Question

The minimum force required to start pushing a body up a rough (frictional coefficient $u$) inclined plane is $a$ while the minimum force needed to prevent it from sliding down is $b$. If the inclined plane makes an angle $\theta$ from the horizontal such that $\tan \theta =2u$, then the ratio $\frac{a}{b}$ is:

• A. $1$
• B. $2$
  
• C. $3$
  
• D. $4$
  

Answer 1

The correct option is C $3$

$tan\theta =2u⟹u=\frac{tan\theta }{2}$
To prevent it from sliding, we have to work against sliding due to gravity but frictional force will support us.
$b=mgsin\theta -umgcos\theta ⟹b=mgsin\theta -\frac{mgsin\theta }{2}⟹b=\frac{mgsin\theta }{2}$
To push it upwards, we have to work against both sliding due to gravity and frictional force.
$a=mgsin\theta +umgcos\theta ⟹a=\frac{3}{2}mgsin\theta ⟹\frac{a}{b}=3$