# An object with a mass of 2 kg is on a plane with an incline of  - pi/3 . If it takes 12 N to start pushing the object down the plane and 5 N to keep pushing it, what are the coefficients of static and kinetic friction?

coefficients of static & kinetic friction are $2.955$ & $2.242$ respectively

#### Explanation:

Let ${\mu}_{s}$ & ${\mu}_{k}$ be the coefficients of static & kinetic friction on a plane inclined at an angle $\setminus \theta = \setminus \frac{\pi}{3}$

When the object of mass $m = 2 \setminus k g$ is just to start sliding down the incline under the application of a force $F = 12 \setminus N$, balancing the force along the incline

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

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

setting the corresponding value in above equation, we get

${\mu}_{s} = \setminus \frac{12 + 2 \setminus \cdot 9.81 \setminus \sin \left(\setminus \frac{\pi}{3}\right)}{2 \setminus \cdot 9.81 \setminus \cos \left(\setminus \frac{\pi}{3}\right)}$

$= 2.955$

Similarly, when the object of mass $m = 2 \setminus k g$ is sliding down the incline under the application of a force $F = 5 \setminus N$, balancing the force along the incline

$F + m g \setminus \sin \setminus \theta = {\mu}_{k} m g \setminus \cos \setminus \theta$

${\mu}_{k} = \setminus \frac{F + m g \setminus \sin \setminus \theta}{m g \setminus \cos \setminus \theta}$

setting the corresponding value in above equation, we get

${\mu}_{k} = \setminus \frac{5 + 2 \setminus \cdot 9.81 \setminus \sin \left(\setminus \frac{\pi}{3}\right)}{2 \setminus \cdot 9.81 \setminus \cos \left(\setminus \frac{\pi}{3}\right)}$

$= 2.242$

hence coefficients of static & kinetic friction are $2.955$ & $2.242$ respectively