Three small masses are positioned as follows: 2.0 kg at (0.0 m,0.0 m),2.0 kg at (2.0 m,0.0 m),and 4.0 kg at (2.0 m,1.0 m).Determine the coordinates of the center of mass (or center of gravity)of this system.
A)(0.50 m,1.5 m)
B)(1.5 m,0.50 m)
C)(2.5 m,1.5 m)
D)(2.5 m,0.50 m)

Question

Quizplus · Accepted Answer

The center of mass $x_{cm}$ and $y_{cm}$ can be calculated using the formulas:$$x_{cm} = \frac{\sum m_i x_i}{\sum m_i}$$$$y_{cm} = \frac{\sum m_i y_i}{\sum m_i}$$Given the masses and their coordinates:- Mass 1: $2.0 \, \text{kg}$ at $(0.0 \, \text{m}, 0.0 \, \text{m})$- Mass 2: $2.0 \, \text{kg}$ at $(2.0 \, \text{m}, 0.0 \, \text{m})$- Mass 3: $4.0 \, \text{kg}$ at $(2.0 \, \text{m}, 1.0 \, \text{m})$Calculating $x_{cm}$:$$x_{cm} = \frac{(2.0 \times 0.0) + (2.0 \times 2.0) + (4.0 \times 2.0)}{2.0 + 2.0 + 4.0} = \frac{0 + 4 + 8}{8} = \frac{12}{8} = 1.5 \, \text{m}$$Calculating $y_{cm}$:$$y_{cm} = \frac{(2.0 \times 0.0) + (2.0 \times 0.0) + (4.0 \times 1.0)}{2.0 + 2.0 + 4.0} = \frac{0 + 0 + 4}{8} = \frac{4}{8} = 0.5 \, \text{m}$$Therefore, the coordinates of the center of mass are $(1.5 \, \text{m}, 0.5 \, \text{m})$.

Three Small Masses Are Positioned as Follows: 2