A thin rod of length L is lying along the x-axis with its ends at x = 0 and x = L. Its linear density (mass/length) varies with x as `k((x)/(L))^n` where n can be zero or any positive number. If the position `X_(CM)` of the centre of mass of the rod is plotted against n, which of the following graphs best approximates the dependence of `X_(CM)` on n?

Options

A.
B.
C.
D.

Answer

A

Explanation

(a) When n= 0,x =k where k is a constant. This means
thet the linear mass density is constant. In this cases
the centre of mass will be at the midelle of the rod ie at
`L/2`. Therefore (c ) is ruled out
n is positive and as its value increases, the rate of
increase of linear mass density with increase in
x increase. This shows that the centre of mass will
shift towards that end of the rod where n=L as the
value of n increases. Therefore graph (b) is ruled out.
The linear mass density `lambda = k((x)/(L))^n Here (x)/(L)le1`
With increase in the value of n, the centre of mass shift
towads the end x=L such thet first the shifting is at a
higher rate with increase in the value of n and then the
rate decreases with the value of n.
These characteristics are represented by graph (a).
`x_(CM) = (int_0^Lxdm)/(int_0^ldm) = (int_0^Lx(lambdadx))/(int_0^Llambdadx) = (int_0^Lk((x)/(L))^nxdx)/(int_0^Lk((x)/(L))^ndx)`
`=(k[x^(n+2)/((n+2)L^n)]_0^L)/([(kx^(n+1))/((n+1)L^n)]_0^L)=(L(n+1))/(n+2)`
`For n = 0, x_(CM) = (L)/(2),n = 1`
`x_(CM) = (2L)/(3), n=2, x_(CM) = (3L)/(4),.....`