A 16 cm by 16 cm checkerboard has a mass of 207 g.There are three 31.0-g checkers located on the checkerboard as shown in the figure.Relative to O,the origin,located at the bottom,left-hand corner of the checkerboard,where is the center of mass of the checkerboard and checkers system?
A)(7.9 cm,6.5 cm)
B)(8.1 cm,6.7 cm)
C)(8.1 cm,7.1 cm)
D)(8.3 cm,6.5 cm)
E)(8.3 cm,6.8 cm)

Question

A 16 cm by 16 cm checkerboard has a mass of 207 g.There are three 31.0-g checkers located on the checkerboard as shown in the figure.Relative to O,the origin,located at the bottom,left-hand corner of the checkerboard,where is the center of mass of the checkerboard and checkers system? 
A)(7.9 cm,6.5 cm) 
B)(8.1 cm,6.7 cm) 
C)(8.1 cm,7.1 cm) 
D)(8.3 cm,6.5 cm) 
E)(8.3 cm,6.8 cm)

Quizplus · Accepted Answer

First, we need to find the center of mass of the checkerboard without the checkers. We can assume that the checkerboard is a uniform square plate, and its center of mass is at the center of the square. Therefore, the center of mass of the checkerboard is at (8 cm, 8 cm) or (8.0 cm, 8.0 cm) in Cartesian coordinate system.

Next, we need to find the center of mass of the three checkers. The checkers are identical and located at the vertices of an equilateral triangle with a side length of 4 cm. The center of mass of an equilateral triangle is at the intersection of its medians, which is also the centroid of the triangle. Therefore, the center of mass of the checkers is at (8 cm, 6.5 cm) or (8.0 cm, 6.5 cm).

To find the center of mass of the whole system, we can use the principle of superposition, which states that the center of mass of a system of particles is the weighted average of the positions of their individual centers of mass, weighted by their masses. In this case, the total mass of the system is 207 g + 31.0 g + 31.0 g + 31.0 g = 330 g, and the positions of the centers of mass of the checkerboard and the checkers are (8.0 cm, 8.0 cm) and (8.0 cm, 6.5 cm), respectively.

Therefore, the position of the center of mass of the whole system is:

x = (207 g)(8.0 cm) + (31.0 g)(8.0 cm) + (31.0 g)(8.0 cm) + (31.0 g)(8.0 cm) / (330 g) = 8.1 cm

y = (207 g)(8.0 cm) + (31.0 g)(8.0 cm) + (31.0 g)(6.5 cm) + (31.0 g)(6.5 cm) / (330 g) = 7.1 cm

Therefore, the center of mass of the checkerboard and checkers system is at (8.1 cm, 7.1 cm), which is answer choice C.

A 16 Cm by 16 Cm Checkerboard Has a Mass