# An object with a mass of 10 kg is on a plane with an incline of  - pi/4 . If it takes 12 N to start pushing the object down the plane and 8 N to keep pushing it, what are the coefficients of static and kinetic friction?

Oct 1, 2017

${\mu}_{k} \approx 0.113$
${\mu}_{s} \approx 0.169$

#### Explanation:

The minimum force to start pushing will help to overcome the static friction, so we can calculate the static friction from there. The Gravitational force acting on the block :
${F}_{g} = m \cdot g = 10 \cdot 10 = 100 N$

Taking components along the incline and perpendicular to the incline,  F_g cos θ = F_g cos 45° = 50sqrt(2)
and,  F_g sin θ = F_g sin 45° = 50sqrt(2)

Since there is no motion perpendicular to the incline, the net force in that direction will be zero.

 F_g cos θ = F_n = 50sqrt(2)  ( where ${F}_{n}$ is normal force. )

Static friction, ${f}_{s} = {\mu}_{s} \cdot {F}_{n}$

According to the question, the value of static friction force happens to be 12N.

$\implies 12 = {\mu}_{s} \cdot \left(50 \sqrt{2}\right)$

$\implies {\mu}_{s} \approx 0.169$

Now the kinetic friction will be applicable when the body starts to move.

Kinetic friction, ${f}_{k} = {\mu}_{k} \cdot {F}_{n}$

According to the question, the value of static friction force happens to be 8N.

$\implies 8 = {\mu}_{k} \cdot \left(50 \sqrt{2}\right)$

$\implies {\mu}_{k} \approx 0.113$