A block slides down an inclined plane that makes an angle `theta` with the horizontal. If the coefficient of kinetic friction is `mu`, find the acceleration of the block.

Explanation

First draw a free-body diagram. Notice that in the diagram shown below, the weight of the block, `F_(w)= mg`, has been written in terms of its scalar components : `F_(w)sin theta` parallel to the ramp, and `F_(w) cos theta` normal to the ramp.

The force of friction, `F_(f)` , that acts up the ramp (opposite to the direction in which the block slides ) has magnitude `F_(f)= mu F_(N)`. But the diagram shows that `F_(N)= F_(w) cos theta, " so " F_(f)= mu (mg cos theta)`. Therefore, the net force down the ramp is
`F_(w) sin theta- F_(f)= mg sin theta- mu mg cos theta= mg (sin theta - mu cos theta)`.
Then, setting `F_("net")` equal to ma, we solve for a :
`a=(F_("net"))/(m)=(mg(sin theta - mu cos theta))/(m)`
`=g(sin theta- mu cos theta)`