An object, previously at rest, slides #1 m# down a ramp, with an incline of #(3pi)/8 #, and then slides horizontally on the floor for another #2 m#. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?
WARNING: Long-ish answer
We're asked to find the coefficient of kinetic friction (
We'll split this problem into two parts: the first part is where the object is sliding down the incline, and the second part is where it is sliding across the floor.
The expression for the coefficient of kinetic friction
#f_k#is the magnitude of the retarding friction force acting as it slides down (denoted #f#in the above image)
#n#is the magnitude of the normal force exerted by the incline, equal to #mgcostheta#(denoted #N#in the above image)
The expression for the net horizontal force
And since the normal force
What we can now do is apply a kinematics equation for constant acceleration to find the final velocity
Here, the initial velocity
This velocity is also the initial velocity of the motion along the floor..
As the object slides across the floor, the plane is perfectly horizontal, so the normal force
The net horizontal force
Using Newton's second law again, we can find the floor acceleration
We can now use the same constant-acceleration equation as before, but this time the initial velocity
and the final velocity
Plugging in known values, we have
At this point, we're just solving for
Now, we can divide all terms by
Plugging in the angle
Notice how the coefficient doesn't depend on the mass