Question

For the system of particles described in the Part B, find the x coordinate of the center of mass. Assume that the particle of mass is at the origin and the positive x axis is directed to the right.Express your answer in terms of L.M 2M 3M|------------|------------|L L

Answer 1

The
`x` coordinate of center of the mass from the
`{\rm{M}}\, is 1.33{\rm{ L}}`

The coordinate of center of the mass from the is .

The coordinate of center of the mass from the is .

Step-by-step explanation:

from fig attached, Distance between one particle to another particle is
`{\rm{L}} . {m_1},{m_2},{m_3}` are the mass of different particles. Calculate the coordinates of the center of the mass. Particle of mass is at the origin and the positive
`x` axis is directed to the right

The expression for xx coordinate of center of the mass from the mass is,

`{x_(cm)} = \frac{{{m_1}{x_1} + {m_2}{x_2} + {m_3}{x_3}}}{{{x_1} + {x_2} + {x_3}}}`

Substitute for , for , for , for , for and for in the above equation.

`\begin{array}{c}\\{x_(cm)} = \frac{{\left( {\rm{M}} \right){\rm{0}} + \left( {{\rm{2M}}} \right){\rm{L}} + \left( {{\rm{3M}}} \right)\left( {{\rm{2L}}} \right)}}{{{\rm{M}} + {\rm{2M}} + \left( {{\rm{3M}}} \right)}}\\\\ = \frac{{2{\rm{ML}} + 6{\rm{ML}}}}{{6{\rm{M}}}}\\\\ = \frac{{8{\rm{ML}}}}{{6{\rm{M}}}}\\\\ = 1.33{\rm{ L}}\\\end{array}`

The coordinate of center of the mass from the is .

The center of mass is the unique point which represents the mean position of the matter in a body or system.

The expression for coordinate of center of the mass from the mass is,

`{x_(cm)} = \frac{{{m_1}{x_1} + {m_2}{x_2} + {m_3}{x_3}}}{{{x_1} + {x_2} + {x_3}}}`

​Substitute for , for , for , for , for and for in the above equation.

`\begin{array}{c}\\{x_(cm)} = \frac{{\left( {\rm{M}} \right) - {\rm{L}} + \left( {{\rm{2M}}} \right){\rm{0}} + \left( {{\rm{3M}}} \right)\left( {\rm{L}} \right)}}{{{\rm{M}} + {\rm{2M}} + \left( {{\rm{3M}}} \right)}}\\\\ = \frac{{ - {\rm{ML}} + 3{\rm{ML}}}}{{6{\rm{M}}}}\\\\ = \frac{{{\rm{2ML}}}}{{6{\rm{M}}}}\\\\ = 0.33{\rm{ L}}\\\end{array}`

The coordinate of center of the mass from the is .

The position of the center of mass is at a distance of 0.33L from the mass 2M.

The expression for coordinate of center of the mass from the mass is,

`{x_(cm)} = \frac{{{m_1}{x_1} + {m_2}{x_2} + {m_3}{x_3}}}{{{x_1} + {x_2} + {x_3}}}`

​Substitute for , for , for , for , for and for in the above equation.

`\begin{array}{c}\\{x_(cm)} = \frac{{\left( {\rm{M}} \right) - 2{\rm{L}} + \left( {{\rm{2M}}} \right) - {\rm{L}} + \left( {{\rm{3M}}} \right){\rm{0}}}}{{{\rm{M}} + {\rm{2M}} + \left( {{\rm{3M}}} \right)}}\\\\ = \frac{{ - 2{\rm{ML}} - 2{\rm{ML}}}}{{6{\rm{M}}}}\\\\ = \frac{{ - {\rm{4ML}}}}{{6{\rm{M}}}}\\\\ = 0.63{\rm{ L}}\\\end{array}`

The coordinate of center of the mass from the is .

When mass of the particle
`{\rm{M}}` is the center of the origin that distance between