Find the center of mass of a system composed of three spherical objects with masses of 3.0kg , 1.6kg , and 4.3kg and centers located at (-6.7m , 0), (1.0m , 0), and (2.7m , 0), respectively.
Express your answers using two significant figures separated by a comma.
xCM, yCM
To find the center of mass of a system, we need to consider both the masses and the positions of the objects.
The x-coordinate of the center of mass (xCM) for a system is given by the formula:
xCM = (m1x1 + m2x2 + m3x3) / (m1 + m2 + m3)
where m1, m2, and m3 are the masses of the objects, and x1, x2, and x3 are the x-coordinates of their centers, respectively.
Let's calculate xCM:
xCM = (3.0kg * (-6.7m) + 1.6kg * (1.0m) + 4.3kg * (2.7m)) / (3.0kg + 1.6kg + 4.3kg)
xCM = (-20.1kg·m + 1.6kg·m + 11.6kg·m) / 8.9kg
xCM = -7.0m
Therefore, xCM = -7.0m.
The y-coordinate of the center of mass (yCM) for a system can be considered as 0 since all the objects are located along the x-axis.
Therefore, yCM = 0.
Expressing the answers using two significant figures separated by a comma, we have:
xCM = -7.0m, yCM = 0.
To find the center of mass (CM) of a system, we need to calculate the weighted average of the positions of the individual objects, where the weights are the masses of the objects. For a two-dimensional system like this, we can find the x-coordinate (xCM) and y-coordinate (yCM) separately.
Let's calculate the xCM first:
xCM = (m1x1 + m2x2 + m3x3) / (m1 + m2 + m3)
Where m1, m2, and m3 are the masses of the objects, and x1, x2, and x3 are the x-coordinates of their centers.
Plugging in the given values:
xCM = (3.0kg * (-6.7m) + 1.6kg * (1.0m) + 4.3kg * (2.7m)) / (3.0kg + 1.6kg + 4.3kg)
Simplifying the expression:
xCM = (-20.1kg*m + 1.6kg*m + 11.61kg*m) / 8.9kg
Calculating the numerator:
xCM = (-20.1kg*m + 1.6kg*m + 11.61kg*m) / 8.9kg
= (-7.89kg*m) / 8.9kg
= -0.8854m
Now, let's calculate the yCM:
yCM = (m1y1 + m2y2 + m3y3) / (m1 + m2 + m3)
Since all the objects have their centers located at y = 0, the yCM will be 0.
Therefore, the center of mass is located at (-0.89m, 0), where the xCM is -0.89m and the yCM is 0.