An object with a mass of 4 kg is on a plane with an incline of  - pi/4 . If it takes 7 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?

$\setminus {\mu}_{s} = 1.252$ & $\setminus {\mu}_{k} = 1.18$

Explanation:

The force $F$ required to push an object of mass $m = 4$ kg downward on an inclined plane at an angle $\theta = - \frac{\setminus \pi}{4} = - {45}^{\setminus} \circ$ & coefficient of static friction $\setminus {\mu}_{s}$

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

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

$\setminus {\mu}_{s} = \setminus \frac{7 + 4 \setminus \times 9.81 \setminus \sin {45}^{\setminus} \circ}{4 \setminus \times 9.81 \setminus \cos {45}^{\setminus} \circ}$
$= 1.252$

Similarly, when the motion starts, then the force (F) required to keep the object moving

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

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

$\setminus {\mu}_{k} = \setminus \frac{7 + 4 \setminus \times 9.81 \setminus \sin {45}^{\setminus} \circ}{4 \setminus \times 9.81 \setminus \cos {45}^{\setminus} \circ}$
$= 1.18$