happens concurrently with the slip. In fact, the drop in the friction coefficient in the HS regime is remarkable at the slowest slip rates. We have argued why the slip rate is thus an important parameter to predict the onset of pressure-solution facilitated slip based on the kinetics of mineral dissolution. A comparison between the dissolution kinetics of the mineral and the contact time during slip, as described in this work, should also define the conditions for pressure-solution facilitated slip in natural systems. In agreement with this, there is evidence for the dependence of the pressure solution on the slip rate in macroscopic experiments. For instance, grain size reduction due to fluid-assisted solution transfer was only observed at sufficiently slow sliding (< 30 µm s−1) in calcite-rich gouge experiments16. While the role of ion diffusion and other phenomena, like electrochemical potential gradients54, have been neglected so far, they may either retard or enhance pressure solution. This still requires more focused investigations.

The velocity-strengthening frictional behavior observed in this work under all conditions agrees well with previous studies on water-saturated calcite-rich gouges in the same range of sliding velocities and temperature10,16,55. It is to be noted that these studies also showed a transition into velocity-weakening frictional behavior16,56,57, yet at sliding velocities greater than investigated in this work due to the limitations of our instrument, and hence, a discussion of this behavior is out of the scope of this work. RSF constitutive equations predict that a velocity-strengthening frictional response should favor aseismic creep55, and therefore, calcite gouge would be expected to resist initial rupture nucleation at slow slip rates ( < 10 µm s–1) and over a wide range of contact stresses according to our nanoscale experiments. Nevertheless, prior work has shown that under stress/temperature conditions typical of the occurrence of fluid-induced seismicity, a wide variety of fault materials exhibits velocity-strengthening frictional behavior in the laboratory, which emphasizes the limitations of the standard RSF analysis. For instance, it has been recently reported that a high fluid pore pressure in carbonate-bearing fault gouge resulted in gouge dilation and was able to trigger fast acceleration and earthquake slip, despite its velocity-strengthening frictional behavior6. The effect of the disjoining pressure can be considered analogous to that of the pore fluid overpressure in decreasing the effective normal stress, and thereby in bringing faults closer to failure according to the Coulomb–Mohr relation. Moreover, according to our work, the nanoconfined fluid enables hydration lubrication, which appears as the origin of a dramatic weakening of the fault strength of single-asperity contacts; especially at high contact stresses when hydration lubrication is re-activated by pressure solution of calcite at the multi-asperity contacts. It should be noted that (long-term) slip weakening has been observed to overcome the velocity-strengthening behavior and trigger fault instability58. But even in the case of stable frictional behavior, fault creep—as a result of the significant decrease in frictional strength—will result in stress transfer in the surrounding, and hence, possibly, earthquake triggering on adjacent fault patches that could be more prone to develop frictional instabilities.

In conclusion, the simple geometry in our study and the selected mechanical and chemical conditions have enabled us to evaluate the direct effect of hydration lubrication and pressure solution on the frictional strength of calcite. We have demonstrated that pressure solution of calcite can directly impair the frictional strength of single-asperity contacts providing hydration lubrication, similar to that occurring at much lower contact stresses. This nanoscale evidence can be extended to fault weakening: when a fault undergoes creep slow enough for pressure solution to come into play, a slip can be triggered more favorably, which alters fault dynamics at the level of single-asperities. We thus hypothesize that earthquakes ruptures might be triggered and perhaps propagated by fault weakening induced by pressure solution of calcite asperity contacts. Our results have also shown that the influence of the brine composition on mineral reactivity should be considered. The relative contributions of subcritical crack growth, crystal plasticity, lubrication, and pressure solution to macroscopic friction still remain to be quantified. The range of applied pressures and slip rates investigated in this study is pertinent to earthquake nucleation, while the additional effect of Earth’s crust temperature at greater depths on the proposed mechanisms calls for future studies.

