What causes a rolling object to gain more speed going down a sinusoidal ramp than a linear ramp?

Are they measuring final speed, or are they measuring the TIME it takes?

Don't confuse the two. It is most likely that they both end at the same final velocity, yet, on a concave-up sinusoidal section of ramp, it takes less time.

BTW, was it a concave-up or a concave-down portion of a sinusoid? Shaped like a mound, or shaped like a bowl?

The only way that final speed could be different in either case, is if friction and air drag effects were significant. Because without these effects, then by the trade of GPE to KE, the final speed would be guaranteed to be the same.

I can see a concave-down sinusoid as being a shape which would reduce the "dry friction", for the following reason. The fact that it is moving along a curved path, means that it needs to have a centripetal acceleration towards the center of curvature of that path. This means that the normal force of constraint is unloaded, to less than what is needed for equilibrium alone at each point along the way. This means less normal force, and less corresponding kinetic friction.

Compare this to the straight ramp, and the straight ramp will have one constant normal force, equal to exactly what is needed for equilibrium alone. It never gets "unloaded", because there is no acceleration in any direction other than along its path.

For a concave-up curve, the centripetal acceleration is in an upward direction, and the normal force is therefore loaded beyond what is needed for equilibrium. This would mean more normal force and more kinetic friction, thus less final kinetic energy.

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If however, it is TOTAL TRAVEL TIME that is different, then read below:

There is a famous problem in Differential Equations called the Brachistochrone Problem. It calls for the derivation of the curve that a frictionless sliding object, can slide due to gravity alone, in the shortest amount of time between two points that are not directly above each other.

http://en.wikipedia.org/wiki/Brachistochrone_curve

It turns out that the solution to the Brachistochrone Problem, is an inverted Cycloid. A cycloid is the path that a point on the tip of a wheel will trace as the wheel rolls without slip. Now imagine the wheel rolling without slip, on the ceiling. And imagine the path that it traces out. That is what a cycloid is.

So an inverted cycloid is THE BEST curve for a ramp between two points, if you want to minimize time for an idealized frictionless particle to slide between the points. Given a straight line and the concave-up portion of a sinusoid, the concave-up portion of the sinusoid is closer to the inverted cycloid than a straight line.

no, each thing no count how large falls on the comparable fee of acceleration because of gravity 9.8m/s^2 inertia vs gravity the ramp basically makes that fee slower for each thing yet while speaking a pair of ramp shape make come into play. friction has no longer something to do with momentum