Question

A flat plate is moving normal to its plane through a gas under the action of a constant force \(F\). The gas is kept at a very low pressure. The speed of the plate \(v\) is much less than the average speed \(u\) of the gas molecules. Which of the following options is/are true? [A] The pressure difference between the leading and trailing faces of the plate is proportional to \(u v\) [B] The resistive force experienced by the plate is proportional to \(v\) [C] The plate will continue to move with constant non-zero acceleration, at all times [D] At a later time the external force \(F\) balances the resistive force

Short answer

[A] is plausible, [B] is likely true, [C] is false, [D] is true. The resistive force should indeed increase with \(v\), and the external force eventually balances the resistive force, leading to constant velocity.

Step-by-step solution

Understanding the scenario

First, understand that we have a plate moving through a gas with a constant force applied to it, and we need to analyze the dynamical behavior of the plate to determine the truthfulness of the given statements. We are in a regime where the speed of the plate is much less than the average speed of the gas molecules, and the gas pressure is very low, which indicates we are operating in a regime where molecular flow or free-molecular flow approximations might be valid.

Assessing option [A]

The pressure difference between the leading and trailing faces of the plate could be affected by the relative speed of the plate and the gas molecules. The relative speed increases the momentum transfer between colliding molecules and the plate, and therefore, the pressure difference could indeed depend on both the speed of the gas molecules and the speed of the plate, making it proportional to the product of the two, i.e., proportional to \(uv\). Without further details (such as the exact nature of molecular collisions), this option cannot be completely assessed but seems plausible.

Assessing option [B]

The resistive force, which is often experienced due to friction or drag, in a low-pressure regime, should be proportional to the speed of the plate relative to the gas, assuming the force arises from interactions and relative motion between the gas molecules and the plate. Thus, the slowing force or resistive force should indeed increase with the speed of the plate \(v\), so this option is likely true.

Assessing option [C]

If the plate will continue to move with a constant non-zero acceleration, it implies a continual increase in speed, which would require a continually increasing force due to Newton's second law. However, we're given that the force \(F\) is constant, not increasing. Therefore, eventually, the resistive force and the external force should balance out, resulting in zero acceleration. Thus, this statement is false.

Assessing option [D]

According to Newton's first law of motion, an object will remain at rest or continue to move at a constant velocity unless acted upon by a resultant (net) force. In this case, when the constant external force \(F\) and the resistive force due to interaction with gas molecules balance each other out, the plate will reach equilibrium and move with constant velocity (zero acceleration), not a constant force imbalance. This state is consistent with the scenario described, so this option is true.

Key concepts

Momentum Transfer in Gases

Understanding the concept of momentum transfer in gases is essential when dealing with gas dynamics under various conditions.
In the case of gases, momentum transfer refers to the process by which the molecules of the gas transfer their momentum to other entities, such as particles, surfaces, or other molecules, during collisions. In our exercise, a flat plate moving through a gas experiences this transfer of momentum when gas molecules collide with it. These collisions can exert a force on the plate, which counteracts the motion.

Effect of Molecular Speeds on Momentum Transfer

The average speed of the gas molecules (\(u\)) and the speed of the plate (\(v\)) both play roles in how much momentum is exchanged. Faster-moving molecules will impart more momentum per collision than slower ones. Therefore, if the gas is kept at a very low pressure where molecular interactions are minimal, the majority of the molecular collisions directly affect the plate, leading to a scenario where the pressure difference across the plate becomes proportional to the product \(uv\), validating option [A] from our exercise.

Drag Force in Low-Pressure Gas

Drag force is a resistance force caused by the interaction of an object with a fluid, including gases, in motion. In a low-pressure environment, the drag force becomes less straightforward to describe, due to the reduced number of gas molecules interacting with the object.

The Nature of Drag at Low Pressures

At low pressures, we often deal with what is known as free-molecular flow, where the mean free path of the gas molecules – the average distance a molecule travels before colliding with another molecule – is much longer. This results in fewer intermolecular collisions and more frequent interactions with the surface of the moving plate. As suggested in our exercise, the resistive force experienced by the plate is proportional to its velocity (\(v\)), which aligns with the concept that the drag force in such a scenario increases with the speed of the plate. This supports option [B] and provides insight into how molecular dynamics influence observable macroscopic behavior.

Newton's Laws of Motion

Newton's laws of motion form the foundation for classical mechanics and explain the relationship between a body and the forces acting upon it. Specifically, Newton's second law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration (\(F = ma\)).
In the context of our exercise, a constant force is applied to a plate moving through a gas. According to Newton's second law, if the force is constant and the mass remains unchanged, the acceleration cannot continue to increase indefinitely; it would eventually reach a point where it becomes zero when the force applied is balanced by the resistive force. This contradicts option [C] which incorrectly asserts that the plate will have a non-zero acceleration at all times. Additionally, we can invoke Newton's first law, which leads us to conclude that the plate will eventually move with a constant velocity once the forces balance out, further supporting option [D].

Gas Dynamics

Gas dynamics is the study of the motion of gases and the forces involved. It encompasses a wide range of topics, including the behavior of gases moving at various speeds, the effect of external forces, and the interactions between gas molecules.

Understanding Free-Molecular Flow

When analyzing gas dynamics in the context of our exercise, the concept of free-molecular flow is particularly pertinent. In free-molecular flow, the dynamics of the gas are characterized by rare interactions between molecules due to the low pressure and, consequently, a high mean free path. Alternatively, the dominant interactions are those between the gas molecules and the boundaries—the moving plate in our case.
This nuanced understanding of gas dynamics helps explain the behavior of the moving plate and the forces acting upon it. The intricacies of gas dynamics, such as the transition from a drag force proportional to velocity to a state of equilibrium, provide additional clarity on why option [D] is deemed true in the provided exercise, thereby offering a complete picture of the system's behavior.