## Methods

### Sample preparation

To simulate calcium concentrations of water bodies in nature, from ground water24 to concentrated brines25, calcium chloride CaCl2 (purity > = 99.0%, Sigma-Aldrich) was dissolved at room temperature in ultrapure water (18.2 MΩ cm resistivity) to achieve concentrations of 0 mM (no CaCl2 added), 1, 10, 100, and 1 M. All solutions were then equilibrated with excessive calcite (CaCO3) powder overnight (purity > = 99.0%, Sigma-Aldrich) and filtered prior to use. Ion concentrations of the CaCl2 solutions and pH values can be found in Supplementary Table 1. Freshly cleaved Iceland Spar calcite crystals fixed to glass slides were immersed in the solutions with the ($10 1 ̄ 4$) plane exposed to solutions. The home-made fluid cell was covered by a membrane to minimize evaporation during experiments. The calcite surface was equilibrated for 1 h in each solution before performing the friction-force measurements.

### Friction force measurements

A Nanowizard AFM (JPK Instruments, Germany) was used to measure the friction force; the mechanic model is shown in the Supplementary Fig. 1. All the experiments reported here were performed with the same AFM cantilever type (CSC38/no Al, Mikromasch) with normal spring constants ranging between 0.1 and 0.3 N m−1, as determined by the thermal calibration method59. To measure friction, a naturally oxidized silicon tip was slid along the atomically flat calcite ($10 1 ̄ 4$) plane; the sliding direction was maintained constant during each experiment. The tip radius of the reported experiments ranges from 100 to 150 nm, as determined by scanning electron microscopy (SEM) imaging. Considering that the smallest reliable force that the instrument can apply is ~0.2 nN, such large tip radii were selected to allow applying pressures smaller than the solvation pressure60. Nevertheless, replica experiments were conducted with smaller tip sizes to confirm the reproducibility of results. SEM imaging of the AFM tips before and after friction-force measurements showed no significant changes in tip size, thereby excluding any irreversible change of the AFM tip, e.g., by wear or plastic deformation, during the duration of the experiment.

AFM imaging in contact mode was first conducted searching for an atomically smooth 100 nm × 100 nm surface area (RMS roughness < 100 pm, Supplementary Fig. 2). Isothermal friction-force measurements at 25 °C were made on an atomically flat calcite surface along the ($10 1 ̄ 4$) plane as a function of applied load (L) and velocity (v) at each selected solution concentration. Here, the tip slid on the calcite surface in a reciprocating motion, while the lateral deflection in the forward (trace) and backward (retrace) directions was acquired, with trace and retrace scans defining a single friction loop. Figure 5b–d shows representative friction loops, with a positive lateral force in trace and a negative lateral force in retrace over a selected displacement. The friction force was calculated by averaging over the half width of the trace and retrace scans and the standard deviation of the friction force was determined over ten loops. Replica experiments with different sliding lengths (100 nm, 200 nm and 1 µm) did not lead to any significant difference in the experimental results. The data shown in the main text correspond to a sliding length of 100 nm. The load-dependent friction-force measurements were performed at sliding velocities of 0.2 and 2 µm s–1, which were maintained constant by a piezo motor. Here, the load was ramped up from 0.5 to 20 nN. In these experiments, the coefficient of friction (µ) was obtained by calculating the slope of the friction vs. the load for each stress regime. The velocity-dependent friction-force measurements were conducted while the applied load was maintained constant; following loads were selected: 0.5, 1, 2 nN for LS regime, 5, 10, 15, 20 nN for IS regime, and 20, 30, 40 and 50 nN for HS regime. The sliding velocity was increased stepwise from 0.05 to 10 µm s–1 at each selected load. The Hertzian average contact stress in each regime is 191 MPa (for 2 nN), 413 MPa (for 20 nN), and 733 MPa (for 50 nN), respectively (Supplementary Table 2). For both load- and velocity-dependent measurements, ten friction loops were measured at each velocity and each load. Replica experiments were performed at three different spots on each calcite sample to confirm reproducibility.

### Data availability

All relevant data in this work are available from the authors per request.

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## Acknowledgements

The authors would like to thank Prof. Dag Kristian Dysthe for his constructive suggestions and the fruitful discussion. The authors also thank Josue Lopez for providing the Matlab code to process the friction data. This work is supported by the National Science Foundation Grant CMMI-1435920.

## Author information

### Affiliations

1. #### Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 205 N. Matthews Avenue, Urbana, IL, 61801, USA

• Yijue Diao
•  & Rosa M. Espinosa-Marzal

### Contributions

Y.D. performed experiments. R.M.E.-M. designed the research and supervised the graduate student. Y.D. and R.M.E.-M. analyzed the data and wrote the manuscript.

