🔑:To solve this problem, we'll use the concept of time dilation, which is a fundamental aspect of special relativity. Time dilation states that time appears to pass more slowly for an observer in motion relative to a stationary observer. The amount of time dilation depends on the relative velocity between the two observers.Let's denote the following:* Δt' = time measured by the spaceship's clock (proper time) = 1 week = 604,800 seconds* Δt = time measured on Earth (coordinate time)* v = relative velocity between the spaceship and Earth = 0.99c (99% of the speed of light)* c = speed of light = approximately 299,792,458 meters per second* γ = Lorentz factor, which describes the time dilation effectThe Lorentz factor is given by:γ = 1 / sqrt(1 - v^2/c^2)Plugging in the values, we get:γ = 1 / sqrt(1 - (0.99c)^2/c^2)= 1 / sqrt(1 - 0.9801)= 1 / sqrt(0.0199)= 1 / 0.1413≈ 7.088Now, we can use the time dilation equation to relate the proper time (Δt') to the coordinate time (Δt):Δt = γ * Δt'Substituting the values, we get:Δt = 7.088 * 604,800 seconds≈ 4,293,091 secondsThere are 86,400 seconds in a day, so:Δt ≈ 4,293,091 seconds / 86,400 seconds/day≈ 49.7 daysTherefore, approximately 49.7 days will have passed on Earth while the spaceship was traveling for 1 week (604,800 seconds) according to its own clock.As for the astronaut's aging, since time dilation causes time to pass more slowly for the astronaut, they will have aged less than someone who remained on Earth. The aging effect is directly proportional to the time dilation factor. Since the astronaut experienced 1 week of time, while 49.7 days passed on Earth, the astronaut will have aged approximately:Aging = Δt' / Δt * Aging on Earth= 604,800 seconds / 4,293,091 seconds * Aging on Earth≈ 0.141 * Aging on EarthIn other words, the astronaut will have aged about 14.1% of the amount that someone on Earth would have aged during the same period. If we assume that the person on Earth aged 49.7 days, the astronaut will have aged approximately:Astronaut's aging ≈ 0.141 * 49.7 days≈ 7 daysSo, the astronaut will have aged about 7 days, while approximately 49.7 days will have passed on Earth.Keep in mind that these calculations are based on the assumptions of special relativity and neglect any effects of general relativity, which might become significant at very high speeds or in strong gravitational fields.

🔑:## Step 1: Understand the components of the problemTo derive the Lorentz invariance of electric charge, we need to consider the conservation of Maxwell's 4-current, the correctness of Lorentz transformations, and the assumption that charge densities and currents vanish at infinity. Maxwell's 4-current is given by (J^mu = (rho, mathbf{J})), where (rho) is the charge density and (mathbf{J}) is the current density.## Step 2: Apply the conservation of Maxwell's 4-currentThe conservation of Maxwell's 4-current is expressed as (partial_mu J^mu = 0). This equation implies that the total charge in a closed system is constant over time, as it is a statement of the continuity equation in four-dimensional spacetime.## Step 3: Consider the Lorentz transformationLorentz transformations relate the spacetime coordinates of an event in one inertial frame to those in another. For a boost in the x-direction, the transformation of the coordinates is given by (t' = gamma(t - vx/c^2)), (x' = gamma(x - vt)), (y' = y), and (z' = z), where (gamma = 1/sqrt{1 - v^2/c^2}). The transformation of the 4-current components must be considered to understand how charge and current densities transform under Lorentz transformations.## Step 4: Transform the 4-currentUnder a Lorentz transformation, the charge and current densities transform as follows: (rho' = gamma(rho - vJ_x/c^2)) and (J_x' = gamma(J_x - vrho)), with (J_y') and (J_z') transforming similarly. This shows how the charge and current densities in one frame are related to those in another frame.## Step 5: Integrate the 4-current over a closed surface at infinityTo prove the invariance of electric charge, we need to show that the integral of the current density over a closed surface at infinity vanishes. This involves integrating the 4-current (J^mu) over a 3-dimensional hypersurface in spacetime that encloses a 4-dimensional volume. Given that charge densities and currents vanish at infinity, the integral of the current density over such a surface will be zero.## Step 6: Apply Gauss's theorem for the 4-currentGauss's theorem in four dimensions relates the integral of the 4-current over a closed 3-dimensional hypersurface to the charge enclosed within that hypersurface. Since the charge densities and currents are assumed to vanish at infinity, the total charge enclosed within any surface at infinity is invariant, implying the Lorentz invariance of electric charge.The final answer is: boxed{0}

