Question

show that centre of mass of a uniform rod of mass M and lenght L lies at the middle point of the road.

Answer 1

Let  l= length of rod

      m= mass of the rod.

Let consider one end of the rod to be origin i.e (0,0) and henceforth the other end will lie at ( l,0) when placed along x- axis.

Let at x distance from the origin dm mass is of rod element of length dx. (i.e very small mass of rod )

Mass per unit lenght = m/l

Then small mass dm (of lenght dx)= m.dx/l

Center of mass = ∫x.dm/m

                          =∫ x . m.dx/m.l

                           = ∫ x.dx/l

                           = [x^2/2] /l   ( limits from 0 to l )

                           = l^2/2l

                           = l/2

Hence Proved.

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Answer 2

Let us consider that the length of rod is l and mass is M.

And consider a small length dx

dm= M/l×dx

Xcm=1/M[dm

Put the value of dm

Xcm=1/M[M/l×dx

Now take limits 0 to l

=1/l[×dx

=1/l[x²/2

=1/l×l²/2

=l/2

Hence centre of mass of a uniform rod is l/2