View source for Forest UCM Ch3 CoM

The Center of mass

=Definition of the Center of Mass=

The position <math>\vec R</math> of the center of mass is given by

:<math>\vec{R} = \frac{\sum_i^N m_i \vec{r}_i}{\sum_i^N m_i}</math>


The center of mass is given as the sum of the position of each object in the system weighted by the objects mass.


For a rigid object the location of the center of mass is given by


:<math>\vec{R} = \frac{1}{M} \int \vec{r} dm</math>

=Example 1: CM of three particles=

Calculate the location of the center of mass given the three particles below

:<math>\vec{r}_1 = (1,1,0) = \hat i + \hat j</math>
:<math>\vec{r}_2 = (1,-1,0) = \hat i - \hat j</math>
:<math>\vec{r}_3 = (0,0,0) = \vec{0}</math>
when 
:<math>m_1 = m_2 = 3 m_3</math>


let
<math> m_3 = M</math>

:<math>\vec{R} = \frac{3M (1) + 3 M (1) + M(0)}{9M} \hat i  + \frac{3M (1) + 3 M (-1) + M(0)}{9M} \hat j</math>
:<math>\vec{R} = \frac{6}{9} \hat i  + \frac{0}{9} \hat j</math>

=Example 2: CM of a flat disk=

=Example 3: CM of a semicircle=


A semicircle of radius R lies in the xy plane with its center at the origin and a diameter lying along the x axis.  Use polar coordinates to locte the position of the center of mass of the semicircle.


Assume

:<math>M =</math> mass of the semicircle
:<math>\sigma =</math> the mass density

:<math>\vec{R} = \frac{1}{M} \int \sigma \vec{r} dA</math>

=Problem 3-19=

[[Forest_UCM_MnAM#Center_of_Mass]]