Question

The block of mass '$m$' initially at $x = 0$ is acted upon by a horizontal force $F = a - bx$ as shown in the figure. The coefficient of friction between the surfaces of contact is $\mu $ . The net work done on the block is zero if the block travels a distance of ________.

• A. $( a -\mu mg ) / 2 b$
• B. $( a -\mu mg ) / b$
• C. $2( a -\mu mg ) / b$
• D. $( a -2 \mu mg ) / b$

Answer 1

Answer: (C)

$f=a-b x $
$f_{n_{e f}}=a-b x-f_{s}$
For block to move
at $x=0$
$\mathrm{a}-\mathrm{b}_{\mathrm{x}}>\left(\mathrm{f}_{\mathrm{s}}\right)_{\max }=\mu \mathrm{mg}$
$a>\mu m g$,
So we are assuming motion starts at $x=0$
$w =\int( a - bx -\mu mg ) dx$
$o = ax -\frac{ bx ^{2}}{2}-\mu mgx$
$\Rightarrow x \left( a -\frac{ bx }{2}-\mu mg \right)=0$
$x=0$
$a-\frac{b x}{2}-\mu m g=0$
$\frac{ bx }{2}=( a -\mu mg )$
$x =\frac{2}{ b }( a -\mu mg )$