Rolling Marble on Track · CAP 2002 #2
Physics · mechanics
Problem
A demonstration apparatus sits on a table in the lab. It consists of a metal track along which a perfectly spherical marble can roll without slipping. In one run, the marble is released from rest at a height h above the table on the left section, rolls down one side and then up the other side without slipping, briefly stopping when it has reached h₁. Assuming the table to be horizontal and neglecting air drag as well as any energy loss due to rolling, [Figure: A track with angles φ and θ, showing initial height h and final height h₁] (a) h₁ < h always, because the friction that keeps the ball rolling must dissipate energy. (b) h₁ = h, since total mechanical energy is conserved. (c) h₁ = h only if tilt angles φ and θ are equal. (d) h₁ = h only if tilt angle φ is larger than θ.Intuitive explanation
The key insight is distinguishing between rolling friction (which enables rolling without slipping) and sliding friction (which dissipates energy). Since the problem states no energy loss and the marble rolls without slipping on a frictionless track, mechanical energy—including rotational kinetic energy—is fully conserved. The marble converts potential energy to both translational and rotational motion on the way down, then back to potential energy on the way up. The track geometry (angles φ and θ) doesn't affect this energy balance; what matters is whether the marble has enough total energy to reach height h₁, which it does if energy is truly conserved.
Intuitive explanation
The key insight is distinguishing between rolling friction (which enables rolling without slipping) and sliding friction (which dissipates energy). Since the problem states no energy loss and the marble rolls without slipping on a frictionless track, mechanical energy—including rotational kinetic energy—is fully conserved. The marble converts potential energy to both translational and rotational motion on the way down, then back to potential energy on the way up. The track geometry (angles φ and θ) doesn't affect this energy balance; what matters is whether the marble has enough total energy to reach height h₁, which it does if energy is truly conserved.