Show that the centre of mass of uniform rod of mass M and length L lies at the middle point of the rod.

Explanation

Suppose a rod of mass M and length L is lying along the x - axis with ts one end at x = 0 and the other end at x = L .
Then , mass per unit length of the rod = `M/L `

Here , the mass of the element pQ of length d situated at `x = x ` is `dm = M/L dx`
The cooredinates of the element PQ are (x,0,0) . Therefore , x - coordinate of CM of the rod will be
` x_(CM) = (overset (L) underset (0)intx dm )/(int dm ) = (overset(L) underset (0) int(x) (M/L dx))/M = 1/L overset (L) underset (0) int xdx = L/2 `
` y_(CM) = (intydm)/(intdm) = 0 and " Similarly " , z_(CM) = 0 `
i.e the coordinate of CM of teh rod are `(L/2 ,0,0)` Thus the CM lies at the centre of the rod .