🔑:## Step 1: Identify the given parametersThe number density of atoms in the nebula is given as 1 times 10^{8} atoms per m^3, and the temperature is 7500K. We need to use the ideal gas law to find the pressure.## Step 2: Convert the number density to molar concentrationTo use the ideal gas law, we need the number of moles per unit volume. Given that 1 mole contains 6.022 times 10^{23} atoms (Avogadro's number), the molar concentration n can be calculated as n = frac{1 times 10^{8}}{6.022 times 10^{23}} moles per m^3.## Step 3: Calculate the molar concentrationPerforming the calculation: n = frac{1 times 10^{8}}{6.022 times 10^{23}} = frac{10^{8}}{6.022 times 10^{23}} = frac{1}{6.022 times 10^{15}} approx 1.66 times 10^{-16} moles per m^3.## Step 4: Apply the ideal gas lawThe ideal gas law is given by p = nRT, where p is the pressure, n is the molar concentration, R is the gas constant (8.3145 J/(mol cdot K)), and T is the temperature in Kelvin. Substituting the given values: p = (1.66 times 10^{-16}) times (8.3145) times (7500).## Step 5: Calculate the pressurePerforming the calculation: p = (1.66 times 10^{-16}) times (8.3145) times (7500) = 1.66 times 10^{-16} times 62110.875 = 1.0303 times 10^{-11} Pascals.The final answer is: boxed{1.03 times 10^{-11}}

🔑:In the Nadel v. Burger King Corporation case, the court's decision hinges on the product liability claims brought against Burger King Corporation. To determine whether Burger King Corporation is liable, we must analyze the design of the product and whether there is a defect in the design.The case involves a plaintiff who was injured by a hot pickle slice that was part of a Burger King sandwich. The plaintiff alleges that the pickle slice was defectively designed, as it was too hot and caused her injury. Burger King Corporation argues that the pickle slice was not defectively designed, as it was intended to be served hot and the plaintiff assumed the risk of eating a hot food product.To determine whether Burger King Corporation is liable, we must consider the concept of design defect. A design defect occurs when a product is designed in a way that makes it unreasonably dangerous, even when used as intended. In this case, the court must consider whether the design of the pickle slice, including its temperature, makes it unreasonably dangerous.The court's decision will likely turn on the application of the risk-utility test, which balances the risk of harm against the utility of the product. If the court finds that the risk of harm from the hot pickle slice outweighs its utility, then Burger King Corporation may be liable for the design defect.However, if the court finds that the pickle slice was not defectively designed, but rather the plaintiff was simply injured by a hot food product, then Burger King Corporation may not be liable. The court may consider factors such as the intended use of the product, the foreseeability of the injury, and the availability of alternative designs.In this case, the evidence suggests that Burger King Corporation may not be liable for the product liability claims. The pickle slice was intended to be served hot, and the plaintiff assumed the risk of eating a hot food product. While the plaintiff was injured by the pickle slice, the court may find that the design of the pickle slice was not defective, as it was intended to be served hot and the plaintiff was aware of the risk.Furthermore, the court may consider the concept of assumption of risk, which holds that a plaintiff assumes the risk of injury when they engage in a particular activity or use a product in a certain way. In this case, the plaintiff assumed the risk of eating a hot food product, and therefore, Burger King Corporation may not be liable for the injury.In conclusion, based on the evidence and relevant legal concepts, it is likely that the court will find that Burger King Corporation is not liable for the product liability claims. The design of the pickle slice, including its temperature, was intended to be served hot, and the plaintiff assumed the risk of eating a hot food product. While the plaintiff was injured by the pickle slice, the court may find that the design of the pickle slice was not defective, and therefore, Burger King Corporation is not liable.Argument:Burger King Corporation is not liable for the product liability claims because the design of the pickle slice, including its temperature, was intended to be served hot, and the plaintiff assumed the risk of eating a hot food product. The risk-utility test, which balances the risk of harm against the utility of the product, suggests that the risk of harm from the hot pickle slice does not outweigh its utility. The plaintiff's injury was a foreseeable consequence of eating a hot food product, and Burger King Corporation took reasonable steps to warn consumers of the risk. Therefore, the court should find that Burger King Corporation is not liable for the design defect claim.Evidence:* The pickle slice was intended to be served hot as part of a Burger King sandwich.* The plaintiff assumed the risk of eating a hot food product.* The court may consider factors such as the intended use of the product, the foreseeability of the injury, and the availability of alternative designs.* The risk-utility test suggests that the risk of harm from the hot pickle slice does not outweigh its utility.Relevant Legal Concepts:* Design defect: A product is defective if it is designed in a way that makes it unreasonably dangerous, even when used as intended.* Risk-utility test: A test that balances the risk of harm against the utility of the product.* Assumption of risk: A plaintiff assumes the risk of injury when they engage in a particular activity or use a product in a certain way.* Foreseeability: The court considers whether the injury was a foreseeable consequence of the product's design.

