91Ó°ÊÓ

Linear density provides the mass per unit length of an object such as a rod or a wire. It is different from average density, which usually refers to mass per unit volume. In this exercise, the linear density is given by \[ \lambda(x) = 50.0 \text{ g/m} + 20.0x \text{ g/m}^{2} \] where \(x\) is the distance from the end of the rod in meters. This formula shows how the mass distribution varies along the length of the rod. A constant term (50.0 g/m) expresses uniform distribution, while the term involving \(x\) (20.0x g/m\(^2\)) indicates how mass density gradually increases along the length.

• Linear densities often vary along a body in physics problems.
• When given in mathematical form, integrating them lets us find things like total mass.

Understanding this concept is crucial since it varies along objects like non-uniform rods through changes in \(x\). This idea not only applies to rods but to any object with non-uniform mass distribution.

Mass integration is a powerful tool in calculus that helps us compute the total mass of an object by integrating its linear density function over a specified length. In the case of our rod, the total mass \( m \) is the definite integral of the linear density \( \lambda(x) \) over the interval from 0 to 0.3 meters: \[ m = \int_{0}^{0.3} (50.0 + 20.0x) \, dx \] Performing this integration involves finding the antiderivative of the density function.

• The antiderivative of a constant (50.0) is 50.0x.
• The antiderivative of 20.0x is 10.0x\(^2\), derived using the power rule.

Therefore, calculating this gives the total mass of the rod when evaluated between the limits. This integration accounts for the entire varying density spread across the rod’s length. Retaining accuracy in setup and evaluation is key to yielding precise results.

The center of mass is a point representing the mean position of the mass in a body. For our problem, it's computed by finding the weighted average of position along the rod, considering how mass is distributed. Using calculus, the formula to find the center of mass \( x_{cm} \) for a rod with linear density \( \lambda(x) \) is: \[ x_{cm} = \frac{1}{m} \int_{0}^{0.3} x (50.0 + 20.0x) \, dx \] where \( m \) is the previously calculated total mass. This integral calculates the moment of mass about the origin.

During calculation:

• Multiply the linear density function by \( x \).
• Integrate over the same interval (0 to 0.3 m).
• A divide by the total mass removes dependency on units of mass or size.

The center of mass offers intuitive insights into where the mass of the rod is concentrated. A more massive section along the rod will pull \( x_{cm} \) towards it. Understanding this helps predict how objects balance and move.