Center of Mass

Center of mass is the point where the whole mass of the system can be supposed to be concentrated and the motion of the system can be defined in terms of the center of mass.

Position of center of mass for the two-particle system:

Consider a system of two point masses $m_{1}$and $m_{2}$,whose position vectors at time $t$ with respect to the origin O are ${r_{1}}$ and ${r_{2}}$ respectively as shown below.

Figure:15.a

The total force ${F_{1}} $ that is acting on point mass $m_{1}$ consists of two parts.

<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><msub><mover><mi>F</mi><mo>&#x21C0;</mo></mover><mn>1</mn></msub><mfenced><mrow><mi>t</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>l</mi></mrow></mfenced></msub><mo>=</mo><msub><msub><mover><mi>F</mi><mo>&#x21C0;</mo></mover><mn>1</mn></msub><mfenced><mrow><mi>e</mi><mi>x</mi><mi>t</mi></mrow></mfenced></msub><mo>+</mo><msub><mover><mi>F</mi><mo>&#x21C0;</mo></mover><mn>12</mn></msub></math>

Where ${F}_{1(ext)}$= external force acting on the system

${F}_{12}$= Force due to $m_{2}$.

Similarly for point mass $m_{2}$,

<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><msub><mover><mi>F</mi><mo>&#x21C0;</mo></mover><mn>2</mn></msub><mfenced><mrow><mi>t</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>l</mi></mrow></mfenced></msub><mo>=</mo><msub><msub><mover><mi>F</mi><mo>&#x21C0;</mo></mover><mn>2</mn></msub><mfenced><mrow><mi>e</mi><mi>x</mi><mi>t</mi></mrow></mfenced></msub><mo>+</mo><msub><mover><mi>F</mi><mo>&#x21C0;</mo></mover><mn>21</mn></msub></math>

The equation of motion for point mass can be obtained using the second law of motion,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mfenced><mrow><msub><mi>m</mi><mn>1</mn></msub><msub><mover><mi>v</mi><mo>&#x2192;</mo></mover><mn>1</mn></msub></mrow></mfenced></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mover><mi>F</mi><mo>&#x2192;</mo></mover><mrow><mn>1</mn><mo>&#xA0;</mo><mfenced><mrow><mi>t</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>l</mi></mrow></mfenced></mrow></msub></math> …………….. (1)

And for point mass $m_{2}$,

 ……………….. (2)

Adding equations (1) and (2), we get,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mfenced><mrow><msub><mi>m</mi><mn>1</mn></msub><msub><mover><mi>v</mi><mo>&#x2192;</mo></mover><mn>1</mn></msub></mrow></mfenced></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mo>d</mo><mfenced><mrow><msub><mi>m</mi><mn>2</mn></msub><msub><mover><mi>v</mi><mo>&#x2192;</mo></mover><mn>2</mn></msub></mrow></mfenced></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mover><mi>F</mi><mo>&#x2192;</mo></mover><mrow><mn>1</mn><mo>&#xA0;</mo><mfenced><mrow><mi>t</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>l</mi></mrow></mfenced></mrow></msub><mo>+</mo><msub><mover><mi>F</mi><mo>&#x2192;</mo></mover><mrow><mn>2</mn><mo>&#xA0;</mo><mfenced><mrow><mi>t</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>l</mi></mrow></mfenced></mrow></msub><mspace linebreak="newline"/><mi>o</mi><mi>r</mi><mo>,</mo><mo>&#xA0;</mo><mfrac><mrow><mo>d</mo><mfenced><mrow><msub><mi>m</mi><mn>1</mn></msub><msub><mover><mi>v</mi><mo>&#x2192;</mo></mover><mn>1</mn></msub><mo>+</mo><msub><mi>m</mi><mn>2</mn></msub><msub><mover><mi>v</mi><mo>&#x2192;</mo></mover><mn>2</mn></msub></mrow></mfenced></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mover><mi>F</mi><mo>&#x2192;</mo></mover><mrow><mn>1</mn><mo>&#xA0;</mo><mfenced><mrow><mi>t</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>l</mi></mrow></mfenced></mrow></msub><mo>+</mo><msub><mover><mi>F</mi><mo>&#x2192;</mo></mover><mrow><mn>2</mn><mo>&#xA0;</mo><mfenced><mrow><mi>t</mi><mi>o</mi><mi>t</mi><mi>a</mi><mi>l</mi></mrow></mfenced></mrow></msub><mspace linebreak="newline"/><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><msub><mover><mi>F</mi><mo>&#x2192;</mo></mover><mrow><mn>1</mn><mo>&#xA0;</mo><mfenced><mrow><mi>e</mi><mi>x</mi><mi>t</mi></mrow></mfenced></mrow></msub><mo>+</mo><msub><mover><mi>F</mi><mo>&#x2192;</mo></mover><mn>12</mn></msub><mo>+</mo><msub><mover><mi>F</mi><mo>&#x2192;</mo></mover><mrow><mn>2</mn><mo>&#xA0;</mo><mfenced><mrow><mi>e</mi><mi>x</mi><mi>t</mi></mrow></mfenced></mrow></msub><mo>+</mo><msub><mover><mi>F</mi><mo>&#x2192;</mo></mover><mn>21</mn></msub></math> ……………….. (3)

Also, we have from Newton’s third law of motion,

<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi>F</mi><mo>&#x21C0;</mo></mover><mn>21</mn></msub><mo>=</mo><mo>-</mo><msub><mover><mi>F</mi><mo>&#x21C0;</mo></mover><mn>12</mn></msub></math>

Then from equation (3), we have,

 …………………… (4)

