Problem: Three beads are placed along a thin rod. The first bead, of mass m1 = 24 g, is placed a distance d1 = 1.1 cm from the left end of the rod. The second bead, of mass m2 = 12 g, is placed a distance d2 = 1.9 cm to the right of the first bead. The third bead, of mass m3 = 56 g, is placed a distance d3 = 3.9 cm to the right of the second bead. Assume an x-axis that points to the right.A. Find the center of mass, in centimeters, relative to the left end of the rod.B. Write a symbolic equation for the location of the center of mass of the three beads relative to the center bead, in terms of the variables given in the problem statement.C. Find the center of mass, in centimeters, relative to the middle bead.

FREE Expert Solution

X-coordinate of the center of mass:

$\overline{){{\mathbf{x}}}_{\mathbf{c}\mathbf{m}}{\mathbf{=}}\frac{{\mathbf{m}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{m}}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{\mathbf{m}}_{\mathbf{n}}{\mathbf{x}}_{\mathbf{n}}}{{\mathbf{m}}_{\mathbf{1}}\mathbf{+}{\mathbf{m}}_{\mathbf{2}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{\mathbf{m}}_{\mathbf{n}}}}$

A.

The center of mass of the system relative to the left en of the rod is:

$\begin{array}{rcl}{\mathbf{x}}_{\mathbf{c}\mathbf{m}}& \mathbf{=}& \frac{{\mathbf{m}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{m}}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{m}}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{3}}}{{\mathbf{m}}_{\mathbf{1}}\mathbf{+}{\mathbf{m}}_{\mathbf{2}}\mathbf{+}{\mathbf{m}}_{\mathbf{3}}}\\ & \mathbf{=}& \frac{{\mathbf{m}}_{\mathbf{1}}{\mathbf{d}}_{\mathbf{1}}\mathbf{+}{\mathbf{m}}_{\mathbf{2}}\mathbf{\left(}{\mathbf{d}}_{\mathbf{1}}\mathbf{+}{\mathbf{d}}_{\mathbf{2}}\mathbf{\right)}\mathbf{+}{\mathbf{m}}_{\mathbf{3}}\mathbf{\left(}{\mathbf{d}}_{\mathbf{1}}\mathbf{+}{\mathbf{d}}_{\mathbf{2}}\mathbf{+}{\mathbf{d}}_{\mathbf{3}}\mathbf{\right)}}{{\mathbf{m}}_{\mathbf{1}}\mathbf{+}{\mathbf{m}}_{\mathbf{2}}\mathbf{+}{\mathbf{m}}_{\mathbf{3}}}\\ & \mathbf{=}& \frac{\mathbf{\left(}\mathbf{24}\mathbf{\right)}\mathbf{\left(}\mathbf{1}\mathbf{.}\mathbf{1}\mathbf{\right)}\mathbf{+}\mathbf{\left(}\mathbf{12}\mathbf{\right)}\mathbf{\left(}\mathbf{1}\mathbf{.}\mathbf{1}\mathbf{+}\mathbf{1}\mathbf{.}\mathbf{9}\mathbf{\right)}\mathbf{+}\mathbf{\left(}\mathbf{56}\mathbf{\right)}\mathbf{\left(}\mathbf{1}\mathbf{.}\mathbf{1}\mathbf{+}\mathbf{1}\mathbf{.}\mathbf{9}\mathbf{+}\mathbf{3}\mathbf{.}\mathbf{9}\mathbf{\right)}}{\mathbf{24}\mathbf{+}\mathbf{12}\mathbf{+}\mathbf{56}}\end{array}$

Problem Details

Three beads are placed along a thin rod. The first bead, of mass m1 = 24 g, is placed a distance d1 = 1.1 cm from the left end of the rod. The second bead, of mass m2 = 12 g, is placed a distance d2 = 1.9 cm to the right of the first bead. The third bead, of mass m3 = 56 g, is placed a distance d3 = 3.9 cm to the right of the second bead. Assume an x-axis that points to the right.

A. Find the center of mass, in centimeters, relative to the left end of the rod.

B. Write a symbolic equation for the location of the center of mass of the three beads relative to the center bead, in terms of the variables given in the problem statement.

C. Find the center of mass, in centimeters, relative to the middle bead.