A 27.7-cm-by-27.7-cm checkerboard has a mass of 103.1 g. There are four 27.7-g checkers located on the checkerboard, as shown in the figure. Relative to the origin located at the bottom left corner of the checkerboard, where is the center of mass of the checkerboard-checkers system?
To find the center of mass of the checkerboard-checkers system, we need to consider the masses and positions of all the components.
First, let's calculate the mass of the checkerboard. The given mass of the checkerboard is 103.1 g.
Next, let's calculate the position of the center of mass of the checkerboard. Since the checkerboard is a square with dimensions 27.7 cm by 27.7 cm, the center of mass of the checkerboard is at its geometrical center. The geometric center of a square is located at the intersection of its diagonals, so the center of mass of the checkerboard is at a distance of half its side length from both the bottom left and top right corners. Therefore, the position of the center of mass of the checkerboard is (27.7 cm/2, 27.7 cm/2) = (13.85 cm, 13.85 cm).
Now, let's consider the checkers. There are four checkers, each with a mass of 27.7 g. To find the center of mass of the checkers, we need to consider their individual masses and positions.
Since the figure is not provided, I will assume that the four checkers are located at the four corners of the checkerboard. In this case, the positions of the checkers relative to the origin are as follows:
- Top right corner: (27.7 cm, 27.7 cm)
- Top left corner: (0 cm, 27.7 cm)
- Bottom right corner: (27.7 cm, 0 cm)
- Bottom left corner: (0 cm, 0 cm)
To find the total mass of the checkers, we add the masses of the four checkers: 4 * 27.7 g = 110.8 g.
To find the center of mass of the checkers, we calculate the weighted average of their positions, where the weights are the masses of the checkers.
The x-coordinate of the center of mass of the checkers is given by:
(X1*M1 + X2*M2 + X3*M3 + X4*M4) / (M1 + M2 + M3 + M4),
where X1, X2, X3, X4 are the x-coordinates of the checker positions, and M1, M2, M3, M4 are their respective masses.
Substituting the respective values, we get:
(27.7 cm * 27.7 g + 0 cm * 27.7 g + 27.7 cm * 27.7 g + 0 cm * 27.7 g)/(27.7 g + 27.7 g + 27.7 g + 27.7 g) = 27.7 cm
The y-coordinate of the center of mass of the checkers is given by:
(Y1*M1 + Y2*M2 + Y3*M3 + Y4*M4) / (M1 + M2 + M3 + M4),
where Y1, Y2, Y3, Y4 are the y-coordinates of the checker positions.
Substituting the respective values, we get:
(27.7 cm * 27.7 g + 27.7 cm * 27.7 g + 0 cm * 27.7 g + 0 cm * 27.7 g)/(27.7 g + 27.7 g + 27.7 g + 27.7 g) = 27.7 cm
Therefore, the center of mass of the checkerboard-checkers system is located at (13.85 cm, 13.85 cm) with respect to the origin at the bottom left corner of the checkerboard.