Three particles of mass \(m\) are placed at \((-a,-a)\), \((a,-a)\), and \((0, a)\). (a) Find the center of the mass of this system. (b) Find the moment of inertia \(I_{\mathrm{cm}}\) around the center of mass of the system for an axis along the \(z\)-axis. (c) Find the moment of inertia \(I_{0, z}\) for an axis along the \(z\)-axis through the origin. (d) Find the moment of inertia \(I_{0, x}\) for an axis along the \(x\)-axis through the origin. (e) Find the moment of inertia \(I_{0, z}\) for an axis along the \(y\)-axis through the origin.

Given three particles of mass \(m\) at positions \((-a, -a)\), \((a, -a)\), and \((0, a)\). Need to find the center of mass, moments of inertia around different axes, including the center of mass and the origin.

The center of mass (CM) coordinates \((x_{\text{cm}}, y_{\text{cm}})\) can be found using the formula: \[x_{\text{cm}} = \frac{1}{M} \sum_{i}m_ix_i\] \[y_{\text{cm}} = \frac{1}{M} \sum_{i}m_iy_i\] where \(M\) is the total mass. For three particles: \[x_{\text{cm}} = \frac{m(-a) + m(a) + m(0)}{3m} = 0\] \[y_{\text{cm}} = \frac{m(-a) + m(-a) + m(a)}{3m} = -\frac{a}{3}\] Thus, the center of mass is at \((0, -\frac{a}{3})\).

The moment of inertia (\(I_{\text{cm}}\)) around the z-axis through the center of mass is: \[I_{\text{cm}} = m[(x_1 - x_{\text{cm}})^2 + (y_1 - y_{\text{cm}})^2] + m[(x_2 - x_{\text{cm}})^2 + (y_2 - y_{\text{cm}})^2] + m[(x_3 - x_{\text{cm}})^2 + (y_3 - y_{\text{cm}})^2]\] With \(x_{\text{cm}} = 0\) and \(y_{\text{cm}} = -\frac{a}{3}\): \[I_{\text{cm}} = m[(-a-0)^2 + (-a+\frac{a}{3})^2] + m[(a-0)^2 + (-a+\frac{a}{3})^2] + m[(0-0)^2 + (a+\frac{a}{3})^2 ]\] \[I_{\text{cm}} = m[a^2 + (\frac{4a}{3})^2] + m[a^2 + (\frac{4a}{3})^2] + m[0 + (\frac{4a}{3})^2]\] \[I_{\text{cm}} = m[a^2 + \frac{16a^2}{9}] + m[a^2 + \frac{16a^2}{9}] + m[\frac{16a^2}{9}]\] \[= m [a^2 (1 + \frac{16}{9})] + m[a^2 (1+ \frac{16}{9})] + m[\frac{16a^2}{9}]\] \[= m [a^2 (\frac{25}{9})] + m [a^2 (\frac{25}{9})] + m [\frac{16a^2}{9}]\] \[= \frac{25a^2}{3} + \frac{16a^2}{3}\] \[= \frac{91a^2}{9}\]

The moment of inertia (\(I_{0,z}\)) around the z-axis through the origin is: \[I_{0,z} = m(x_1^2 + y_1^2) + m(x_2^2 + y_2^2) + m(x_3^2 + y_3^2)\] Given positions: \[I_{0,z} = m(a^2 + a^2) + m(a^2 + a^2) + m(a^2)\] \[I_{0,z} = 3ma^2\]

The moment of inertia (\(I_{0,x}\)) around the x-axis through the origin is: \[I_{0,x} = m(y_1^2) + m(y_2^2) + m(y_3^2)\] \[I_{0,x} = m(a^2) + m(a^2) + m(a^2)\] \[I_{0,x} = 3ma^2\]

The moment of inertia (\(I_{0,y}\)) around the y-axis through the origin is: \[I_{0,y} = m(x_1^2) + m(x_2^2) + m(x_3^2)\] \[I_{0,y} = m(a^2) + m(a^2) + m(0^2)\] \[I_{0,y} = 2ma^2\]