Question

Assuming that there is no transformation of mass to energy or vice versa, it is possible to write a continuity equation for mass. ( $a$ ) If we use the continuity equation for charge as our model, what quantities correspond to $\mathbf{J}$ and ${\rho }_v$ ? (b) Given a cube $1\mathrm{cm}$ on a side, experimental data show that the rates at which mass is leaving each of the six faces are $10.25,-9.85,1.75,-2.00$, $-4.05$, and $4.45\mathrm{mg}/\mathrm{s}$. If we assume that the cube is an incremental volume element, determine an approximate value for the time rate of change of density at its center.

Short answer

The quantities that correspond to J and 𝜌_v in the continuity equation for mass are mass current density (𝑱_m) and mass density (𝜌_m), respectively. 𝑱_m represents the mass flow per time per area through a given surface, and 𝜌_m represents the mass per unit volume at a given position and time. What is the time rate of change of density at the center of a 1 cm cube given the mass rates at which mass is leaving each face of the cube? The time rate of change of density at the center of the 1 cm cube is approximately -0.0005 kg/m³/s.

Step-by-step solution

Quantity corresponding to $\mathbf{J}$ and ${\rho }_v$ in the continuity equation for mass

When using the continuity equation for charge as our model, let's consider the following correspondence relationship: 1. The current density $\mathbf{J}$ corresponds to the mass current density ${\mathbf{J}}_{\mathbf{m}}$, where ${\mathbf{J}}_{\mathbf{m}}(x,y,z,t)$ represents the mass flow per time per area through a given surface at position $(x,y,z)$ and time $t$. 2. The volume charge density ${\rho }_v$ corresponds to mass density ${\rho }_m$, where ${\rho }_m(x,y,z,t)$ represents the mass per unit volume at position $(x,y,z)$ and time $t$. With these correspondences, we can establish the continuity equation for mass that follows: $$\frac{\partial {\rho }_m}{\partial t}+\mathrm{\nabla }\cdot {\mathbf{J}}_{\mathbf{m}}=0$$

Applying the continuity equation for mass to the cube

To determine the time rate of change of density at the center of a 1 cm cube, we need to find $\frac{\partial {\rho }_m}{\partial t}$ at the center. Given the rates at which mass is leaving each face of the cube $(10.25,-9.85,1.75,-2.00,-4.05,4.45)$ mg/s, we'll use the principle of conservation of mass: $$\frac{\partial {\rho }_m}{\partial t}V=-\sum _{i=1}^6{M}_i$$ where: - $V$ is the volume of the cube, $1{\text{cm}}^3={10}^{-3}{\text{m}}^3$ - $M_i$ is the mass leaving the $i$-th face of the cube (given in mg/s) - In order to sum the given values, convert all mass rates to kg/s: ${10}^{-6}\text{kg/mg}$ Therefore: $$\frac{\partial {\rho }_m}{\partial t}=-\frac{\sum _{i=1}^6{M}_i}{V}$$

Calculating the time rate of change of density

Now we can calculate the time rate of change of density at the center of the cube: $$\frac{\partial {\rho }_m}{\partial t}=-\frac{(10.25-9.85+1.75-2.00-4.05+4.45)\times {10}^{-6}}{{10}^{-3}}\frac{\text{kg}}{{\text{m}}^3\text{s}}$$ $$\frac{\partial {\rho }_m}{\partial t}=-0.0005\frac{\text{kg}}{{\text{m}}^3\text{s}}$$ Hence, the approximate value for the time rate of change of density at the center of the cube is $-0.0005\frac{\text{kg}}{{\text{m}}^3\text{s}}$.

Key concepts

Understanding Mass Density

The concept of mass density is vital in understanding how mass is distributed in a given volume. Mass density, often represented by ${\rho }_m$, is defined as the amount of mass per unit volume. Simply put, it tells us how much mass is packed into a particular space at a given time.

• Mathematically, it is expressed as ${\rho }_m=\frac{m}{V}$, where $m$ is the mass and $V$ is the volume.
• In the scenario where mass is conserved and does not change into other forms like energy, mass density becomes essential to tracking how mass may shift over time and space.

In essence, mass density gives us insight into the concentration of mass within a specific region and helps predict movements and changes in the physical world based on this distribution.

Exploring Mass Flow

Mass flow refers to the movement of mass from one point to another. In physics, when we discuss mass flow, we're often talking about how mass is transferred across surfaces.

• Mass flow is integral in systems where mass conservation is key, as it helps us understand where mass is being directed over time.
• The mass current density, denoted by ${\mathbf{J}}_{\mathbf{m}}$, describes the flow more formally as the mass flowing per unit area per unit time.

Understanding mass flow involves calculating the mass leaving or entering through various boundaries, and it's crucial for solving problems relating to the conservation of mass.

Unpacking Mass Current Density

Mass current density, represented by ${\mathbf{J}}_{\mathbf{m}}$, is closely tied to mass flow, but it offers a more detailed view. This measurement specifies how much mass passes through a given surface area in a given time period, typically expressed in units like $\frac{\text{kg}}{{\text{m}}^2\cdot \text{s}}$.

• This is particularly useful in the continuity equation, where ${\mathbf{J}}_{\mathbf{m}}$ becomes a critical factor in determining changes in mass density over time.
• The equation $abla\cdot {\mathbf{J}}_{\mathbf{m}}$ helps express whether mass is diverging or converging within a particular volume.

By analyzing mass current density, we gain a deeper understanding of how and where mass density is changing, pinpointing shifts within any given system of interest.

Grasping the Conservation of Mass

The conservation of mass is a fundamental principle stating that mass in a closed system remains constant over time, regardless of the processes acting within the system. This means mass cannot be created or destroyed, only transferred or transformed.

• It directly relates to the continuity equation for mass: $\frac{\partial {\rho }_m}{\partial t}+abla\cdot {\mathbf{J}}_{\mathbf{m}}=0$.
• This equation guarantees that any change in mass density within a given volume is accounted for by the flux of mass across the boundary.

Understanding and applying the conservation of mass is crucial for solving problems in physics, engineering, and environmental science. It provides reassurance that any analysis performed on a system accurately respects this enduring principle.