The mass density formula of a one-dimensional object, like a thin rod or wire with varying density, is given by:
m = ∫ab ρ(x) dx
Where:
- m represents the total mass of the object.
- ρ(x) is the linear mass density function, which describes how the density varies along the object's length. It essentially tells you the mass per unit length at a specific point x.
- x is the position along the one-dimensional object.
- a and b are the limits of integration, representing the starting and ending points of the object along the x-axis.
- ∫ represents the integral, which sums up the mass contributions from each infinitesimally small segment of the object.
- dx represents an infinitesimally small length element along the object.
Explanation:
The formula essentially calculates the total mass by integrating the linear mass density function over the length of the one-dimensional object. Think of it as dividing the object into infinitely small segments of length dx. The mass of each tiny segment is approximately ρ(x) * dx. By summing up the masses of all these tiny segments (through integration), you obtain the total mass of the object.
Example:
Imagine a thin rod of length 5 meters. Its linear mass density varies according to the function ρ(x) = x2 kg/m, where x is the distance from one end of the rod. To find the total mass of the rod, you would integrate the density function from x = 0 to x = 5:
m = ∫05 x2 dx = [x3/3]05 = (53/3) - (03/3) = 125/3 kg ≈ 41.67 kg
Therefore, the total mass of the rod is approximately 41.67 kg.
Constant Density Case:
If the linear mass density is constant (ρ(x) = ρ), the formula simplifies to:
m = ∫ab ρ dx = ρ ∫ab dx = ρ(b - a) = ρL
Where L = (b - a) is the length of the object. This makes intuitive sense: the total mass is simply the constant density multiplied by the length of the object.
