Question

A man is pushing a box of mass $m$ along the ground with a force $F$. The coefficient of kinetic friction between the box and the ground is ${\mu }_k$. The box is accelerating, but then the person stops pushing and the box slides to a halt. The person then starts pushing on the box again with a force $F$, but the box doesn't budge. The maximum coefficient of static friction between the box and the ground is ${\mu }_s$. Identify which of the following statements is true?

• A. $F>\frac{{\mu }_k}{{\mu }_s}$
• B. ${\mu }_k>{\mu }_s$
• C. ${\mu }_{k}mg<F\le {\mu }_{s}mg$
• D. ${\mu }_{s}mg=F$
• E. The scenario described is physically impossible

Answer 1

The correct option is C ${\mu }_{k}mg<F\le {\mu }_{s}mg$

When the box is moving ,force applied by man is greater than frictional force , that's why the box is capable to move .Now due to movement , kinetic frictional force is in action i.e.

Kinetic frictional force < Force applied

or ${\mu }_{k}R={\mu }_{k}mg<F$

Now the box is not moving it means static friction is in action and this static frictional force is greater than or atleast equal to force applied by man because static frictional force is a self adjusting force ,

${\mu }_{s}mg\ge F$

as a result ${\mu }_{k}mg<F\le {\mu }_{s}mg$