To find the mass of an object using a density function, you need to integrate the density function over the volume of the object.
Here's a breakdown, covering both constant and varying density scenarios:
1. Constant Density:
If the object has a constant density (D) throughout its volume (V), the mass (m) is simply:
*m = D V**
2. Variable Density:
When the density varies throughout the object, you need to use integration. Here's how to approach it depending on the object's geometry:
a) One-Dimensional Object (e.g., Thin Rod):
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Imagine a thin rod where the density changes along its length (x-axis). The density is given by the function ρ(x).
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To find the mass (m) of the rod between positions x=a and x=b, integrate the density function:
m = ∫ab ρ(x) dx
This integral calculates the sum of the density multiplied by an infinitesimally small length element (dx) along the rod.
b) Two-Dimensional Object (e.g., Thin Plate):
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Consider a thin plate where the density varies across its surface. The density is given by the function ρ(x, y).
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To find the mass (m) of the plate, integrate the density function over the area (A) of the plate:
m = ∬A ρ(x, y) dA
Where dA represents an infinitesimally small area element (e.g., dx dy in Cartesian coordinates). This is a double integral.
c) Three-Dimensional Object:
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For a three-dimensional object with a density function ρ(x, y, z), you'll need a triple integral over the volume (V):
m = ∭V ρ(x, y, z) dV
Where dV represents an infinitesimally small volume element (e.g., dx dy dz in Cartesian coordinates).
Example (One-Dimensional):
Suppose a rod has a length from x = 0 to x = 2 meters. Its density is given by ρ(x) = x2 kg/m. To find the mass of the rod:
m = ∫02 x2 dx = [x3/3]02 = (8/3) - 0 = 8/3 kg
Therefore, the mass of the rod is 8/3 kg.
Key Considerations:
- Units: Ensure that the units of the density function and the volume element (dx, dA, dV) are consistent so that the final mass is in the desired unit (e.g., kg).
- Coordinate System: Choose the coordinate system (Cartesian, cylindrical, spherical, etc.) that best suits the geometry of the object to simplify the integration process.
- Limits of Integration: Carefully define the limits of integration (a and b for 1D, the area A for 2D, and the volume V for 3D) to cover the entire object.
