🔑:## Step 1: Determine the critical buckling load formula for a column with one end fixed and the other end free.The critical buckling load (P_cr) for a column with one end fixed and the other end free is given by the formula P_cr = (2.046 * π^2 * E * I) / L^2, where E is the Young's modulus, I is the moment of inertia about the weak axis, and L is the length of the column.## Step 2: Express the moment of inertia (I) in terms of the breadth (b) and thickness (t) of the I-beam.For an I-beam, the moment of inertia about the weak axis (I) can be approximated as I = (1/12) * b * t^3 + 2 * (1/12) * (b/2) * t^3, considering the web and two flanges. However, for an I-beam with equal thickness on the web and flanges, and given that d = 2b (depth is twice the breadth, implying the height of the web is b), the formula simplifies further. The total moment of inertia about the weak axis (I) for such a section, considering the contribution of both the web and the flanges, can be simplified to I = (1/12) * b * t_web^3 + 2 * (1/12) * b_flange * t_flange^3, but since t_web = t_flange = t, and knowing the web's height equals b, we simplify this to I = (1/12) * b * t^3 + 2 * (1/12) * (b/2) * t^3 for the flanges' contribution. However, this simplification does not fully apply as we must consider the actual distribution of material in an I-beam for accurate calculations.## Step 3: Correctly calculate the moment of inertia (I) for the I-beam section.Given the I-beam's geometry, where the web thickness is 10mm and the flange thickness is also 10mm, and knowing d = 2b, we need to correctly calculate I. The moment of inertia about the weak axis for an I-beam is I = (b * t_web^3)/12 + 2 * [(b/2) * t_flange^3]/12, but this formula does not fully capture the I-beam's geometry. A more accurate approach involves considering the I-beam as composed of two flanges and a web, where the moment of inertia about the weak axis is more accurately represented by the formula specific to the geometry of an I-beam, taking into account the web and flange dimensions correctly.## Step 4: Apply the correct formula for the moment of inertia of an I-beam about its weak axis.For an I-beam with equal thickness for the web and flanges, and where d = 2b, the correct formula considering the web and flange contributions should be directly applied to calculate I. However, the provided simplification in step 2 and step 3 may not directly apply due to the complexity of calculating the moment of inertia for an I-beam section. The general formula for the moment of inertia of an I-beam about its weak axis involves the dimensions of the flanges and the web, and it is crucial to apply the correct formula considering the given dimensions and the relationship between d and b.## Step 5: Calculate the minimum section requirements using the critical buckling load formula.Given P_cr = 92 kN, E = 200 GN/m^2, and L = 6 m, we can rearrange the critical buckling load formula to solve for I: I = (P_cr * L^2) / (2.046 * π^2 * E). Substituting the given values, we find I = (92000 * 6^2) / (2.046 * π^2 * 200000000000) = (92000 * 36) / (2.046 * π^2 * 200000000000).## Step 6: Perform the calculation for the moment of inertia (I).I = (3312000) / (2.046 * π^2 * 200000000000) = 3312000 / (2.046 * 9.8696 * 200000000000) = 3312000 / 4012936960000 ≈ 8.25 * 10^-7 m^4.## Step 7: Determine the relationship between b and d, and express d in terms of b.Given d = 2b, and knowing the I-beam's dimensions are constrained by this relationship, we can proceed to find the minimum section requirements based on this constraint and the calculated moment of inertia.## Step 8: Use the calculated moment of inertia (I) to find the minimum breadth (b) of the I-beam.The formula for the moment of inertia of an I-beam about its weak axis, considering the web and flange thicknesses are equal (t), is complex and involves the dimensions of both the web and the flanges. However, given the simplification that the depth (d) is twice the breadth (b), and knowing the thickness (t) is 10mm, we aim to find the minimum 'b' that satisfies the calculated I, keeping in mind the actual formula for I of an I-beam may require a more detailed analysis of the section's geometry.## Step 9: Apply the correct approach to find the minimum section requirements.Given the complexity of directly calculating 'b' from the simplified formula for I, and considering the need for a detailed geometric analysis of the I-beam, the approach should involve using the critical buckling load to find the required moment of inertia, and then applying the geometric constraints (d = 2b, t = 10mm) to determine the minimum section dimensions that satisfy the calculated I, ensuring the I-beam can withstand the given load without buckling.The final answer is: boxed{0.1}

