Step 1: Understanding the Concept:
To find the distance between the center of mass (COM) and point P, we define a 2D coordinate system, calculate the coordinates of the COM using the mass distribution, identify the coordinates of P, and then use the Euclidean distance formula.
Step 2: Key Formula or Approach:
Coordinates of COM:
\[ X_{\text{com}} = \frac{\sum m_i x_i}{\sum m_i}, \quad Y_{\text{com}} = \frac{\sum m_i y_i}{\sum m_i} \]
Distance formula:
\[ d = \sqrt{(X_{\text{com}} - X_p)^2 + (Y_{\text{com}} - Y_p)^2} \]
Step 3: Detailed Explanation:
Let the apex of the triangle (15 kg) be at the origin \((0,0)\). The triangle has equal side lengths of \(10 \, \text{cm}\) and an apex angle of \(120^\circ\). The base angles are \(30^\circ\).
Coordinates of the masses:
- \(m_1 = 15 \, \text{kg}\) at \((0, 0)\)
- \(m_2 = 2 \, \text{kg}\) at \((-10\sin 60^\circ, -10\cos 60^\circ) = (-5\sqrt{3}, -5)\)
- \(m_3 = 3 \, \text{kg}\) at \((10\sin 60^\circ, -10\cos 60^\circ) = (5\sqrt{3}, -5)\)
Calculate COM:
\[ X_{\text{com}} = \frac{15(0) + 2(-5\sqrt{3}) + 3(5\sqrt{3})}{15 + 2 + 3} = \frac{5\sqrt{3}}{20} = \frac{\sqrt{3}}{4} \]
\[ Y_{\text{com}} = \frac{15(0) + 2(-5) + 3(-5)}{20} = \frac{-25}{20} = -1.25 = -\frac{5}{4} \]
\(\text{COM} = \left( \frac{\sqrt{3}}{4}, -\frac{5}{4} \right)\).
Point P is the midpoint of the median from the origin to the base midpoint.
Base midpoint \(M = \left( \frac{-5\sqrt{3} + 5\sqrt{3}}{2}, \frac{-5 + -5}{2} \right) = (0, -5)\).
Midpoint of median (origin to \(M\)):
\[ P = \left(0, -2.5\right) = \left(0, -\frac{5}{2}\right) \]
Calculate the distance \(d\):
\[ d = \sqrt{\left(\frac{\sqrt{3}}{4} - 0\right)^2 + \left(-\frac{5}{4} - \left(-\frac{5}{2}\right)\right)^2} \]
\[ d = \sqrt{\left(\frac{\sqrt{3}}{4}\right)^2 + \left(\frac{5}{4}\right)^2} = \sqrt{\frac{3}{16} + \frac{25}{16}} = \sqrt{\frac{28}{16}} \]
\[ d = \sqrt{1.75} \approx 1.32 \, \text{cm} \]
Step 4: Final Answer:
The distance is approximately \(1.32\).