### Competing interests

The authors declare no competing interests.

### Corresponding author

Correspondence to Rosa M. Espinosa-Marzal.


text:
Homework Help: Newton III problem involving two objects in space

1. Mar 7, 2009

angotta

1. The problem statement, all variables and given/known data

An 93.0 spacewalking astronaut pushes off a 620 satellite, exerting a 120 force for the 0.470 it takes him to straighten his arms

2. Relevant equations

FA on S=FS on A
Xf=Xi+Vi$$\Delta$$t+1/2a($$\Delta$$t)2

Those are the equations I think I use. And initial velocity should be zero, also for time I subtracted the time of the astronaut pushing off the total time

3. The attempt at a solution

I found the acceleration of the astronaut to be 1.29
and the acceleration of the satellite to be 0.194

Then I tried to fill those into the kinematic equations but I got a way large answer. I just need some guidance on where to start and such.

2. Mar 7, 2009

Staff: Mentor

Please state the problem exactly as given. What are you asked to find?

3. Mar 7, 2009

angotta

Im sorry we are asked to find how far apart are the astronaut and the satellite are after 1.50 ?

4. Mar 7, 2009

Staff: Mentor

Good. Those are the accelerations during the push. Now use kinematics to solve for how fast they are going at the end of the push.

(Once the push is over, they move at constant speed. Then you can calculate how far they get from the starting point.)

5. Mar 7, 2009

angotta

Thank you I found the answer to be 62.8 m which was correct.

6. Mar 7, 2009

lubuntu

Ah Newton the Third, his laws are much less well known than his grandfather's though ;)

format: plain


text:
# -1 (a) Find the eigenvalues and associated eigenvectors of A % -1 (b)Theorem: Determining an Eigenvalue from an Eigenvector. Ffxis

###### Question:

-1 (a) Find the eigenvalues and associated eigenvectors of A % -1 (b)Theorem: Determining an Eigenvalue from an Eigenvector. Ffxis an eigenvector of a matrix A then its corresponding eigenvalue is given by Ax 1 = This quotient is called the Rayleigh quotient Verify the eigenvector results above using the Rayleigh quotient. (c) Find the matrix P (also called the modal matrix) that will diagonalize A through the similarity transformation: P-'AP Verify the Cayley Hamilton Theorem for A in Matlab: (d) What is A-1 using the Cayley Hamilton Theorem. Verify your result using Matlab' $inv function (e) What is A" using the diagonalized form of A. Verify your result in Matlab. (0) What is exp(A) using the diagonalized form of A. Verify your result using Matlab'$ expm function. Prove that the eigenvalues of A" are A5,45,13 where A,,2z,Az are the eigenvalues of A Prove that the eigenvalues of A-!are 41' ,45', 45' . (h) What is the property of A that is equal to the product of eigenvalues: A,,kz,A,

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text:
## Notes About Using amsmath split Environment In Hebrew Documents

Recently I’ve worked on a Hebrew document in LaTeX and wanted to use the split environment to typeset some multiline formula. The document which compiled just fine till that point, failed to compile with the following error:

Package amsmath Error: \begin{split} won't work here.


text:
# Are there countably or uncountably many infinite subsets of the positive even integers?

Let $S$ be the set of all infinite subsets of $\mathbb N$ such that $S$ consists only of even numbers.

Is $S$ countable or uncountable?

I know that set $F$ of all finite subsets of $\mathbb N$ is countable but from that I am not able to deduce that $S$ is uncountable since it looks hard to find a bijection between $S$ and $P(\mathbb N)\setminus F$. Also I am not finding the way at the moment to find any bijection between $S$ and $[0,1]$ to show that $S$ is uncountable nor I can find any bijection between $S$ and $\mathbb N$ or $S$ and $\mathbb Q$ to show that it is countable. So I am thinking is there some clever way to show what is the cardinality of $S$ by avoiding bijectivity arguments?

So can you help me?

Notice that by dividing by two, you get all infinite subsets of $\mathbb{N}$. Now to make a bijection from $]0,1]$ to this set, write real numbers in base two, and for each real, get the set of positions of $1$ in de binary expansion.

