You find the mass by integrating the density function over the given length or volume of the object.
Let's break this down:
Mass Calculation with Density Functions
In many scenarios, particularly when dealing with objects that have variable density, we can't just multiply the density by the length or volume to get the mass. We must use integration to account for the varying density.
One-Dimensional Objects (Rods)
As mentioned in the reference, for a thin rod oriented along the x-axis, if the density (ρ) is constant, the mass is calculated simply by multiplying the density and the length of the rod:
(b − a)ρ
Where:
- b is the ending x-coordinate of the rod
- a is the starting x-coordinate of the rod
- ρ is the constant density (mass per unit length)
However, if the density varies along the rod’s length, we represent the density as ρ(x), where x is the position along the rod. Then, to find the mass of the rod, we integrate the density function over the length of the rod:
Mass = ∫ab ρ(x) dx
Here:
- ρ(x) represents the density function, where x is position along the x-axis
- a and b are the lower and upper limits of integration, representing the start and end positions of the rod respectively.
Example:
Imagine a rod that is 5 cm long, starts at position x=0, and has a density function defined as ρ(x) = x2. To calculate the mass, we would integrate this density function from x=0 to x=5.
Two-Dimensional Objects (Plates)
For a thin plate, we would integrate over the area of the plate. Here, the density is usually given in mass per unit area, denoted as σ(x,y), and the calculation will be a double integral:
Mass = ∬ σ(x,y) dA
Here, σ(x,y) is the density at position (x,y), and we would integrate over the area of the plate.
Three-Dimensional Objects (Volumes)
Similarly for a 3D object, we would perform a triple integral:
Mass = ∭ ρ(x,y,z) dV
Here, ρ(x,y,z) is the density at position (x,y,z), and we would integrate over the volume of the object.
General Approach
The general principle for any dimension is that the mass of an object is the integral of its density function over its volume. The key is to:
- Determine the density function. Whether it is ρ(x), σ(x,y), or ρ(x,y,z).
- Identify the limits of integration. Based on the geometry of the object.
- Perform the integration. To determine the mass.
| Object | Density Function | Mass Calculation |
|---|---|---|
| 1D (Rod) | ρ(x) | ∫ab ρ(x) dx |
| 2D (Plate) | σ(x,y) | ∬ σ(x,y) dA |
| 3D (Volume) | ρ(x,y,z) | ∭ ρ(x,y,z) dV |
In summary, to find the mass using the density function, integrate the density function over the object's spatial dimensions. This integration method is crucial for accurately calculating the mass of objects with non-uniform density.
