A 32-cm-by-32-cm checkerboard has a mass of 120 g. There are four 19-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?
The coordinates of the checkers
1(6;14)
2 (14;22)
3(22;10
4(30;30)
5(16;16) checkerboard
x(cm) = {m•6+m•14+m•22+m•30+M•16}/{4•m+M} =
={72•19+16•120}/4•19+120 =16.78 cm,
y(cm) = {m•14+m•22+m•10+m•30+M•16}/{4•m+M} =
={76•19+16•120}/4•19+120 =17.16 cm.
C.M. (16.78; 17.16)
Well, considering the checkers are weighty players on the checkerboard, we need to find where their combined center of mass is located.
Let's do some calculations. The checkerboard has a mass of 120 g, and each checker has a mass of 19 g. That gives us a total mass of 76 g for the checkers.
Since the checkers are evenly distributed on the board, we can assume that their combined center of mass would be right in the middle of the board. So, if we divide the length and width of the board by 2, we can pinpoint the center.
(32 cm / 2) = 16 cm
(32 cm / 2) = 16 cm
Therefore, the center of mass of the checkerboard-checkers system would be located at the coordinates (16 cm, 16 cm) relative to the origin.
Just make sure the checkers don't turn the checkered world upside down with their thrilling moves!
To determine the center of mass of the checkerboard-checkers system, we need to consider the mass and position of each component.
Let's assume that the origin is located at the bottom left corner of the checkerboard, and the x-axis extends horizontally to the right while the y-axis extends vertically upwards.
First, let's calculate the position of the center of mass for the checkerboard itself:
Mass of the checkerboard = 120 g
Dimensions of the checkerboard = 32 cm by 32 cm
The position of the center of mass for a rectangle can be calculated using the formula:
x_cm = (x1 * m1 + x2 * m2 + ... + xn * mn) / (m1 + m2 + ... + mn)
y_cm = (y1 * m1 + y2 * m2 + ... + yn * mn) / (m1 + m2 + ... + mn)
For the checkerboard, its mass (m) is 120 g, and its dimensions are 32 cm by 32 cm.
The center of the checkerboard is located at (16 cm, 16 cm).
Next, let's calculate the position of the center of mass for each checker:
Mass of each checker = 19 g
From the figure, let's assume the positions of the checkers are as follows:
Checker 1: (4 cm, 4 cm)
Checker 2: (4 cm, 28 cm)
Checker 3: (28 cm, 4 cm)
Checker 4: (28 cm, 28 cm)
To find the center of mass for the checkerboard-checkers system, we need to consider the mass and position of each component.
The total mass of the system is calculated by adding the masses of the checkerboard and the checkers:
Total mass = Mass of the checkerboard + (4 * Mass of each checker)
Total mass = 120 g + (4 * 19 g)
Total mass = 196 g
To calculate the position of the center of mass, we use the formula mentioned earlier:
x_cm = (x1 * m1 + x2 * m2 + ... + xn * mn) / (m1 + m2 + ... + mn)
y_cm = (y1 * m1 + y2 * m2 + ... + yn * mn) / (m1 + m2 + ... + mn)
x_cm = (16 cm * 120 g + 4 cm * 19 g + 4 cm * 19 g + 28 cm * 19 g + 28 cm * 19 g) / 196 g
x_cm = (1920 + 76 + 76 + 532 + 532) / 196
x_cm = 3136 / 196
x_cm = 16 cm
y_cm = (16 cm * 120 g + 4 cm * 19 g + 28 cm * 19 g + 4 cm * 19 g + 28 cm * 19 g) / 196 g
y_cm = (1920 + 76 + 532 + 76 + 532) / 196
y_cm = 3136 / 196
y_cm = 16 cm
Therefore, the center of mass of the checkerboard-checkers system relative to the origin located at the bottom left corner of the checkerboard is at (16 cm, 16 cm).
To find the center of mass of the checkerboard-checkers system, we need to consider both the mass and the position of each component.
First, let's calculate the total mass of the system. We are given that the checkerboard has a mass of 120 g. There are also four checkers, each with a mass of 19 g. So the total mass of the system is:
120 g (checkerboard mass) + 4 * 19 g (checker masses) = 196 g
Next, we need to determine the position of the center of mass. We'll assign coordinates to the center of the bottom left square on the checkerboard as (0,0) and use a Cartesian coordinate system.
We can treat the checkerboard as a rectangular plate with uniform density, so its center of mass is at its geometrical center. The dimensions of the checkerboard are 32 cm by 32 cm, so its center of mass is located at (16 cm, 16 cm).
Now, let's consider the checkers. Since all four checkers have the same mass, their center of mass will lie at the average position of their individual centers of mass.
Assuming the checkers are placed symmetrically on the checkerboard, we can estimate the center of mass of the checkers to be at the midpoint of the line segment connecting the centers of two diagonally opposite checkers.
Using the dimensions of the checkerboard, each square is 16 cm by 16 cm. Since the checkers are positioned on the 16-cm mark from the origin in both the x and y directions, we can estimate their center of mass to be located at (16 cm, 16 cm).
To calculate the overall center of mass of the system, we find the weighted average of the individual centers of mass, taking into account the mass of each component. Since the mass of the checkerboard is greater than that of the checkers, the center of mass of the system will be closer to the center of the checkerboard.
In summary, the estimated center of mass of the checkerboard-checkers system, relative to the origin at the bottom left corner of the checkerboard, is approximately (16 cm, 16 cm).