You have to write numbers of the form $\frac{n}{2^p}$ with infinitely many $1$ digits (they have two binary expansions, one finite, one infinite). Otherwise, the image of such a real by this application would not fall into the set of infinite sequences of integers (it would have only finitely many $1$).

• Note that $0.01111\ldots_2 = 0.1_2$. Of course your argument can be easily fixed, but the function you describe is not a bijection. – dtldarek Mar 25 '13 at 14:02
• That's exactly why I wrote one need to use the infinite expansion when there are two possible binary expansions. Why do you think it's not a bijection? – Jean-Claude Arbaut Mar 25 '13 at 14:04
• You are right, I should refresh my browser more often (sometimes AJAX does not work). – dtldarek Mar 25 '13 at 14:06
• But you are right, I had to include 1 in the interval, to take care of image of $\mathbb{N}$. Now I think everything is fixed. – Jean-Claude Arbaut Mar 25 '13 at 14:08
• Thank you, it surely looks good! – user67878 Mar 25 '13 at 14:08

The set of even numbers is countably infinite, and so equinumerous to $\Bbb N$ (explicit bijection given by $n\mapsto\frac12n$). With $F$ the set of finite subsets of $\Bbb N$, as you say, we can use the map $f(n)=\frac12n$ to get a bijection $g:S\to P(\Bbb N)\setminus F$, given by $$g(A)=\{f(n):n\in A\}.$$ I leave it to you to prove that it's bijective.

• So, if I understood your argument well, it is the case that if there is a bijection between $A$ and $B$ then there is a bijection between $R$ and $S$ where $R$ is the set of all infinite subsets of $A$ and $S$ is the set of all infinite subsets of $B$? – user67878 Mar 25 '13 at 13:57
• Absolutely right, and you can prove that as a general statement using the same idea that I outlined above. – Cameron Buie Mar 25 '13 at 13:59
• Do you mean by the "general statement" that it works for every cardinality of $A$ and $B$ if we precisely enough define cardinality of $R$ and $S$, which is not needed in my case because every infinite subset in my problem must be of countable infinity? – user67878 Mar 25 '13 at 14:05
• What a flaw by me! It is fairly "obvious" what is the set of all infinite subsets of some set so in my previous comment maybe it is better to put away the word "precise". And you already answered that it is possible so it looks like I am asking the same question as in the first comment. It was a wave of confusion few minutes ago so it is clearer now for me to see it should work for every cardinality. Thank you! – user67878 Mar 25 '13 at 14:14
• So, if I ask you what approach would you choose for yourself, a set theory in which that statement is true, or set theory in which that statement is not true, what would you choose? You can choose it without an axiom of choice! ;.D – user67878 Mar 27 '13 at 19:44

Use the Cantor-Bernstein theorem, that is (in the simplified version)

$$|A| \leq |B| \text{ and } |B| \leq |A| \text{ implies } |A| = |B|.$$

We have that $S \subset P(\mathbb{N})$, so $|S| \leq |P(\mathbb{N})|$. Moreover, we have an injection $f : P(\mathbb{N}) \to S$ given by

$$f(A) = \{4k \mid k \in A\} \cup \{4k+2 \mid k \in \mathbb{N}\}.$$

Hence, we have $|P(\mathbb{N})| \leq |S|$, so $|S| = |P(\mathbb{N})|$.

I hope this helps ;-)

• Yes, you all got +1 for the effort! – user67878 Mar 25 '13 at 14:02
• @Thus As there might be many solutions, it would be still of benefit to you to know the Cantor-Bernstein theorem. Sometimes bijections are troublesome to construct (see, e.g. bijection between $P(\mathbb{N})$ and $\mathbb{R}$), while usually there is a much more simple way to construct two injections. – dtldarek Mar 25 '13 at 14:14
• I know about the theorem but didn´t look in that direction that it could be easier to construct two injections instead of one bijection, not because of $1<2$ :-D – user67878 Mar 25 '13 at 14:21

Hint: If $\{x_n\mid n\in\Bbb N\}$ is a set of even integers then $\{\frac{x_n}2\mid n\in\Bbb N\}$ is a sequence of integers. Show that this operation is in fact a bijection between $S$ and the set of all infinite subsets of integers. The latter is more familiar.

