Three masses are connected by massless rigid rods. Rods AC and CB have length l= 16.6 cm. If massA=115.0 g, massB=206.0 g, and massC=310.0 g, what are the coordinates of the center of mass? (Enter your answer for the x coordinate first, followed by the y coordinate.)

draw the picure. Write coordinates for all the masses (use trig, if necessary).

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To find the coordinates of the center of mass, we need to calculate the weighted average of the positions of the masses.

First, let's assign coordinates to the masses. We can choose any convenient coordinate system, but for simplicity, let's place mass A at the origin (0, 0).

The x-coordinate of the center of mass (CMx) can be found using the formula:

CMx = (mAx * xA + mBx * xB + mCx * xC) / (mAx + mBx + mCx)

Similarly, the y-coordinate of the center of mass (CMy) can be found using the formula:

CMy = (mAy * yA + mBy * yB + mCy * yC) / (mAy + mBy + mCy)

Here, xA, xB, and xC are the x-coordinates of mass A, B, and C respectively, and similarly, yA, yB, and yC are the y-coordinates of mass A, B, and C respectively.

Since mass A is placed at (0, 0), we only need to find the x and y coordinates of masses B and C.

Given that rods AC and CB have a length of l = 16.6 cm, we can use right-angled triangles to find the x and y coordinates.

For mass B:
xB = l
yB = 0

For mass C:
xC = l * cos(60°) = l/2
yC = l * sin(60°) = l * (√3 / 2)

Now, let's substitute the values into the formulas for CMx and CMy:

CMx = (mAx * 0 + mBx * l + mCx * (l/2)) / (mAx + mBx + mCx)
CMy = (mAy * 0 + mBy * 0 + mCy * (l * (√3 / 2))) / (mAy + mBy + mCy)

Substituting the given masses:
CMx = (115.0 * 0 + 206.0 * l + 310.0 * (l/2)) / (115.0 + 206.0 + 310.0)
CMy = (115.0 * 0 + 206.0 * 0 + 310.0 * (l * (√3 / 2))) / (115.0 + 206.0 + 310.0)

Simplifying the equations further, we get:
CMx = (206.0l + 155.0l) / 631.0
CMy = (310.0l * (√3 / 2)) / 631.0

Calculating the values:
CMx = (206.0l + 155.0l) / 631.0 = 361.0l / 631.0
CMy = (310.0l * (√3 / 2)) / 631.0 = 310.0l√3 / (2 * 631.0)

Finally, substitute the value of l (16.6 cm) to get the numerical coordinates of the center of mass:

CMx = 361.0 * 16.6 / 631.0 ≈ 9.478 cm (rounded to 3 decimal places)
CMy = 310.0 * 16.6 * √3 / (2 * 631.0) ≈ 15.992 cm (rounded to 3 decimal places)

Therefore, the coordinates of the center of mass are approximately (9.478, 15.992) cm.