Question

The density of a \(1.00-\mathrm{m}\) long rod can be described by the linear density function \(\lambda(x)=\) \(100 \cdot \mathrm{g} / \mathrm{m}+10.0 x \mathrm{~g} / \mathrm{m}^{2}\) One end of the rod is positioned at \(x=0\) and the other at \(x=1.00 \mathrm{~m} .\) Determine (a) the total mass of the rod, and (b) the center-of-mass coordinate.

Short answer

Answer: (a) The total mass of the rod is 105 g. (b) The center-of-mass coordinate is approximately 0.54 m.

Step-by-step solution

Find the total mass of the rod m

To find the total mass of the rod, we must integrate the linear density function λ(x) over the length of the rod, from x=0 to x=1 m. The total mass m can be calculated by integrating the function: $$m = \int_{0}^{1} \lambda(x)dx$$ Now, plug in the given linear density function λ(x) = 100g/m + 10.0x(g/m²): $$m = \int_{0}^{1} (100 + 10x)dx$$

Evaluate the integral for the mass

Integrate the function with respect to x: $$m = \left[100x + \frac{10x^2}{2}\right]_{0}^{1}$$ Evaluate the integral at the limits: $$m = (100(1) + \frac{10(1)^2}{2}) - (100(0) + \frac{10(0)^2}{2})$$ $$m = 100 + 5 = 105 \mathrm{~g}$$ The total mass of the rod is 105 g.

Find the center-of-mass coordinate x_cm

To find the one-dimensional center-of-mass coordinate x_cm, we can use the formula: $$x_{\mathrm{cm}} = \frac{1}{m} \int_{0}^{1} x\lambda(x) dx$$ Plug in the given linear density function λ(x) = 100g/m + 10.0x(g/m²), and the total mass m = 105g: $$x_{\mathrm{cm}} = \frac{1}{105} \int_{0}^{1} x(100 + 10x) dx$$

Evaluate the integral for x_cm

Expand the integrand and integrate the function with respect to x: $$x_{\mathrm{cm}} = \frac{1}{105} \int_{0}^{1} (100x + 10x^2) dx$$ $$x_{\mathrm{cm}} = \frac{1}{105} \left[\frac{100x^2}{2} + \frac{10x^3}{3}\right]_{0}^{1}$$ Evaluate the integral at the limits: $$x_{\mathrm{cm}} = \frac{1}{105}\left(\frac{100(1)^2}{2} + \frac{10(1)^3}{3}\right) - \frac{1}{105}\left(\frac{100(0)^2}{2} + \frac{10(0)^3}{3}\right)$$ $$x_{\mathrm{cm}} = \frac{1}{105}(50 + \frac{10}{3})$$ Now, simplify the expression for x_cm: $$x_{\mathrm{cm}} \approx \frac{1}{105}(56.67) \approx 0.54 \mathrm{~m}$$ The center-of-mass coordinate is approximately 0.54 m. So the results are: (a) The total mass of the rod is 105 g. (b) The center-of-mass coordinate is approximately 0.54 m.

Key concepts

Linear Density Function

Understanding the concept of a linear density function is crucial in calculating varying densities along an object. A linear density function tells us how the mass is distributed along a specific length of a material or object. In our specific exercise, the linear density function is given as \[ \lambda(x) = 100\, \text{g/m} + 10.0x\, \text{g/m}^2 \]This implies that the density starts at 100 g/m and increases by 10 g/m² for each additional meter along the rod. Consequently, as you move along the rod from one end to the other, the density isn't constant—it increases linearly with distance.
Understanding how mass distribution varies can be essential in many real-world applications, such as engineering and material fabrication.

Integration in Physics

Integration in physics is a mathematical technique often used to find quantities when dealing with continuous distributions or changes. In the case of our rod, we need to integrate to find the total mass because the density isn't uniform—it changes along the rod. The integration of the linear density function over the specified length of the rod enables us to calculate the total mass:\[ m = \int_{0}^{1} \lambda(x)\, dx = \int_{0}^{1} (100 + 10x)\, dx \]Integration helps by adding up the infinitely small pieces of mass along the rod, taking into account the changing density.
This technique is not only applicable to mass calculations but also to other areas in physics, such as finding electric charge, gravitational force, and fields, among others.

Mass Calculation

To determine the total mass of the rod, we integrate the linear density function along the length of the rod. The process of integration involves finding the antiderivative of the linear density function and then evaluating it over the limits of the rod:\[ m = \left[ 100x + \frac{10x^2}{2} \right]_{0}^{1} = 100 + 5 = 105 \text{ g} \]This calculation shows that the total mass is 105 g. It's important to follow each step, as integration accumulates small changes in density over the entire length. Once the integration results are evaluated at specific boundaries, you can obtain a concrete value representing the full mass, essential in many physical computations.

Physics Problem Solving

Physics problem-solving often requires a step-by-step approach to translate a problem statement into calculations. In this exercise, solving for the center of mass requires understanding the distribution of mass and performing integration. First, ensure you understand the problem and what each function and formula represents.

• Identify the need for integration when dealing with variable densities or quantities.
• Calculate intermediate values like total mass before approaching complex concepts like the center of mass.
• Use known formulas like the center of mass formula, integrating within appropriate limits to find desired parameters.

Finally, translating these steps into the computation reveals that sometimes breaking complex problems into smaller parts helps manage and solve them efficiently.
In this case, systematically integrating and interpreting each outcome was key to finding both the total mass and the center of mass successfully.