🔑:To address the question of what happens to the temperature of the gas in a container placed on a fast-moving train, we need to consider the principles of thermodynamics and the behavior of gases in motion. The key concept here is the distinction between the temperature of the gas, which is a measure of the average kinetic energy of the gas molecules, and the kinetic energy associated with the bulk motion of the gas as a whole (i.e., the motion of the train).## Step 1: Understanding Temperature and Kinetic EnergyThe temperature of a gas is directly related to the average kinetic energy of its molecules. This relationship is given by the equation (KE = frac{3}{2}kT), where (KE) is the average kinetic energy of a molecule, (k) is Boltzmann's constant, and (T) is the temperature in Kelvin. This equation shows that temperature is a measure of the thermal motion of the gas molecules.## Step 2: Considering the Motion of the TrainWhen the train moves, the container and the gas within it are also in motion. However, this bulk motion of the gas does not directly affect the temperature of the gas. The temperature of the gas is determined by the random, thermal motion of its molecules, not by the directed motion of the gas as a whole.## Step 3: Applying the Concept of Relative MotionFrom the perspective of an observer on the train (or within the container), the gas molecules are moving randomly in all directions, but there is no net flow of gas in any direction. The motion of the train adds a component of velocity to each gas molecule in the direction of the train's motion, but this does not change the distribution of velocities among the molecules or their average kinetic energy in the direction perpendicular to the train's motion.## Step 4: Conclusion on Temperature ChangeSince the temperature of the gas is determined by the average kinetic energy of the molecules due to their random motion, and the motion of the train does not alter this random motion or the average kinetic energy of the molecules, the temperature of the gas remains unchanged. The energy associated with the bulk motion of the gas (due to the train's motion) is not thermal energy and does not contribute to the temperature of the gas.The final answer is: boxed{No change}

🔑:## Step 1: Determine the mass of the rotor bladeTo find the mass of the rotor blade, we need to calculate the total length of the blade and multiply it by the mass per meter. The blade extends from 0.1 m to 2 m, so its length is 2 - 0.1 = 1.9 m. Given the mass per meter is 2 kg, the total mass of the blade is 1.9 m * 2 kg/m = 3.8 kg.## Step 2: Calculate the centripetal force exerted on the bladeThe centripetal force (F) exerted on an object is given by the formula F = (m * v^2) / r, where m is the mass of the object, v is its velocity, and r is the radius of the circular path. However, since we're dealing with angular velocity (ω), it's more convenient to use the formula F = m * ω^2 * r. For a rotor blade, the force varies along its length, but for simplicity, we can calculate the force at the tip (where r = 2 m) and consider the average effect. However, the correct approach involves integrating the force over the length of the blade, considering the differential mass elements and their distances from the axis of rotation.## Step 3: Integrate the centripetal force over the length of the bladeThe differential mass element (dm) of the blade can be expressed as dm = μ * dr, where μ is the mass per unit length (2 kg/m) and dr is the differential length element. The centripetal force (dF) on this differential mass element is dF = dm * ω^2 * r = μ * dr * ω^2 * r. To find the total centripetal force, we integrate dF from r = 0.1 m to r = 2 m.## Step 4: Perform the integrationThe total centripetal force (F) is given by the integral of dF from 0.1 to 2, which is F = ∫[0.1,2] μ * ω^2 * r * dr. Substituting μ = 2 kg/m, we get F = 2 * ω^2 * ∫[0.1,2] r * dr.## Step 5: Solve the integralThe integral of r * dr from 0.1 to 2 is (1/2) * r^2 | from 0.1 to 2 = (1/2) * (2^2 - 0.1^2) = (1/2) * (4 - 0.01) = (1/2) * 3.99 = 1.995. So, F = 2 * ω^2 * 1.995.## Step 6: Simplify the expression for FF = 2 * ω^2 * 1.995 = 3.99 * ω^2.The final answer is: boxed{3.99 omega^2}