🔑:## Step 1: Convert the Hubble parameter (H0) from years to secondsFirst, we need to convert the Hubble parameter from years to seconds, since the rate of distance increase will be in centimeters per second. There are approximately 3.1557 x 10^7 seconds in a year. Given H0 is the reciprocal of 13.8 billion years, H0 in years is 1 / 13.8 billion years. Converting this to seconds: H0 = (1 / 13.8 billion years) * (1 year / 3.1557 x 10^7 seconds) = 1 / (13.8 * 3.1557 x 10^7) seconds = 1 / 4.354 x 10^17 seconds ≈ 2.297 x 10^-18 per second.## Step 2: Calculate the rate of distance increase between points A and BThe Hubble parameter (H0) is related to the rate of expansion of the universe by the equation v = H0 * d, where v is the velocity (or rate of distance increase) between two points, and d is the distance between them. Given that the distance between A and B is 100 million light-years, we first need to convert this distance into centimeters because the answer requires the rate in centimeters per second. Since 1 light-year is approximately 9.461 x 10^15 meters, 100 million light-years is 100,000,000 * 9.461 x 10^15 meters = 9.461 x 10^22 meters. To convert meters to centimeters, we multiply by 100 (since 1 meter = 100 centimeters), resulting in 9.461 x 10^24 centimeters.## Step 3: Apply the Hubble equation to find the rate of distance increaseNow, we can use the Hubble equation v = H0 * d, with H0 in per second and d in centimeters, to find v in centimeters per second. Substituting the values we have: v = (2.297 x 10^-18 per second) * (9.461 x 10^24 centimeters).## Step 4: Calculate the final value for the rate of distance increasePerforming the multiplication: v = 2.297 x 10^-18 * 9.461 x 10^24 = 2.173 x 10^7 centimeters per second.The final answer is: boxed{21730000}

🔑:## Step 1: Understanding Topological Insulators and Edge StatesTopological insulators are materials that are insulators in the bulk but have conducting states at their edges or surfaces. These edge states are protected by time reversal symmetry, which means that as long as this symmetry is preserved, the edge states remain conducting.## Step 2: Role of Time Reversal SymmetryTime reversal symmetry is a fundamental concept in physics that states if a system is symmetric under the reversal of time, then certain properties of the system remain unchanged. In the context of topological insulators, this symmetry is crucial because it protects the edge states from being localized by non-magnetic impurities or other perturbations that also respect time reversal symmetry.## Step 3: Scenario Where Edge State Could Be CompromisedThe edge state of a topological insulator could be compromised in a scenario where time reversal symmetry is broken. This can happen through the introduction of magnetic impurities or an external magnetic field. When time reversal symmetry is broken, the protection of the edge states is lifted, and these states can become localized or gapped, leading to a loss of conductivity at the edge.## Step 4: Theoretical Basis for Robustness Against PerturbationsThe theoretical basis for the robustness of edge states against perturbations that obey time reversal symmetry lies in the topological nature of these states. The edge states in topological insulators are characterized by a topological invariant, which is a mathematical object that remains unchanged under continuous deformations of the system's parameters as long as the gap in the bulk does not close and time reversal symmetry is preserved. This means that as long as the perturbation does not break time reversal symmetry, it cannot cause the edge states to become gapped or localized.## Step 5: Conclusion on Robustness and Potential CompromiseIn conclusion, the edge states of topological insulators are robust against perturbations that obey time reversal symmetry due to their topological nature and the protection afforded by this symmetry. However, these states can be compromised if time reversal symmetry is broken, such as through the introduction of magnetic fields or impurities.The final answer is: boxed{Magnetic impurities or fields}

🔑:## Step 1: Calculate the torque required to move the doorFirst, we need to calculate the angular acceleration (α) of the door. We can use the equation θ = ω₀t + (1/2)αt², where θ is the final angle (90 degrees or π/2 radians), ω₀ is the initial angular velocity (0, since it starts from rest), and t is the time (30 seconds). Rearranging the equation to solve for α, we get α = 2θ/t² = 2*(π/2)/(30)² = π/900 rad/s².## Step 2: Calculate the torque requiredThe torque (τ) required to produce this angular acceleration can be found using the equation τ = Iα, where I is the rotational inertia of the door (8.7 x 10^4 kg*m²). So, τ = (8.7 x 10^4 kg*m²)*(π/900 rad/s²) = 8.7*π/9 kg*m²/s² = 3.04 kg*m²/s².## Step 3: Calculate the force requiredNow, we can use the equation τ = rF to solve for the force (F) required, where r is the distance from the axis of rotation (the hinge) to the point where the force is applied (the outer edge of the door, which is half of its width, so r = 2.4/2 = 1.2 m). Rearranging the equation to solve for F, we get F = τ/r = (3.04 kg*m²/s²)/(1.2 m) = 2.53 kg*m/s² = 2.53 N.The final answer is: boxed{2.53}

🔑:A potentially electrifying scenario! Throwing a switched-on hair dryer into water while it's still plugged into a regular plug circuit can lead to a series of events with varying outcomes. Here's a breakdown of what might happen:Initial Event:1. The hair dryer, being an electrical appliance, will continue to operate when submerged in water, at least initially.2. Water is an excellent conductor of electricity, and the hair dryer's electrical components will be in contact with the water.Possible Outcomes:Scenario 1: Immediate Short Circuit and Breaker Trip1. The water will create a conductive path between the hair dryer's live electrical components and the surrounding water, causing a short circuit.2. The short circuit will lead to a significant increase in current flow, which will be detected by the circuit breaker.3. The breaker will trip, disconnecting the power supply to the hair dryer and preventing further electrical flow.4. In this scenario, the risk of electrical shock to humans is minimized, as the breaker will have interrupted the power supply.Scenario 2: Delayed Short Circuit and Breaker Trip1. The hair dryer's insulation and protective coatings may initially prevent a short circuit, delaying the electrical shock.2. However, as the water penetrates the appliance, the insulation will eventually fail, causing a short circuit.3. The breaker will still trip, but with a delay, potentially allowing a brief period of electrical shock exposure.4. The risk of electrical shock to humans is higher in this scenario, as the delayed tripping of the breaker may allow for brief exposure to live electrical components.Scenario 3: Ground Fault Circuit Interrupter (GFCI) Trip1. If the circuit is protected by a Ground Fault Circuit Interrupter (GFCI), which is designed to detect ground faults, the GFCI will trip and disconnect the power supply.2. GFCIs are more sensitive than regular breakers and can detect even small ground faults, reducing the risk of electrical shock.3. In this scenario, the GFCI will trip quickly, minimizing the risk of electrical shock to humans.Potential Risks to Human Safety:1. Electrical Shock: The most significant risk is electrical shock, which can occur if a person comes into contact with the water or the hair dryer while it's still energized.2. Drowning: If the person is shocked and becomes incapacitated, they may drown if they're in a bathtub or a pool.3. Fire: Although less likely, a short circuit can potentially cause a fire, especially if the hair dryer's electrical components overheat or ignite nearby flammable materials.Prevention and Safety Measures:1. Never throw an electrical appliance into water while it's still plugged in.2. Always unplug appliances before submerging them in water or exposing them to moisture.3. Ensure that your home is equipped with GFCI-protected circuits, especially in areas where water is present (e.g., bathrooms, kitchens).4. Regularly inspect your electrical appliances and cords for damage or wear, and replace them if necessary.Remember, electrical safety is paramount, and it's always better to err on the side of caution when dealing with electrical appliances and water.