• Thank you! I have a question that I can put in a comment and I believe that you could solve it, do you want of me to ask you? – user67878 Mar 25 '13 at 14:28
• If it is related, sure. – Asaf Karagila Mar 25 '13 at 14:29
• I will answer it too, I just want of you to confirm is my line of reasoning correct. Let $S$ be the set of all subsets of $\mathbb N$ such that every set that belongs to $S$ consists only of numbers that are pairwise coprime (relatively prime). So I was thinking of this. The set of all prime numbers $P$ is in $S$ and it is infinite. Because the $P$ is infinite and every infinite subset of $P$ is also obviously consisted of coprime numbers then $S$ is uncountable because the $P$ is countable so it has uncountable number of countable subsets. Is this good reasoning? – user67878 Mar 25 '13 at 14:43
• If it is then I understood your arguments for the question I asked. – user67878 Mar 25 '13 at 14:43
• Yes, it is an excellent reasoning. – Asaf Karagila Mar 25 '13 at 14:48


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# Carrying capacity

The carrying capacity of an environment is the maximum population size of a biological species that can be sustained by that specific environment, given the food, habitat, water, and other resources available. In population ecology, carrying capacity is defined as the environment's maximal load, which is different from the concept of population equilibrium, which may be far below an environment's carrying capacity.[1] The effect of carrying capacity on population dynamics may be modelled with a logistic function.

The specific reason why a population stops growing is known as a limiting or regulating factor.[citation needed]

Reaching carrying capacity through a logistic growth curve

The difference between the birth rate and the death rate is the "natural increase". If the population of a given organism is below the carrying capacity of a given environment, this environment could support a positive natural increase; should it find itself above that threshold the population typically decrease. Thus, the carrying capacity is the maximum number of individuals of a species that an environment can support.[citation needed]

Population size decreases above carrying capacity due to a range of factors depending on the species concerned, but can include insufficient space, food supply, or sunlight. The carrying capacity of an environment may vary for different species.[citation needed]

The term carrying capacity was originally used to address matters related to international shipping in the 1840’s.[2] It then became a term used in livestock management in the 1870’s before finally becoming a staple term used to define the biological limits of a natural system related to population size in the 1950’s.[2]

## Origins

In terms of population dynamics, the concept of carrying capacity was not explicitly used in 1838 by the Belgian mathematician Pierre François Verhulst when he first published his equations based on research on modelling population growth,[3] but was popularised in the 1920 by the American biostatisticians Raymond Pearl and Lowell Reed.[citation needed]

The origins of the English term "carrying capacity" are uncertain, with researchers variously stating that it was used "in the context of international shipping"[4] or that it was first used during 19th-century laboratory experiments with micro-organisms.[5] A recent review finds the first use of the term in an 1845 report by the US Secretary of State to the US Senate.[4]

## Factors

In the standard ecological algebra as illustrated in the simplified Verhulst model of population dynamics, carrying capacity is represented by the constant K:

${\displaystyle {\frac {dN}{dt}}=rN\left(1-{\frac {N}{K}}\right)}$

where N is the population size, r is the maximum growth rate, K is the carrying capacity of the local environment, and dN/dt, the derivative of N with respect to time t, is the rate of change in population with time. Thus, the equation relates the growth rate of the population N to the current population size, incorporating the effect of the two constant parameters r and K. (Note that decrease is negative growth.) The choice of the letter K came from the German Kapazitätsgrenze (capacity limit).

This equation is a modification of the original Verhulst model:

${\displaystyle {\frac {dN}{dt}}=rN-\alpha N^{2}}$[3]

In this equation, the carrying capacity K, ${\displaystyle N^{*}}$, is

${\displaystyle N^{*}={\frac {r}{\alpha }}.}$
This is a graph of the population due to the logistic curve model. When the population is above the carrying capacity it decreases, and when it is below the carrying capacity it increases.

