A block weighing #4 kg# is on a plane with an incline of #(pi)/2# and friction coefficient of #1#. How much force, if any, is necessary to keep the block from sliding down?
1 Answer
Explanation:
We're asked to find the necessary force (if such a force is necessary) needed to keep an object on an incline at rest.
If the object is at rest, then it is in equilibrium, and the net force on it is zero.
Let's take a look at the forces acting on the block parallel to the ramp (which I'll call the

gravitational force (acting downward), equal to
#mgsintheta# 
friction force (directed upward because it opposes the direction of sliding), equal to
#f = mun = mumgcostheta#
The net force equation for the block is
#ul(sumF_x = overbrace(mumgcostheta)^"upward force"  overbrace(mgsintheta)^"downward force"#
We're given

#m = 4# #"kg"# 
#theta = pi/2# 
#mu = 1# 
and
#g = 9.81# #"m/s"^2#
Plugging these in:
#sumF_x = (1)(4color(white)(l)"kg")(9.81color(white)(l)"m/s"^2)cos[pi/2]  (4color(white)(l)"kg")(9.81color(white)(l)"m/s"^2)sin[pi/2]#
#= color(red)(ul(39.24color(white)(l)"N"#
That is, the net force acting on the object is
So, to prevent the object from sliding (or falling in this case, because the angle is