Question

What minimum force is required to move it up the incline with constant speed?

• A. $\frac{2}{3}mg$
• B. $\frac{mg}{2}$
• C. $\frac{7mg}{6}$
• D. $\frac{5mg}{6}$

Answer 1

The correct option is D $\frac{5mg}{6}$

There will be three equations.

1) Initially at rest: ${\mu }_{s}mgcos30\ge mgsin30\Rightarrow 2{\mu }_k\ge tan30\Rightarrow {\mu }_k\ge \frac{1}{2\sqrt{3}}$

2) Start moving when F applied: $F+mgsin30={\mu }_{s}mgcos30\Rightarrow F=mg(\sqrt{3}{\mu }_k+0.5)$

3) Block is sliding with constant speed: $F=mgsin30-{\mu }_{k}mgcos30\Rightarrow F=\frac{mg}{2}(1-\sqrt{3}{\mu }_k)$

From (2) & (3): $\frac{mg}{2}(1-\sqrt{3}{\mu }_k)=mg(\sqrt{3}{\mu }_k+0.5)\Rightarrow 2{=3\sqrt{3}\mu }_k\Rightarrow {\mu }_k=\frac{2}{3\sqrt{3}}$

Therefore the minimum value of force required to move it up the incline with constant speed:

$F=\frac{mg}{2}(1+\sqrt{3}\frac{2}{3\sqrt{3}})=\frac{5mg}{6}$.