When the Verhulst model is plotted into a graph, the population change over time takes the form of a sigmoid curve, reaching its highest level at K. This is the logistic growth curve and it is calculated with:

${\displaystyle f(x)={\frac {L}{1+e^{-k(x-x_{0})}}},}$

where

e is the natural logarithm base (also known as Euler's number),
x0 is the x value of the sigmoid's midpoint,
L is the curve's maximum value,
K is the logistic growth rate or steepness of the curve [6] and
${\displaystyle f(x_{0})=L/2.}$

The logistic growth curve depicts how population growth rate and the carrying capacity are inter-connected. As illustrated in the logistic growth curve model, when the population size is small, the population increases exponentially. However, as population size nears the carrying capacity, the growth decreases and reaches zero at K.[7]

What determines a specific system's carrying capacity involves a limiting factor which may be something such as available supplies of food, water, nesting areas, space or amount of waste that can be absorbed. Where resources are finite, such as for a population of Osedax on a whale fall or bacteria in a petridish, the population will curve back down to zero after the resources have been exhausted, with the curve reaching its apogee at K. In systems in which resources are constantly replenished, the population will reach its equilibrium at K.[citation needed]

Software is available to help calculate the carrying capacity of a given natural environment.[8]

## Humans

Several estimates of the carrying capacity of the earth for humans have been made with a wide range of population numbers. A 2001 UN report said that two-thirds of the estimates fall in the range of 4 billion to 16 billion with unspecified standard errors, with a median of about 10 billion.[9] Some of these issues have been studied by computer simulation models such as World3.

The application of the concept of carrying capacity for the human population, which exists in a non-equilibrium, has been criticized for not successfully being able to model the processes between humans and the environment.[10]

## References

1. ^ Hui, C (2006). "Carrying capacity, population equilibrium, and environment's maximal load". Ecological Modelling. 192 (1–2): 317–320. doi:10.1016/j.ecolmodel.2005.07.001.
2. ^ a b Berkshire encyclopedia of sustainability. Great Barrington, MA: Berkshire Pub. Group. 2010–2012. ISBN 978-1-933782-01-0. OCLC 436221172.
3. ^ a b Verhulst, Pierre-François (1838). "Notice sur la loi que la population poursuit dans son accroissement" (PDF). Correspondance Mathématique et Physique. 10: 113–121. Retrieved 3 December 2014.
4. ^ a b Sayre, N. F. (2008). "The Genesis, History, and Limits of Carrying Capacity". Annals of the Association of American Geographers. 98: 120–134. doi:10.1080/00045600701734356. S2CID 16994905.
5. ^ Zimmerer, Karl S. (1994). "Human Geography and the "New Ecology": The Prospect and Promise of Integration" (PDF). Annals of the Association of American Geographers. 84: 108–125. doi:10.1111/j.1467-8306.1994.tb01731.x.
6. ^ Verhulst, Pierre-François (1845). "Recherches mathématiques sur la loi d'accroissement de la population" [Mathematical Researches into the Law of Population Growth Increase]. Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Bruxelles. 18: 1–42. Retrieved 2013-02-18.
7. ^ Swafford, Angela Lynn. "Logistic Population Growth: Equation, Definition & Graph." Study.com. N.p., 30 May 2015. Web. 21 May 2016. "Logistic Population Growth - Boundless Open Textbook." Boundless. N.p., n.d. Web. 21 May 2016.
8. ^ Martire, Salvatore; Castellani, Valentina; Sala, Serenella (2015). "Carrying capacity assessment of forest resources: Enhancing environmental sustainability in energy production at local scale". Resources, Conservation and Recycling. 94: 11–20. doi:10.1016/j.resconrec.2014.11.002.
9. ^ "UN World Population Report 2001" (PDF). p. 31. Retrieved 16 December 2008.
10. ^ Cliggett, Lisa (2001). "Carrying Capacity's New Guise: Folk Models for Public Debate and Longitudinal Study of Environmental Change". Africa Today. 48: 3–19. doi:10.1353/at.2001.0003. S2CID 143983509.


text:
# Force Applied -- General doubt about pushing 2 blocks on a floor with friction

1. Apr 13, 2015

### gracy

Force" F"is applied on M1 .Although force applied is not directly on M2,but it will also experience this force" F" via M1,right?

But I don't think it is correct.Because it implies that no matter how big the force on M1,the net force on M1 will always remain zero.And there will be unbalanced force "F" on block M2.It is like we are pushing M1 by force "F"but it is in it's place(not moving)instead M2 moves.Note that between M1 and M2 there will be frictional force ,as they are pressing against each other but it will be along the surface of contact and as there is no force to counter in y direction the frictional force will not come into picture.
I am not posting this in homework section because It is not my homework .It's my conceptual doubt.
Thanks.