Now,

<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi>v</mi><mo>&#x2192;</mo></mover><mn>1</mn></msub><mo>=</mo><mfrac><mrow><mi>d</mi><msub><mover><mi>r</mi><mo>&#x2192;</mo></mover><mn>1</mn></msub></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>&#xA0;</mo><mi>a</mi><mi>n</mi><mi>d</mi><mo>&#xA0;</mo><msub><mover><mi>v</mi><mo>&#x2192;</mo></mover><mn>2</mn></msub><mo>=</mo><mfrac><mrow><mi>d</mi><msub><mover><mi>r</mi><mo>&#x2192;</mo></mover><mn>2</mn></msub></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></math>

So, from equation (4), we get,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mspace linebreak="newline"/><mo>&#xA0;</mo><mspace linebreak="newline"/><mfrac><mrow><mo>&#xA0;</mo><msup><mi>d</mi><mn>2</mn></msup></mrow><mrow><mi>d</mi><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mfenced><mrow><msub><mi>m</mi><mn>1</mn></msub><msub><mover><mi>r</mi><mo>&#x2192;</mo></mover><mn>1</mn></msub><mo>+</mo><msub><mi>m</mi><mn>2</mn></msub><msub><mover><mi>r</mi><mo>&#x2192;</mo></mover><mn>2</mn></msub></mrow></mfenced><mo>=</mo><mover><mi>F</mi><mo>&#x2192;</mo></mover><mspace linebreak="newline"/><mi>o</mi><mi>r</mi><mo>,</mo><mo>&#xA0;</mo><mover><mi>F</mi><mo>&#x2192;</mo></mover><mo>=</mo><mi>M</mi><mfrac><mrow><mo>&#xA0;</mo><msup><mi>d</mi><mn>2</mn></msup></mrow><mrow><mi>d</mi><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mfrac><mfenced><mrow><msub><mi>m</mi><mn>1</mn></msub><msub><mover><mi>r</mi><mo>&#x2192;</mo></mover><mn>1</mn></msub><mo>+</mo><msub><mi>m</mi><mn>2</mn></msub><msub><mover><mi>r</mi><mo>&#x2192;</mo></mover><mn>2</mn></msub></mrow></mfenced><mi>M</mi></mfrac></math>

This equation is clearly the equation of motion of a hypothetical object of mass $M=m_{1}+m_{2}$.

The position of this point at any time is given by the position vector ${R}_{CM}$such that

<math xmlns="http://www.w3.org/1998/Math/MathML"><mspace linebreak="newline"/><mo>&#xA0;</mo><msub><mover><mi>R</mi><mo>&#x2192;</mo></mover><mrow><mi>C</mi><mi>M</mi></mrow></msub><mo>=</mo><mfrac><mfenced><mrow><msub><mi>m</mi><mn>1</mn></msub><msub><mover><mi>r</mi><mo>&#x2192;</mo></mover><mn>1</mn></msub><mo>+</mo><msub><mi>m</mi><mn>2</mn></msub><msub><mover><mi>r</mi><mo>&#x2192;</mo></mover><mn>2</mn></msub></mrow></mfenced><mi>M</mi></mfrac></math>

${R}_{CM}$is the center of mass of the two-particle system.

If $m_{1}=m_{2}$then,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mspace linebreak="newline"/><mo>&#xA0;</mo><msub><mover><mi>R</mi><mo>&#x2192;</mo></mover><mrow><mi>C</mi><mi>M</mi></mrow></msub><mo>=</mo><mfrac><mfenced><mrow><msub><mover><mi>r</mi><mo>&#x2192;</mo></mover><mn>1</mn></msub><mo>+</mo><msub><mover><mi>r</mi><mo>&#x2192;</mo></mover><mn>2</mn></msub></mrow></mfenced><mn>2</mn></mfrac></math>

Thus, the center of mass of two equal masses lies exactly at the center of the line joining the two masses.

If $(x_{1, }y_{1})$ and $(x_{2, }y_{2})$ are the coordinates of the locations of two particles , then the coordinates of their center of mass is given by,

<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mrow><mi>C</mi><mi>M</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>m</mi><mn>1</mn></msub><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>m</mi><mn>2</mn></msub><msub><mi>x</mi><mn>2</mn></msub></mrow><mrow><msub><mi>m</mi><mn>1</mn></msub><mo>+</mo><msub><mi>m</mi><mn>2</mn></msub></mrow></mfrac><mo>&#xA0;</mo><mi>a</mi><mi>n</mi><mi>d</mi><mo>&#xA0;</mo><msub><mi>y</mi><mrow><mi>C</mi><mi>M</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>m</mi><mn>1</mn></msub><msub><mi>y</mi><mn>1</mn></msub><mo>+</mo><msub><mi>m</mi><mn>2</mn></msub><msub><mi>y</mi><mn>2</mn></msub></mrow><mrow><msub><mi>m</mi><mn>1</mn></msub><mo>+</mo><msub><mi>m</mi><mn>2</mn></msub></mrow></mfrac></math>

The center of mass for the many-particle system is given by,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mspace linebreak="newline"/><mo>&#xA0;</mo><msub><mi>R</mi><mrow><mi>C</mi><mi>M</mi></mrow></msub><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><munderover><mo>&#x2211;</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover></mstyle><msub><mi>m</mi><mi>i</mi></msub><msub><mi>r</mi><mi>i</mi></msub></mrow><mrow><mstyle displaystyle="true"><munderover><mo>&#x2211;</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover></mstyle><msub><mi>m</mi><mi>i</mi></msub></mrow></mfrac></math>

Points to remember:

  1. The position of the center of mass is independent of the coordinate system.
  2. The position of the center of mass depends upon the shape of the body and the distribution of mass across the body.
  3. Symmetrical bodies in which the distribution of mass is homogeneous, the center of mass coincides with the geometrical center.

 

Please Share