Problem 66 - Master Resource Book in JEE Main Physics

Question

A closed compartment containing gas is moving with some acceleration in horizontal direction. Neglect effect of gravity. Then, the pressure in the compartment is (a) same everywhere (b) lower in front side (c) lower in rear side (d) lower in upper side

Step-by-Step Solution

Answer

The pressure in the compartment is lower in the front side (b).

Step 1: Analyze the System

The compartment is moving horizontally with acceleration, and there's no effect of gravity to consider. This means that pressure differences will depend solely on the acceleration of the system.

Step 2: Consider Inertial Forces

As the compartment accelerates in the horizontal direction, the gas inside will experience an inertial force opposite to the direction of acceleration. This force can affect the distribution of gas molecules within the compartment.

Step 3: Apply the Concept of Effective Force on Gas

When the compartment accelerates, due to the inertial force, gas molecules will tend to accumulate on the side opposite to the direction of acceleration. This creates a pressure gradient within the compartment.

Step 4: Determine Pressure Differences

Since molecules accumulate on the side opposite to acceleration, pressure at the rear side (opposite to the direction of acceleration) will be higher, while pressure at the front side will be lower.

Step 5: Conclusion

Based on the previous analysis, the pressure in the compartment will be lower in the front side because of the accumulation of gas molecules toward the rear side.

Key Concepts

Inertial Forces

Inertial forces come into play when an object or system is accelerating. These forces arise because the gas molecules tend to resist any change in motion.
For example, when a container filled with gas accelerates to the right, the gas inside experiences an inertial force to the left. This resistance to movement causes a redistribution of gas molecules.
In essence, inertial forces are not actual forces but rather the perceptible effects of a change in an object's motion. These effects stem from Newton's first law of motion, which states that an object in motion stays in motion unless acted upon by an external force.

• As the compartment moves, these forces push the gas in the opposite direction of acceleration.
• This results in a dynamic shift in how gas molecules are spread throughout the compartment.

Understanding inertial forces is crucial for determining how pressure changes within the compartment when exposed to acceleration.

Pressure Gradient

A pressure gradient is essentially a difference in pressure across a given area or volume. Within a moving compartment, this is created by the effects of inertial forces on the gas inside.
When a compartment accelerates, it causes gas molecules to be displaced towards the region of lower inertial force. This sets up a pressure gradient, where:

• Pressure is higher on the side where molecules accumulate (opposite to the direction of acceleration).
• Pressure is lower on the side towards which the compartment is accelerating.

The magnitude of this pressure gradient depends on several factors, including:

• The rate of acceleration of the compartment.
• The specific properties of the gas involved, such as its density.

A clear grasp of pressure gradients is essential to understand how different regions within the compartment experience varying pressures.

Acceleration Effects on Gases

Acceleration affects gases by altering their distribution within a moving compartment. When acceleration is involved, gas molecules tend to shift their position, impacting the internal pressure in notable ways.
Upon acceleration, gases experience:

• A skewed distribution, with molecules drifting towards the area opposite to the acceleration direction.
• A pressure imbalance, where pressure decreases in the direction of acceleration.

This behavior can be visualized as a tilt in the even distribution of gas that occurs when the system is at rest. With more molecules collecting on one side, pressure distribution becomes uneven.
Understanding these effects is vital, as they dictate how gases behave in dynamic environments, not just in theory but also in practical applications like fluid dynamics in engines and atmospheric science.
By grasping these concepts, one can predict changes in pressure and gas behavior in accelerating systems more accurately.