2. Apr 13, 2015

### Staff: Mentor

As you seem to have realized, this is not the case. The force F is applied to M1. In turn, M1 will apply a different force on M2. That force will be less than F.

3. Apr 13, 2015

### gracy

Than why in this case

There is friction between the two blocks as well as between the block and wall.

4. Apr 13, 2015

### Staff: Mentor

In this case, the blocks are not accelerating.

5. Apr 13, 2015

### gracy

This seems to me the result ,not the reason.

6. Apr 13, 2015

### gracy

Why https://physicsforums-bernhardtmediall.netdna-ssl.com/data/attachments/65/65001-ae59a43002c3519191eb12aaa6fd31ab.jpg [Broken]
is applicable in this case
https://physicsforums-bernhardtmediall.netdna-ssl.com/data/attachments/65/65022-752f6be69d8fa34ccdbce044f6fd5318.jpg [Broken]
but not in

Last edited by a moderator: May 7, 2017
7. Apr 13, 2015

### Staff: Mentor

Not sure what you are looking for. Given the force F applied, you then deduce any other forces using Newton's laws.

What applies in both cases are Newton's laws.

Last edited by a moderator: May 7, 2017
8. Apr 13, 2015

### gracy

ok.I can point out.
but in the case when blocks are pushed against wall
the force will be equal to F.Why?That's what I am looking for.

9. Apr 13, 2015

### Staff: Mentor

In the first case, the blocks are accelerating. Thus, there must be a net force on each block.

In the second case, when pushed against the wall, the blocks are not accelerating. Thus, there must be no net force on them. (Which implies that the force between the blocks must equal the applied force.)

10. Apr 13, 2015

### gracy

That means for block M1 there will be less acceleration than M2 ,if F is not much larger than force on M2?

11. Apr 13, 2015

### mattt

Expanding what Doc Al just said:

in "the accelerating case" (and taking the horizontal axe oriented to the right) you have:

$F = (M_1 + M_2) a$

$F-R = M_1 a$

$R = M_2 a$

R is (the magnitude of) the force the second block exerts on the first block and also it is (the magnitude of) the force the first block exerts on the second block.

In the "against the wall case" you have:

$F - N = (M_1 + M_2) a = 0$

$F - R = M_1 a = 0$

$R - N = M_2 a = 0$

N is the "Normal Force" the wall exerts on the second block.

12. Apr 13, 2015

### mattt

No, look at my previous post.

13. Apr 13, 2015

### gracy

So,you mean acceleration will be same for both the blocks.

14. Apr 13, 2015

### gracy

But you did not mention anything about friction in the accelerating case" (and taking the horizontal axe oriented to the right).

15. Apr 13, 2015

### mattt

Do you mean friction between each block and the floor? You can add that force too. Tell me if you want me to write the above equations adding the friction force between each block and the floor.

16. Apr 13, 2015

### gracy

Yes.So nice of you.But I am afraid that I would be not showing any effort if I ask you to write all equations for me.Better,you guide me how to write them then I will write them by myself.

17. Apr 13, 2015

### mattt

Well, first thing I should have asked you is, what have you already studied?

Do you know Newton laws, applied to one particle and to system of particles?

18. Apr 13, 2015

### gracy

Yes.As far as I know,I know.

19. Apr 13, 2015

### mattt

I have to go back to work right now, but if you understood my previous post ( # 11 ) then it won't be that difficult to you to add friction forces between each block and the floor.

The best way (when you are learning) is drawing a picture and drawing in it each and every (vector) force, and then to apply Newton laws, to the whole "block1 + block2", and/or (depending on what you want) to each block singled out.

Try to do it, and tomorrow I'll come back here again to see.

20. Apr 13, 2015

### gracy

OK.I will try.Thanks again.


text:
Search

# Timed Choices Ren'py Code

This is a short note about how we got timed choices to work in our game.

Below is a sample of the code we used in one of the exam scenes in the game:

After the first three paragraphs, you only need to write the rest of the code (starting from "label questiontime1:") to summon the timer in whichever scene you want from then on.

Under "menu" you would write the usual choices. Don't forget to write "hide screen countdown" after, below the choice, so the timer bar will disappear after the player has chosen a choice.

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