🔑:To calculate the mass of air inside the bicycle tyre, we need to follow these steps:1. Calculate the volume of the tyre: Approximate the tyre as a torus and calculate its volume.2. Apply the ideal gas law: Use the ideal gas law to find the number of moles of air in the tyre.3. Calculate the mass of air: Use the average molecular weight of air to find the mass of air in the tyre. Step 1: Calculate the volume of the tyreThe volume (V) of a torus can be approximated by the formula:[V = 2pi^2Rr^2]where (R) is the radius of the torus (the distance from the center of the torus to the center of the cross-section circle), and (r) is the radius of the cross-section circle.Given:- Diameter between the centers of the two cross-section circles (which is (2R)) = 630mm, so (R = 315mm = 0.315m)- Diameter of the cross-section circle ((2r)) = 30mm, so (r = 15mm = 0.015m)Substitute these values into the volume formula:[V = 2pi^2 times 0.315m times (0.015m)^2][V = 2pi^2 times 0.315m times 0.000225m^2][V = 2 times (pi)^2 times 0.000071125m^3][V approx 2 times 9.8696 times 0.000071125m^3][V approx 0.0014107m^3] Step 2: Apply the ideal gas lawThe ideal gas law is given by:[PV = nRT]where:- (P) is the pressure of the gas (in Pascals, Pa),- (V) is the volume of the gas (in cubic meters, (m^3)),- (n) is the number of moles of gas,- (R) is the gas constant ((R = 8.3145 , text{J/mol·K})),- (T) is the temperature of the gas in Kelvin (K).Given:- (P = 500kPa = 500,000Pa),- (V approx 0.0014107m^3) (from Step 1),- (T = 20°C = 293K) (room temperature).Rearrange the ideal gas law to solve for (n):[n = frac{PV}{RT}]Substitute the given values:[n = frac{500,000Pa times 0.0014107m^3}{8.3145 , text{J/mol·K} times 293K}][n = frac{500,000 times 0.0014107}{8.3145 times 293}][n = frac{705.35}{2433.0835}][n approx 0.2901 , text{mol}] Step 3: Calculate the mass of airThe average molecular weight of air is approximately (28.97 , text{g/mol}). To find the mass of air in the tyre, multiply the number of moles of air by the average molecular weight of air:[m = n times M]where (m) is the mass of air, (n) is the number of moles of air, and (M) is the average molecular weight of air.Substitute the values:[m = 0.2901 , text{mol} times 28.97 , text{g/mol}][m approx 8.406 , text{g}]Therefore, the mass of air inside the bicycle tyre is approximately 8.41 grams.

🔑:When the outer gimbal is rotated, a reaction torque is exerted on the inner gimbal, causing it to rotate about a vertical axis. The spinning axis of the gyroscope also rotates, due to the conservation of angular momentum. The gyroscope's tendency to maintain its original plane of rotation results in it processing about the vertical axis, a motion known as precession.

🔑:## Step 1: Understanding the Initial ConditionsThe problem describes a large, perfectly round sphere filled with water and several marbles at the 'bottom' of the sphere in outer space, where gravity from other bodies is negligible. This means the primary forces acting on the marbles and the water are due to the rotation or shaking of the sphere itself.## Step 2: Effect of Rotation on the MarblesWhen the sphere is rotated, the water and the marbles inside experience a centrifugal force due to the rotation. This force pushes objects away from the center of rotation. Since the marbles are denser than water, they would initially be at the 'bottom' of the sphere due to the lack of external gravity. As the sphere rotates, the centrifugal force acts on both the water and the marbles, but its effect is more pronounced on the marbles due to their density and size.## Step 3: Movement of Marbles During RotationAs the sphere rotates, the marbles, due to the centrifugal force, would move away from the center of the sphere and towards the inner surface of the sphere. However, because the sphere is filled with water, the marbles' movement would be influenced by the viscosity and the flow of water around them. The marbles would tend to distribute themselves around the inner surface of the sphere in a pattern that reflects the balance between the centrifugal force and the resistance from the water.## Step 4: Influence of the Force Applied to Rotate the SphereThe force applied to rotate the sphere directly influences the centrifugal force experienced by the marbles. A stronger rotational force results in a greater centrifugal force, which in turn causes the marbles to move more rapidly towards the inner surface of the sphere. The distribution and the speed at which the marbles move to their new positions are directly related to the speed and acceleration of the sphere's rotation.## Step 5: Behavior of Marbles if the Sphere is Strongly ShakenIf the sphere is strongly shaken instead of rotated, the marbles would experience a series of rapid changes in direction and acceleration. This would result in a chaotic movement of the marbles within the sphere, as they would be subjected to varying forces in different directions. The water's viscosity and the marbles' density would still play a role in their movement, but the unpredictable nature of the shaking would make their paths highly irregular. The marbles might collide with each other and the inner surface of the sphere, leading to a complex and dynamic behavior.The final answer is: boxed{They move to the equator of the sphere and possibly collide with each other.}

