# An object with a mass of 7 kg is on a ramp at an incline of pi/12 . If the object is being pushed up the ramp with a force of  6 N, what is the minimum coefficient of static friction needed for the object to remain put?

Feb 2, 2016

$\mu = \left(\frac{6}{m g \cos \left(\frac{\pi}{12}\right)} - \tan \left(\frac{\pi}{12}\right)\right)$

#### Explanation:

From FBD:
${F}_{i} = {F}_{f} + {F}_{| \setminus |} = \mu {F}_{N} + m g \sin \alpha$
${F}_{i} = \mu m g \cos \alpha + m g \sin \alpha$
$6 = \mu m g \cos \left(\frac{\pi}{12}\right) + m g \sin \left(\frac{\pi}{12}\right)$
Solve for $\mu$
$\mu = \frac{6 - m g \sin \left(\frac{\pi}{12}\right)}{m g \cos \left(\frac{\pi}{12}\right)}$
$\mu = \left(\frac{6}{m g \cos \left(\frac{\pi}{12}\right)} - \tan \left(\frac{\pi}{12}\right)\right)$
Now pull out your calculator and voila!