Three masses are connected by massless rigid rods. Rods AC and CB have length l= 44.8cm. If massA=280.0g, massB=386.0g, and massC=739.0g, what are the coordinates of the center of mass? (Enter your answer for the x coordinate first, followed by the y coordinate.)
To find the coordinates of the center of mass, we need to first calculate the position of each mass relative to a coordinate system, and then use the formula for center of mass.
Let's assume that the origin of our coordinate system is at point C, where mass C is located. For simplicity, we can take the x-axis parallel to rod AC and the y-axis parallel to rod CB.
Now, the x-coordinate of the center of mass (Xcm) can be calculated using the formula:
Xcm = (m1x1 + m2x2 + m3x3) / (m1 + m2 + m3)
where m1, m2, and m3 are the masses of A, B, and C respectively, and x1, x2, and x3 are their respective x-coordinates.
Similarly, the y-coordinate of the center of mass (Ycm) can be calculated using the formula:
Ycm = (m1y1 + m2y2 + m3y3) / (m1 + m2 + m3)
where y1, y2, and y3 are the respective y-coordinates of masses A, B, and C.
To determine the x and y coordinates for each mass:
- The x-coordinate for mass A is -l/2 since point A is located l/2 units to the left of point C.
- The y-coordinate for mass A is 0 since it lies on the x-axis.
- The x-coordinate for mass B is l/2 since point B is located l/2 units to the right of point C.
- The y-coordinate for mass B is 0 since it lies on the x-axis.
- Since point C is at the origin, the coordinates for mass C are both 0.
Now we can substitute the given values into the center of mass formulas:
Xcm = (mAx + mBx + mCx) / (mA + mB + mC)
= [(280.0g)(-l/2) + (386.0g)(l/2) + (739.0g)(0)] / (280.0g + 386.0g + 739.0g)
Ycm = (mAy + mBy + mCy) / (mA + mB + mC)
= [(280.0g)(0) + (386.0g)(0) + (739.0g)(0)] / (280.0g + 386.0g + 739.0g)
Now, let's substitute the values and calculate:
Xcm = [(280.0g)(-l/2) + (386.0g)(l/2)] / (280.0g + 386.0g + 739.0g)
Ycm = [(280.0g)(0) + (386.0g)(0) + (739.0g)(0)] / (280.0g + 386.0g + 739.0g)
Calculating the values:
Xcm = (140.0g - 193.0g) / 1405.0g
Ycm = 0 / 1405.0g
Simplifying:
Xcm = -53.0g / 1405.0g
Ycm = 0
Therefore, the coordinates of the center of mass are approximately (-0.038 g/cm, 0).