🔑:The phenomenon of a mercury droplet splitting into two when subjected to an electric field is a complex process involving electromigration, electrowetting, and the magnetic field generated by the current. Here's a detailed description of the roles of these factors and the experimental conditions that could influence this phenomenon:Electromigration:Electromigration is the movement of ions or charged particles in a conductor (in this case, the mercury droplet) under the influence of an electric field. When an electric field is applied to the mercury droplet, the ions within the droplet experience a force that drives them towards the electrode of opposite polarity. This movement of ions can lead to a redistribution of charge within the droplet, causing it to deform and eventually split.Electrowetting:Electrowetting is the change in the wetting properties of a liquid (mercury) on a surface (the electrode) in response to an electric field. When an electric field is applied, the mercury droplet can spread or contract on the electrode surface, depending on the polarity of the field. This change in wetting behavior can influence the shape and stability of the droplet, making it more susceptible to splitting.Magnetic field generated by the current:When an electric current flows through the mercury droplet, it generates a magnetic field. This magnetic field can interact with the electric field and the motion of the ions within the droplet, influencing the splitting process. The magnetic field can induce a Lorentz force on the moving ions, which can contribute to the deformation and splitting of the droplet.Interplay between electromigration, electrowetting, and magnetic field:The interplay between these factors can lead to the splitting of the mercury droplet. As the electric field is applied, electromigration drives the ions towards the electrode, causing a redistribution of charge within the droplet. Electrowetting changes the wetting properties of the mercury on the electrode surface, which can lead to a deformation of the droplet. The magnetic field generated by the current can further enhance the deformation and splitting of the droplet by interacting with the moving ions.Experimental conditions that could influence the phenomenon:1. Electrode geometry: The shape and size of the electrodes can influence the electric field distribution and the resulting electromigration and electrowetting effects. For example, a smaller electrode gap can lead to a stronger electric field and more pronounced electromigration.2. Applied potential: The magnitude and polarity of the applied potential can control the strength of the electric field and the resulting electromigration and electrowetting effects. A higher applied potential can lead to a more rapid splitting of the droplet.3. Initial vs. steady-state current: The initial current can influence the rate of electromigration and the resulting deformation of the droplet. The steady-state current can affect the stability of the droplet and the splitting process.4. Mercury droplet size and shape: The size and shape of the mercury droplet can influence its stability and susceptibility to splitting. Smaller droplets may be more prone to splitting due to their higher surface-to-volume ratio.5. Electrolyte properties: The properties of the electrolyte (e.g., conductivity, viscosity) can influence the electromigration and electrowetting effects. A more conductive electrolyte can lead to a stronger electric field and more pronounced electromigration.6. Temperature: Temperature can affect the viscosity and surface tension of the mercury, influencing its stability and susceptibility to splitting.To experimentally observe the splitting of a mercury droplet, researchers can use a setup consisting of two electrodes (e.g., platinum or gold) with a small gap between them. A mercury droplet can be placed on one of the electrodes, and an electric field can be applied by connecting the electrodes to a power source. The droplet's behavior can be observed using high-speed cameras or other imaging techniques. By varying the experimental conditions, such as electrode geometry, applied potential, and initial vs. steady-state current, researchers can study the influence of these factors on the splitting phenomenon.

🔑:## Step 1: Calculate time dilation factor for the train relative to the groundThe time dilation factor, γ (gamma), is given by the equation γ = 1 / sqrt(1 - v^2/c^2), where v is the velocity of the train (0.6c) and c is the speed of light. Substituting v = 0.6c into the equation gives γ = 1 / sqrt(1 - (0.6c)^2/c^2) = 1 / sqrt(1 - 0.36) = 1 / sqrt(0.64) = 1 / 0.8 = 1.25.## Step 2: Determine the time elapsed on the train from the perspective of an observer on the groundFrom the perspective of an observer on the ground, time appears to pass more slowly on the train due to time dilation. The bomb's timer is 5 minutes, but because of time dilation, it will appear to take longer to expire from the ground observer's perspective. The time dilation factor (γ = 1.25) means that for every 1 minute that passes on the train, 1.25 minutes will pass on the ground.## Step 3: Calculate the time on the ground when the bomb would explode from the ground observer's perspectiveSince the clock on the ground reads 4:55 PM when the front of the train passes point A, and considering the bomb's 5-minute timer, we need to calculate how long it will take for the bomb to explode from the ground observer's perspective. The bomb's timer is subject to time dilation, so the 5 minutes of bomb timer time corresponds to 5 * γ = 5 * 1.25 = 6.25 minutes of ground time.## Step 4: Determine if the bomb will be deactivated before it explodes from the ground observer's perspectiveThe ground clocks are set to deploy the blade at 5:00 PM, which is 5 minutes after 4:55 PM. Since the bomb's explosion is delayed by time dilation, the ground observer sees the bomb exploding 6.25 minutes after 4:55 PM, which would be 5:01.25 PM. The blade will deploy at 5:00 PM, which is before the bomb explodes from the ground observer's perspective.