Answer
The center of mass will move horizontally a distance of 0.700 meters.
The center of mass will move vertically a distance of 0.700 meters.
Work Step by Step
Let the position of the hinge be the origin.
$x_{cm} = \frac{m_1x_1+m_2x_2+m_3x_3}{m_1+m_2+m_3}$
$x_{cm} = \frac{(4.00~kg)(-0.75~m)+0+0}{4.00~kg+3.00~kg+2.00~kg}$
$x_{cm} = -0.333~m$
$y_{cm} = \frac{m_1y_1+m_2y_2+m_3y_3}{m_1+m_2+m_3}$
$y_{cm} = \frac{(4.00~kg)(0~m)+(3.00~kg)(-0.90~m)+(2.00~kg)(-1.80~m)}{4.00~kg+3.00~kg+2.00~kg}$
$y_{cm} = -0.700~m$
The coordinates of the center of mass are:
(-0.333 m, -0.700 m)
We can find the center of mass after the vertical bar has been pivoted.
$x_{cm} = \frac{m_1y_1+m_2y_2+m_3y_3}{m_1+m_2+m_3}$
$x_{cm} = \frac{(4.00~kg)(-0.75~m)+(3.00~kg)(0.90~m)+(2.00~kg)(1.80~m)}{4.00~kg+3.00~kg+2.00~kg}$
$x_{cm} = 0.367~m$
$y_{cm} = 0$
The center of mass will move horizontally a distance of (0.367 m) - (-0.333 m), which is 0.700 meters.
The center of mass will move vertically a distance of (0 m) - (-0.700 m), which is 0.700 meters.
