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 #4 N# to keep pushing it, what are the coefficients of static and kinetic friction?

1 Answer
Feb 21, 2017

Answer:

The static coefficient is 1.17; the kinetic coefficient is 1.06.

Explanation:

The object on an incline has one component of the force of gravity pulling it down the slope, usually referred to as #F_(||)#. As the diagram below shows, it is equal to #mgsintheta#

![http://thecraftycanvas.com/library/online-learning-tools/physics-homework-helpers/incline-force-calculator-problem-solver/]!

In this problem, the applied force also acts down the incline, and only friction acts up the ramp to oppose these two forces.

The force of friction is given by #F_f = muF_N = mumgcostheta#

Therefore the free body equation is

#F_"net" = mumgcostheta - mgsintheta - F_"applied"#

Since there is no acceleration, #F_"net"=0# and the equation becomes

#mumgcostheta - mgsintheta - F_"applied"=0#

Inserting what we know, I will find #mu_s# first:

#mu_s(10)(9.8)cos( (pi)/4) - (10)(9.8)sin ((pi)/4) - 12 = 0#

Since #cos(pi/4) = sin(pi/4) = .707#

#mu_s(69.3)-(69.3)-12=0#

#mu_s=81.3-:69.3 = 1.17#

The calculation of #mu_k# is similar, using 4N in place of the 12 N applied force:

#mu_k(10)(9.8)cos( (pi)/4) - (10)(9.8)sin ((pi)/4) - 4 = 0#

#mu_k(69.3)-(69.3)-4=0#

#mu_k=73.3-:69.3 = 1.06#