Question

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

Answer 1

Consider your situation,

where:

`theta = pi/4 = 45°`

Moreover, recall,

`SigmavecF = ma`

Let's determine the normal force,

`SigmaF_y = F_N - mgcostheta = 0`
`=> F_N = mgcostheta`

and recall that to overcome the static friction a force equal to that friction must be exerted on the object.

Hence,

`SigmaF_x = F_P + mgsintheta = mu_s * F_N`

`=> mu_s = (F_P +mgsintheta)/(mgcostheta) approx 1.25`

If we wish to keep pushing the object down the incline, we need to push it with a force greater than or equal to its kinetic friction. Assuming that we are pushing the object with the least amount of force before friction overcomes us, then,

`SigmaF_x = F_P + mgsintheta = mu_k * F_N`

`=> mu_k = (F_P + mgsintheta)/(mgcostheta) approx 1.22`

Note I assumed gravity aided us in each case due to the incline. The negative angle of inclination is kind of a curve ball that gets students who don't know the material that well, it kind of looks like this on a coordinate plane,

Thus, I took it to mean in that direction, rather than the one depicted above.