Problem 35 - Physics for Scientists and Engineers with Modern Physics

Question

(1) What are the dimensions of density, which is mass per volume?

Step-by-Step Solution

Answer

The dimensions of density are \([M L^{-3}]\).

Step 1: Identify the Formula for Density

Density is defined as mass per unit volume. The formula for density \( \rho \) is given by \( \rho = \frac{{m}}{{V}} \), where \( m \) is the mass and \( V \) is the volume.

Step 2: Analyze the Dimensions of Mass

The dimension of mass is represented as \([M]\). This notation indicates that mass has one fundamental dimension, denoted as \( M \).

Step 3: Analyze the Dimensions of Volume

Volume is the space occupied by a substance and is typically represented in terms of length. The dimensional formula for volume is \([L^3]\) because it involves three linear dimensions (i.e., length, width, and height).

Step 4: Compute the Dimensions of Density

To find the dimensions of density, divide the dimensional representation of mass \([M]\) by that of volume \([L^3]\). Thus, the dimensions of density are \( \left[ \frac{M}{L^3} \right] = [M L^{-3}] \).

Step 5: State the Dimensions of Density

The dimensions of density are expressed in terms of mass and volume as \([M L^{-3}]\). This indicates that density is dependent on the mass dimension and inversely proportional to the cube of the length dimension.

Key Concepts

Mass Per Volume

Density tells us how much mass is packed into a given volume. It is a measure of how much stuff there is in a certain amount of space. Imagine packing a suitcase for a trip. If you pack more clothes into the same suitcase, the suitcase becomes more dense. This is similar to the concept of density in physics.
To find density, we use the formula:

• Density, \( \rho = \frac{m}{V} \)

Here, \( m \) represents mass and \( V \) stands for volume. This formula tells us the amount of mass \( m \) in a specific volume \( V \). This relationship helps us understand how tightly matter is clustered together. Knowing this, if mass increases but the volume stays the same, density increases.If volume increases but mass stays the same, density decreases.

Dimensional Analysis

Dimensional analysis is like a tool that helps us check our math in physics. It ensures that equations or expressions make sense with respect to the physical dimensions involved. Using dimensions allows us to confirm if our calculations are correct and consistent. In the case of density, we need to consider both dimensions of mass and volume. This gives us clarity about what exactly our calculations refer to. This is crucial in maintaining accuracy, especially in scientific equations.
By analyzing dimensions in equations, we can:

• Verify if an equation is dimensionally correct.
• Convert units appropriately.
• Identify any possible errors in a formula or calculation.

Therefore, dimensional analysis helps keep the computations aligned with physical realities.

Fundamental Dimensions

In physics, every physical quantity is expressed in terms of fundamental dimensions. The fundamental dimensions include things like:

• Mass (\( [M] \))
• Length (\( [L] \))
• Time (\( [T] \))

These dimensions form the basis from which more complex units are derived. For density, the critical fundamental dimension we consider is mass, represented as \( [M] \). Knowing this aids us in understanding how other dimensions, like volume, relate and interact with mass, forming a coherent understanding of the physical world.

Volume Dimensions

Volume describes how much space a substance or object occupies, and it is intrinsically linked to length. Think about a box; it has dimensions of length, width, and height. When calculating volume, these three dimensions are multiplied, giving the dimensional formula \( [L^3] \). This means we consider three linear (length-based) dimensions creating a three-dimensional space. When calculating density, the volume dimension \( [L^3] \) comes into play as we use it in the denominator: \( [M L^{-3}] \). This shows us that density decreases if the volume increases, assuming mass stays constant. Being familiar with volume dimensions helps us visualize and quantify the capacity of an object, which is essential in understanding how much matter it can hold.