Inclines: No Friction

When analyzing objects on inclines, we use a tilted coordinate system with axes parallel and perpendicular to the surface. This simplifies the problem because motion happens along the incline.

The weight $\vec{W} = m\vec{g}$ acts vertically downward. We decompose it into components along the tilted axes.

Choice of positive direction: The sign of components depends on our coordinate choice. We can define positive as either up the ramp or down the ramp for the parallel axis.

When positive is down the ramp:

• Parallel component: $W_{\parallel} = +mg\sin\theta$ (positive, down the slope)
• Perpendicular component: $W_{\perp} = -mg\cos\theta$ (into the surface)