Question

In an evacuated enclosure, a vertical cylindrical tank of diameter \(D\) is sealed by a 3.00-kg circular disk that can move up and down without friction. Beneath the disk is a quantity of ideal gas at temperature \(T\) in the cylinder (Fig. P18.50). Initially the disk is at rest at a distance of \(h\) = 4.00 m above the bottom of the tank. When a lead brick of mass 9.00 kg is gently placed on the disk, the disk moves downward. If the temperature of the gas is kept constant and no gas escapes from the tank, what distance above the bottom of the tank is the disk when it again comes to rest?

Short answer

The disk comes to rest 1 meter above the bottom of the tank.

Step-by-step solution

Analyzing Forces

Initially, the disk is in equilibrium. Its weight and the upward force exerted by the gas below balance each other. The weight of the disk is given by \(m_1g\), where \(m_1 = 3\, \text{kg}\), and \(g = 9.8\, \text{m/s}^2\) is the acceleration due to gravity. Thus, the initial force due to the pressure is \(F_{ ext{initial}} = m_1g = 3 \times 9.8 = 29.4\, \text{N}\).

Calculating Initial Pressure

The pressure exerted by the gas when the disk is at rest initially is calculated as \(P_{ ext{initial}} = \frac{F_{ ext{initial}}}{A}\), where the area \(A\) of the disk is \(\pi \left(\frac{D}{2}\right)^2\). If \(D\) is the diameter of the disk, we use \(A = \pi \left(\frac{D}{2}\right)^2\) to find: \[ P_{ ext{initial}} = \frac{29.4}{0.25\pi D^2} \]

Applying Ideal Gas Law

Initially, the volume of gas is \(V_i = \pi \left(\frac{D}{2}\right)^2 h\). According to the ideal gas law \(PV = nRT\), since \(T\) is constant and there are no gas escapes, the initial state \(P_iV_i\) must equal the final state \(P_fV_f\): \[ (P_{ ext{initial}})(\pi \left(\frac{D}{2}\right)^2 h) = (P_f)(\pi \left(\frac{D}{2}\right)^2 x) \]where \(x\) is the new height of the disk.

New Forces After Adding Mass

When the 9 kg brick is added, the total downward force becomes \(F_{ ext{total}} = (m_1 + m_2)g = (3 + 9) \times 9.8 = 117.6\, \text{N}\). The pressure due to this force when the disk comes to rest is \[P_{ ext{final}} = \frac{117.6}{\pi \left(\frac{D}{2}\right)^2}\]

Equating Initial and Final Conditions

Since the temperature and number of moles are constant, we equate \(P_{ ext{initial}}V_i = P_{ ext{final}}V_f\): \[ 29.4 \times \pi \left(\frac{D}{2}\right)^2 h = 117.6 \times \pi \left(\frac{D}{2}\right)^2 x \] Dividing through by \(\pi \left(\frac{D}{2}\right)^2\), we simplify to:\[ 29.4 \times 4 = 117.6 \times x \] Solving for \(x\), we find \(x = 1\, \text{m}\).

Key concepts

Pressure

Pressure is a fundamental concept in physics, particularly when dealing with gases. It is defined as the force applied perpendicular to the surface of an object per unit area. In the context of the Ideal Gas Law, pressure plays a critical role in describing the behavior of gases under various conditions. For this exercise, the pressure exerted by the ideal gas on the disk changes as additional mass is added.
Initially, the pressure is calculated with the force due to the disk's weight alone. We calculate it using the formula: \( P_{ ext{initial}} = \frac{F_{ ext{initial}}}{A} \), where \( F_{ ext{initial}} \) is the force on the disk, and \( A \) is the disk's area. When the brick is placed on the disk, the pressure changes, as it now supports a greater force.
Understanding how pressure correlates with volume and temperature helps in achieving equilibrium, especially when the system involves movements like in the case of the disk moving up or down the cylindrical tank.

Equilibrium

Equilibrium in this context refers to the state where the forces acting on the disk are balanced. Initially, the disk is in a state of equilibrium with an upward force by the gas balancing its weight. When an additional weight, a brick, is added, it initially disrupts this equilibrium. The gas needs to exert a greater force to achieve a new equilibrium state.
For equilibrium, the downward gravitational force, now increased by the brick’s weight, must be equal to the upward force exerted by the gas. This condition is reached once the disk again comes to rest.
Calculating the initial and final forces determines these equilibrium conditions, allowing us to predict where the disk will stabilize. Identifying equilibrium states helps us apply the Ideal Gas Law effectively in this problem.

Forces

When analyzing forces in this cylindrical tank scenario, it's crucial to identify all forces acting on the disk.
In its initial state, the force in consideration is simply the weight of the disk, calculated as \( m_1g \), where \( m_1 \) is the mass of the disk, and \( g \) is the gravitational acceleration. This gives the base force required to maintain the disk's initial position against the gas pressure.
Adding on the mass of the brick introduces additional force, turning our problem into a multiple-force equilibrium situation. The total weight now acting on the disk becomes \( (m_1 + m_2)g \), with \( m_2 \) being the mass of the brick. The total downward force must equal the upward pressure force exerted by the gas for the disk to be stationary.
Recognizing how these forces interact is critical for properly setting up equations under the Ideal Gas Law and solving for the disk’s final rest position after the brick is added.

Volume

Volume, when discussing gases within the scope of the Ideal Gas Law, directly influences the pressure and temperature dynamics according to the formula \( PV = nRT \). Here, it specifically describes the space beneath the disk containing the gas.
Initially, the volume is calculated as \( \,V_i = \pi \left( \frac{D}{2} \right)^2 h \,\), where \( D \) is the disk's diameter and \( h \) is its height above the tank's bottom. This volume is crucial in determining the initial conditions of the gas' pressure.
When the disk shifts due to the added weight, the new volume below the disk changes, denoted as \( V_f \). We must apply the relationship \( P_iV_i = P_fV_f \) to solve for the disk's new resting height.
Understanding how the volume modifies when conditions change leads us to comprehend how the gas compensates for maintaining pressure and balance in the system.