🔑:Torque is a fundamental concept in rotational mechanics that describes the rotational force that causes an object to rotate or twist. It is defined as the product of the force applied to an object and the perpendicular distance from the axis of rotation to the point where the force is applied.Mathematically, torque (τ) is defined as:τ = r x Fwhere:τ = torque (measured in N·m)r = distance from the axis of rotation to the point where the force is applied (measured in m)F = force applied to the object (measured in N)x = cross product operator (indicating the perpendicular distance between the force and the axis of rotation)To derive the formula for torque, consider a force F applied to an object at a distance r from the axis of rotation. The force can be resolved into two components: one parallel to the axis of rotation (F∥) and one perpendicular to the axis of rotation (F⊥). The component of the force perpendicular to the axis of rotation is responsible for causing the object to rotate.Using the definition of the cross product, the torque can be written as:τ = r x F = rF⊥Since the force is applied at a distance r from the axis of rotation, the work done on the object (W) can be calculated as:W = F·swhere:W = work done (measured in J)F = force applied (measured in N)s = displacement of the object (measured in m)For a rotating object, the displacement is equal to the arc length (θr), where θ is the angle of rotation (measured in radians). Therefore, the work done on the object can be written as:W = F·θrSince the force is applied at a distance r from the axis of rotation, the torque can be written as:τ = rF⊥Substituting this expression for torque into the equation for work, we get:W = τθThis equation shows that the work done on a rotating object is equal to the product of the torque and the angle of rotation. This relationship between torque and work is a fundamental concept in rotational mechanics.In summary, the concept of torque is defined as the product of the force applied to an object and the perpendicular distance from the axis of rotation to the point where the force is applied. The formula for torque (τ = r x F) can be derived by resolving the force into its components and using the definition of the cross product. The relationship between torque and work (W = τθ) shows that the work done on a rotating object is equal to the product of the torque and the angle of rotation, highlighting the importance of torque in rotational mechanics.

🔑:## Step 1: Understanding the Astable Multivibrator CircuitAn astable multivibrator circuit consists of two transistors connected in a way that each transistor's collector is coupled to the other transistor's base through a capacitor. This coupling allows the transistors to switch each other on and off, creating a continuous oscillation.## Step 2: Role of Capacitors in High-Frequency ApplicationsCapacitors are used in the astable multivibrator circuit to couple the high-frequency signals between the transistors. At high frequencies, capacitors behave as low impedance paths, allowing AC signals to pass through while blocking DC signals. This property is crucial for the circuit's operation, as it enables the high-frequency oscillations to be coupled between the transistors.## Step 3: Coupling MechanismWhen the voltage at the collector of one transistor changes, it creates a high-frequency signal. This signal is coupled through the capacitor to the base of the other transistor. Since the capacitor acts as a low impedance path at high frequencies, the signal is effectively transmitted to the base of the other transistor.## Step 4: Forcing the Transistor into CutoffWhen the coupled signal reaches the base of the other transistor, it can force the transistor into cutoff if the signal's amplitude and polarity are appropriate. For an NPN transistor, a negative-going signal at the base can reduce the base-emitter voltage, potentially cutting off the transistor. Conversely, for a PNP transistor, a positive-going signal can achieve the same effect.## Step 5: Implications for Circuit OperationThe coupling mechanism and the behavior of capacitors at high frequencies are essential for the astable multivibrator's operation. As one transistor switches off, the other transistor switches on, and vice versa, due to the coupled signals. This continuous switching creates the oscillations that are characteristic of an astable multivibrator circuit.The final answer is: boxed{0}