🔑:The increasing strength of dark energy over time is a fascinating concept that can have significant implications for the universe's evolution, particularly when considering the behavior of black holes and the overall topology of the universe. Let's dive into the potential effects of dark energy on the singularity of a black hole and the universe's topology, as well as the ultimate fate of the universe, taking into account Hawking radiation.Effects on Black Hole Singularity:1. Singularity avoidance: Dark energy's increasing strength could potentially lead to a scenario where the singularity at the center of a black hole is avoided or altered. This is because dark energy's negative pressure could counteract the gravitational collapse, potentially creating a "fuzzy" or "softened" singularity.2. Modified black hole evaporation: Dark energy's influence on the black hole's environment could affect the rate of Hawking radiation, potentially slowing down or accelerating the evaporation process. This, in turn, could impact the black hole's lifetime and the amount of information lost during the evaporation process.3. Black hole mergers and growth: The increasing strength of dark energy could also influence the merger rates and growth of black holes. As dark energy dominates the universe's expansion, it may lead to a decrease in the merger rates of black holes, potentially affecting the growth of supermassive black holes at the centers of galaxies.Effects on Universe Topology:1. Accelerated expansion: The increasing strength of dark energy would lead to an accelerated expansion of the universe, potentially resulting in a more diffuse and disconnected topology. This could lead to a universe with a "multi-bubble" or "swiss cheese" structure, where vast regions of space are separated by vast distances.2. Topology changes: Dark energy's influence on the universe's large-scale structure could lead to changes in the topology of the universe, potentially creating new connections or "wormholes" between different regions of space. This, in turn, could affect the overall connectivity and structure of the universe.3. Cycles of expansion and contraction: Some models suggest that the increasing strength of dark energy could lead to cycles of expansion and contraction, potentially resulting in a universe with a "cyclic" or "oscillating" topology. This could have significant implications for our understanding of the universe's evolution and the concept of time itself.Ultimate Fate of the Universe:1. Heat death: The increasing strength of dark energy could lead to a scenario where the universe becomes increasingly diffuse and cold, eventually reaching a state known as "heat death." In this scenario, the universe would become a vast, empty expanse with no remaining sources of energy or organization.2. Big Rip: Alternatively, the increasing strength of dark energy could lead to a catastrophic event known as the "Big Rip," where the expansion of the universe becomes so rapid that it tears apart the fabric of space itself, resulting in a universe with a fundamentally different topology.3. Black hole dominated universe: If dark energy's influence on black hole evaporation is significant, it's possible that the universe could become dominated by black holes, potentially leading to a scenario where the universe is composed primarily of black holes and radiation.In conclusion, the increasing strength of dark energy over time could have significant implications for the singularity of black holes and the topology of the universe. The ultimate fate of the universe would depend on the interplay between dark energy, black holes, and the expansion of the universe. Hawking radiation would play a crucial role in shaping the evolution of black holes, and the resulting effects on the universe's topology and ultimate fate would be profound. Further research and simulations are necessary to fully understand the potential consequences of dark energy's increasing strength and its impact on the universe's evolution.