An object with a mass of 6 kg is on a ramp at an incline of pi/12 . If the object is being pushed up the ramp with a force of  3 N, what is the minimum coefficient of static friction needed for the object to remain put?

Jun 1, 2017

The coefficient of static friction is $= - 0.55$

Explanation:

Taking the direction up and parallel to the plane as positive ↗^+

The coefficient of static friction is ${\mu}_{s} = {F}_{r} / N$

Then the net force on the object is

$F = {F}_{r} + W \sin \theta$

$= {F}_{r} + m g \sin \theta$

$= {\mu}_{s} N + m g \sin \theta$

$= m {\mu}_{s} g \cos \theta + m g \sin \theta$

$= m g \left({\mu}_{s} \cos \theta + \sin \theta\right)$

So,

$\frac{F}{m g} = \left({\mu}_{s} \cos \theta + \sin \theta\right)$

${\mu}_{s} \cos \theta = \frac{F}{m g} - \sin \theta$

${\mu}_{s} = \frac{1}{\cos} \theta \cdot \left(\frac{F}{m g} - \sin \theta\right)$

$= \frac{1}{\cos} \left(\frac{\pi}{12}\right) \cdot \left(\frac{3}{6 \cdot 9.8} - \sin \left(\frac{\pi}{12}\right)\right)$

$= - 0.55$