91Ó°ÊÓ

The center of mass of an object is a point representing the mean position of the matter in it. For a rod lying on the x-axis, the center of mass (COM) along the axis can be found using the formula: \(\text{COM} = \frac{1}{M} \int_{a}^{b} x \cdot \delta(x) \, dx\)where \(M\) is the total mass of the rod, \(\delta(x)\) is the density function, and \([a, b]\) is the interval over which the rod extends.

First, we calculate the total mass \(M\) of the rod from \(x=0\) to \(x=\pi\). This is done by integrating the density function:\[M = \int_{0}^{\pi} (2 + \sin x) \, dx\]Calculate the integral:\[M = \left[ 2x - \cos x \right]_{0}^{\pi} = 2(\pi) - (\cos \pi - \cos 0)\]\(M = 2\pi + 1 + 1 = 2\pi + 2\)

Next, we use the total mass \(M\) to find the center of mass by integrating \(x \cdot \delta(x)\).\[\int_{0}^{\pi} x(2 + \sin x) \, dx\]This expands to:\[\int_{0}^{\pi} (2x + x \sin x) \, dx\]Calculate each part separately:1. \(\int_{0}^{\pi} 2x \, dx = \left[ x^2 \right]_{0}^{\pi} = \pi^2\)2. \(\int_{0}^{\pi} x \sin x \, dx\) requires integration by parts.- Let \(u = x\), \(dv = \sin x \, dx\), then \(du = dx\), \(v = -\cos x\).- Using integration by parts:\[\int_{0}^{\pi} x \sin x \, dx = \left[-x \cos x \right]_{0}^{\pi} + \int_{0}^{\pi} \cos x \, dx\]This simplifies to:\[ -x \cos x + \sin x \big|_{0}^{\pi} = (-\pi)(-1) + 0 + 0 - (0 + 0) = \pi\]Thus,\(\int_{0}^{\pi} x(2 + \sin x) \, dx = \pi^2 + \pi\)Now compute the center of mass:\[\text{COM} = \frac{\pi^2 + \pi}{2\pi + 2}\]

Finally, simplify the expression for the center of mass:\[\text{COM} = \frac{\pi(\pi + 1)}{2(\pi + 1)} = \frac{\pi}{2}\]Thus, the center of mass of the rod is at \(x = \frac{\pi}{2}\).