Inclined Plane with Friction

In our analysis of the air cart on the inclined track, we mentioned that a frictionless surface exerts only a normal force on an object. If there is any sideways force, that is supposed to be a friction force . In Figure (1) we show a cart on an inclined plane, with a friction force  included. The normal force is perpendicular to the plane, the friction force is parallel to the plane, and gravity still points down.

Figure 1: Friction force acting on the cart.

To analyze the motion of the cart when acted on by a friction force, we write Newton’s second law in the usual form

..........(1)

The only change is that we have added in the new force .
Since the motion of the cart is still along the plane, it is convenient to take the x axis along the plane as shown in Figure (2). Breaking Equation 1 up into x and y components now gives

...........(2a)

.........(2b)

From 2b we get,
F_n = mg cos θ...........(3)

which is the same result as for the frictionless plane.
The new result comes when we look at motion down the plane. Solving 2a for ax gives

a_x = gsinθ – F_f /m.......... (4)
Not surprisingly, the friction force reduces the acceleration down the plane.

Figure 2