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?

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*g = 10 * 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 * F_n #

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

# implies 12 = mu_s * ( 50 sqrt(2)) #

# implies mu_s ~~ 0.169 #

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

Kinetic friction, #f_k = mu_k * F_n #

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

# implies 8 = mu_k * ( 50 sqrt(2)) #

# implies mu_k ~~ 0.113 #