The center of mass is located at position i EX.5.4. Locate the center of mass of a uniform rod of mass M and length

Explanation

Consider a uniform rod of mass M and length / whose one end coincides with the origin as shown in Figure. The rod is kept along the x axis. To find the center of mass of this rod, we choose an infinitesimally small mass dm of elemental length dr at a distance from the origin
`lambda` is the linear mass density (ie, mass per unit length) of the rod. `lambda=(M)/(l)`
The mass of small element (dm) is `dm, (M)/(l)dx`
Now, we can write the center of mass equation for this mass distribution as,
`X_(CM)=(int x dxm)/( int dm)`
`X_(CM)=( overset(1) underset(0) intx((M)/(l)dx))/(M)=(1)/(l)overset(1) underset(0) intxdx=(1)/(l)[(x^(2))/(2)]_(0)^(l)=(1)/(l)=((t^(2))/(2))`
`X_(CM)=(l)/(2)`
As the `(l)/(2)` position is the geometric center of the rod, it is concluded that the center of mass