Acceleration on an Inclined Plane using the “Wrong” Coordinate System

Neglecting friction, if the roller coaster track is tilted at angle q from horizontal, the gravitational force mg has a component parallel mgsinq parallel to the track, and this is what causes the acceleration. It is the only force parallel to the track. So S F = ma becomes mgsinq =ma, so a = gsinq .

Perpendicular to the plane, there is the normal force and mgcosq in the opposite direction. There is no acceleration ^ to the plane, so these balance out: the normal force is mgcosq .

Now let us look at the trouble we get into if we use the wrong coordinate system. Wrong, that is, in the sense of being too difficult, not incorrect.

Using horizontal and vertical x and y axes, and using N for the normal force, we have

1) N_x = ma_x and

2) N_y - mg = ma_y.

Pretend that we do not know N = mgcosq , because this was learned by using the tilted coordinates. Here is a gimmick to use instead: N is perpendicular to a so their dot product is zero, or

3) N_xa_x + N_ya_y = 0.

Solve equation 3 for N_x and substitute into equation 1, then solve the resulting equation for N_y and substitute into equation 2. We now have an equation with no N. Using
a_x² + a_y² = a² and a_y = -asinq , you will be able to reduce it to

a = mgsinq .

Are we having fun yet, or what? At least now you can appreciate the importance of choosing a coordinate system such that the acceleration is along an axis.

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