## Step 5: Analyze the scenario from the perspective of an observer on the trainFrom the perspective of an observer on the train, time dilation affects the clocks on the ground, making them appear to run slower. The observer on the train sees the ground clocks as desynchronized due to the relativistic effects, with clocks ahead of the train appearing to be slower than the train's clock and clocks behind the train appearing to be faster. However, since the question focuses on the deactivation of the bomb by the blade which is synchronized with the ground clocks, we consider the train's perspective on time passing.## Step 6: Calculate the time elapsed on the train from the train observer's perspectiveFor the observer on the train, the bomb's timer is not affected by time dilation because it is at rest relative to the observer. Thus, the bomb will explode 5 minutes after it is activated, according to the train's clock.## Step 7: Determine if the bomb will be deactivated before it explodes from the train observer's perspectiveSince the train is moving at 0.6c, the clocks on the ground appear to be running slower to the train observer. When the bomb is activated at 4:55 PM ground time, the train observer sees the ground clocks as being slightly behind due to time dilation. However, because the train observer sees the ground clocks as running slower, the 5:00 PM deactivation time on the ground clocks will appear to occur later than 5 minutes after 4:55 PM from the train's perspective. But considering the direct question of whether the bomb will be deactivated, the critical factor is the relative motion and the synchronization of clocks, which suggests a need to focus on the relativistic effects on clock synchronization rather than just time dilation.## Step 8: Reconsider the scenario focusing on clock synchronization and relative motionGiven the infinite synchronized clocks along the track and the train's motion, the key issue is not just time dilation but how the clocks are synchronized in the ground frame and how events are coordinated between the two frames. The clocks on the ground are synchronized in the ground frame, and the bomb's deactivation is set to occur at 5:00 PM in this frame. The train's motion and the time dilation effect mean that the train observer will see the ground clocks as not perfectly synchronized due to the relativity of simultaneity. However, the deactivation mechanism is based on the ground frame's synchronization.The final answer is: boxed{Yes}

🔑:The substance with the highest volumetric heat capacity is liquid hydrogen, with a value of approximately 14.3 J/cm³K at standard temperature and pressure (STP) conditions. However, considering more common and practical materials, water has one of the highest volumetric heat capacities among all substances, with a value of around 4.18 J/cm³K at STP conditions.Compared to water, liquid hydrogen has a volumetric heat capacity about 3.4 times higher. This means that liquid hydrogen can absorb and release more heat energy per unit volume than water, making it an attractive material for certain thermal management applications. However, the extremely low temperature and high reactivity of liquid hydrogen make it difficult to handle and store, limiting its practical use.Pressure can significantly affect the volumetric heat capacity of water. As pressure increases, the density of water increases, which in turn increases its volumetric heat capacity. For example, at a pressure of 1000 bar, the volumetric heat capacity of water is approximately 4.35 J/cm³K, which is about 4% higher than its value at STP conditions. This increase in volumetric heat capacity with pressure makes water an even more effective coolant in high-pressure applications, such as in power plants or industrial processes.Materials with high volumetric heat capacities have numerous potential applications in thermal management, including:1. Cooling systems: High volumetric heat capacity materials can be used as coolants in various applications, such as electronic devices, vehicles, or industrial processes, to efficiently absorb and dissipate heat.2. Thermal energy storage: Materials with high volumetric heat capacities can be used to store thermal energy, which can be released as needed, providing a buffer against temperature fluctuations or peak demand.3. Heat exchangers: High volumetric heat capacity materials can be used to enhance the performance of heat exchangers, allowing for more efficient heat transfer and reduced equipment size.4. Aerospace applications: Materials with high volumetric heat capacities, such as liquid hydrogen, can be used as fuels or coolants in aerospace applications, where high energy density and efficient thermal management are critical.5. Advanced composites: Researchers are exploring the development of advanced composites with high volumetric heat capacities, which could be used in a range of applications, from thermal management in electronics to thermal protection systems in aerospace.In summary, while liquid hydrogen has the highest volumetric heat capacity, water remains an excellent choice for many thermal management applications due to its high volumetric heat capacity, low cost, and ease of handling. The effects of pressure on the volumetric heat capacity of water make it an even more attractive material for high-pressure applications. Materials with high volumetric heat capacities have significant potential for improving thermal management in various industries, and ongoing research aims to develop new materials and applications that can leverage these properties.