Question

Three particles A, B, C with masses of 100 g, 200 g, and 300 g, respectively, are placed at the vertices of an equilateral triangular structure with a side length of 2 m. A is placed at the \( (0, 0) \) position and B is placed at \( (2, 0) \) position in a Cartesian coordinate system. Assume that \( \triangle ABC \) lies parallel to the base. The distance between the center of mass and the position of the particle A is ........... m. (rounded off to 2 decimals)

Hint:

To calculate the distance between a particle and the center of mass, first compute the center of mass coordinates using the mass-weighted average, then apply the distance formula.

Answer 1

Step 1: Understanding the problem.
We are asked to find the distance between the center of mass of the system and particle A. The center of mass of a system of particles is given by the weighted average of the coordinates of the particles: \[ x_{\text{cm}} = \frac{\sum m_i x_i}{\sum m_i}, \quad y_{\text{cm}} = \frac{\sum m_i y_i}{\sum m_i} \] Where \( m_i \) and \( (x_i, y_i) \) are the mass and coordinates of the \( i \)-th particle, respectively.
Step 2: Assigning coordinates and masses.
- Particle A: \( m_A = 100 \, \text{g}, \, (x_A, y_A) = (0, 0) \)
- Particle B: \( m_B = 200 \, \text{g}, \, (x_B, y_B) = (2, 0) \)
- Particle C: \( m_C = 300 \, \text{g}, \, (x_C, y_C) = (1, \sqrt{3}) \) (since the height of an equilateral triangle with side length 2 m is \( \sqrt{3} \))
Step 3: Calculating the center of mass coordinates.
First, calculate \( x_{\text{cm}} \) and \( y_{\text{cm}} \): \[ x_{\text{cm}} = \frac{100(0) + 200(2) + 300(1)}{100 + 200 + 300} = \frac{600 + 300}{600} = \frac{900}{600} = 1.5 \] \[ y_{\text{cm}} = \frac{100(0) + 200(0) + 300(\sqrt{3})}{100 + 200 + 300} = \frac{300\sqrt{3}}{600} = 0.5\sqrt{3} \approx 0.866 \] Step 4: Calculating the distance from A to the center of mass.
The distance \( d \) from particle A to the center of mass is given by the distance formula: \[ d = \sqrt{(x_{\text{cm}} - x_A)^2 + (y_{\text{cm}} - y_A)^2} \] Substituting the values: \[ d = \sqrt{(1.5 - 0)^2 + (0.866 - 0)^2} = \sqrt{(1.5)^2 + (0.866)^2} = \sqrt{2.25 + 0.749} = \sqrt{2.999} \approx 1.73 \, \text{m} \] Final Answer: \[ \boxed{1.73 \, \text{m}} \]