Answer :
The symbolic equation for the location of the center of mass of 3 beads is [tex]x_{cm}[/tex] = m1d1 + m2 (d1 + d2) + m3 ( d1+d2+d3)/ m1 + m2 + m3
Mass of 1st bead = m1 = 22 g
Mass of the 2nd bead = m2 = 12 g
Mass of the 3rd bead = m3 = 51 g
Distance of the 1st bead from left end of the rod = d1 = 1.5 cm
Distance of the 2nd bead from the first bead = d2 = 1.9 cm
Distance of the 3rd bead from the second bead = d3 = 4.5 cm
Location of the center of mass of the three beads relative to the left end of the rod = xcm
We want to find the center of mass relative to the left end of the rod therefore we will need the distances of each bead from the left end of the rod.
Distance of the first bead from the left end of the rod = x1
x1 = d1
Distance of the second bead from the left end of the rod = x2
x2 = d1 + d2
Distance of the third bead from the left end of the rod = x3
x3 = d1 + d2 + d3
[tex]x_{cm}[/tex] = m1 x1 + m2 x2 + m3 x3 / m1 + m2 + m3
[tex]x_{cm}[/tex]= m1(d1) + m2(dı + d2) + m3(d1 + d2 +d3)/ m1 + m2 + m3
[tex]x_{cm}[/tex] = m1d1 + m2 (d1 + d2) + m3 ( d1+d2+d3)/ m1 + m2 + m3
Substituting the values to get a numerical answer,
[tex]x_{cm}[/tex] = (22)(1.5) + (12)(1.5 +1.9) + (51)(1.5 +1.9 +4.5) 22 + 12 + 51
xcm = 5.61 cm
Location of the center of mass of the three beads relative to the center bead = x
Distance of the first bead from the center bead (second bead) = x4
x4 = -d2 (Negative as it is to the left of the second bead)
Distance of the second bead from the center bead (second bead) = x5 = 0 cm
Distance of the third bead from the center bead (second bead) = x6
x6 = d3
x = m1 x4 + m2 x5 + m3 x6/ m1 + m2 + m3
x = m1 ( - d2) + m2 (0) + m3 (d3) / m1 + m2 + m3
x = - m1 d2 + 0 + m3d3 / m1 + m2 + m3
x = m3d3 - m1 d2 / m1 + m2 + m3
Substituting the values to get a numerical answer,
x= (51) (4.5) - (22)(1.9)/22 + 12 + 51
x = 2.21 cm
To learn more about the center of mass
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