Question

The linear mass density of a ladder of length $L$ increases uniformly from one end $A$ to other end $B$, Form an expression for linear mass density as a function of distance $x$ from end $A$ where linear mas density ${\lambda }_0$. The density at one end being twice that of the other end.

• A. $\lambda \left(x\right)={\lambda }_0+\frac{\lambda x}{2L}$
• B. $\lambda \left(x\right)={\lambda }_0+\frac{\lambda x}{L}$
• C. $\lambda \left(x\right)=2{\lambda }_0+\frac{\lambda x}{L}$
• D. $\lambda \left(x\right)={\lambda }_0+\frac{2\lambda x}{L}$

Answer 1

The correct option is B $\lambda \left(x\right)={\lambda }_0+\frac{\lambda x}{L}$

Density at end " A" $={\lambda }_0$
Density at end " B " $($Twice that of end A$)=2{\lambda }_0$
Length of the rod $AB=L$
Change in density over the length $L(2{\lambda }_0-{\lambda }_0)={\lambda }_0$

Change in density per unit length =$\frac{{\lambda }_0}{L}$

Given that linear density increases from end "A" to "B"
Hence density at any point at a distance "x" from end "A" can be given as:
Density at end "A" + (change in linear density)x

$\lambda \left(x\right)={\lambda }_0+\frac{{\lambda }_{0}x}{L}$