The skater on the half-pipe

Problem

The skater on the half-pipe. A skater rides a smooth half-pipe, starting from rest at the top edge. At which point of the descent is the magnitude of the acceleration greatest? Where is it purely radial? Where is it purely tangential? Answer without calculations, reasoning about the direction of the weight force and the shape of the path.

Solution

The two components. Along a curved path the acceleration always splits into two parts:

• tangential $a_t$: changes the magnitude of the velocity; it is linked to the component of the weight along the path;
• radial (centripetal) $a_c = v^2/r$: changes the direction of the velocity; it points towards the centre of curvature.

At the top edge (start). The skater starts from rest: $v = 0$, so $a_c = v^2/r = 0$. There the half-pipe wall is (nearly) vertical, i.e. tangent to the path: the weight force, vertical, is entirely directed along the path. The acceleration is therefore purely tangential — maximum capacity to “pick up speed”, no curvature of the velocity (which does not yet exist).

$\ev{\text{At the top: } a \text{ purely tangential } (v=0 \Rightarrow a_c=0)}$

At the bottom (lowest point). Here the path is horizontal: the tangent is horizontal and the radius (towards the centre of curvature) is vertical, pointing up. The weight force is vertical, so it has no tangential component: $a_t = 0$. Moreover the bottom is the point where the speed $v$ is greatest (the skater has “converted” the whole descent into speed) and the radius of curvature is small, so $a_c = v^2/r$ is large. The acceleration is purely radial (centripetal, pointing up) and it is there that it has maximum magnitude.

$\ev{\text{At the bottom: } a \text{ purely radial and maximum } (a_t=0,\ v \text{ max})}$

In summary. Descending from the edge, the tangential component (which accelerates the skater) decreases while the radial component grows (because $v$ increases): the acceleration gradually rotates from “all tangential” at the top to “all radial” and maximum at the bottom.

Links

Topics: Dinamica Concepts: Accelerazione centripeta · Moto circolare uniforme · Forza peso Skills: Scomposizione vettoriale · Diagramma di corpo libero Methods: Metodo del diagramma di corpo libero