Question

The linear mass density of a ladder of length $l$ increases uniformly from one end $A$ to the other end $B$ ,
(i) Form an expression for linear mass density as a function of distance $x$ from end $A$ where linear mass density ${\lambda _0}$ . The density at one end being twice that of the other end.
(ii) Find the position of the center of mass from end $A$ .

Answer 1

Hint:Use the formula of the linear mass density and find it at the point $A$ itself by substituting the distance and the density to obtain the expression. Then substitute the obtained expression in the value of the distance of the center of mass to obtain the answer.

Formula used:
The formula of the linear mass density is given by
$\lambda = Ax + B$
Where $\lambda $ is the linear mass density and $A\,and\,B$ are the two ends of the ladder.

Complete step by step solution:
It is given that the
Length of the ladder is $l$
The linear mass density is ${\lambda _0}$
At the point $A$ , the distance between the points is zero and the linear mass density $\lambda = {\lambda _0}$. And at point $B$ , at a distance $x$ from the point $A$ , The density at one end is twice the other end, hence $\lambda = 2{\lambda _0}$ .
From these boundary points, we get $A = \dfrac{{{\lambda _0}}}{l}$ and the value of the $B = {\lambda _0}$ . Substituting these in the formula of the linear mass density,
$\lambda = Ax + B$
$\lambda = \dfrac{{{\lambda _0}}}{l}x + {\lambda _0}$
It is known that the
$X = \dfrac{{\int {_0^1xdm} }}{{\int {_0^1dm} }}$ and the $dm = \lambda dx$
Substitute the value of the $\lambda $ in the $dm$ formula,
Hence the $dm$ obtained is $\dfrac{{{\lambda _0}x}}{l} + {\lambda _0}$ .
Substituting the value of the $dm$ in the formula of $X$ .
$X = \dfrac{{\int {_0^1x\left( {\dfrac{{{\lambda _0}x}}{l} + {\lambda _0}} \right)} dx}}{{\int {_0^1\left( {\dfrac{{{\lambda _0}x}}{l} + {\lambda _0}} \right)} dx}}$
By simplifying the above step, we get
$X = \dfrac{{51}}{9}$
Hence the centre of mass is at a distance of $\dfrac{{51}}{9}\,cm$ from the point $A$ .

Note:The integration is done to find the distance of the center of the mass from the given point. Remember the integration formula of the distance and also the linear mass density. The linear mass density is the mass per unit length.