The loop-the-loop: an energy approach

The loop-the-loop, that vertical loop in which a body travels a full circle while staying in contact with the track, unites two ideas that at first sight seem unrelated: the dynamical condition at the top of the loop and the conservation of energy along the whole path. Analysing it with the energy method makes the link between the two transparent.

The starting point is a local condition, valid only at the highest point of the loop. For the body to stay in contact with the track at the top of the loop, its speed cannot drop below a minimum value: at the top it must be at least $v_{\min} = \sqrt{gr}$, where $r$ is the radius of the loop. Below this speed, gravity is no longer entirely “used up” to curve the trajectory, and the body detaches, falling inward. This is the requirement imposed by circular motion at the top, independent of how the body got there.

The natural question is then: from what height must the body start to reach that speed at the top? Here conservation of energy comes into play, bridging the start and the top. Treating the motion as a chain of states, we compare State A (starting from rest, at a height $h$, with energy entirely potential) with State B (top of the loop, at height $2r$, with speed $v_{\min}$). In the absence of friction, mechanical energy is conserved:

$mgh = \tfrac{1}{2}m(gr) + mg(2r)$

On the left is the initial potential energy; on the right, the minimum kinetic energy required at the top plus the potential energy at height $2r$. Solving gives the minimum starting height:

$\ev{h = \frac{5}{2}\,r}$

The result is surprisingly clean: two and a half times the radius of the loop is enough. The strength of this approach lies precisely in the connection. The condition at the top ($v_{\min} = \sqrt{gr}$) supplies the speed value to feed into the energy balance; conservation of energy translates it into a height. Neither one, alone, would answer the question: it’s their combination that does. And if there were friction along the ramp, the balance would require a greater height, to make up for the energy dissipated as heat before reaching the loop.

Links

Topics: Lavoro ed energia Concepts: Conservazione dell’energia meccanica · Energia cinetica · Energia potenziale gravitazionale Skills: Conservazione dell’energia Objects: Giro della morte

Related exercises: Esercizio svolto — altezza minima per il loop · Problema — Palla, molla, attrito e volo (4 stati) · Esercizio svolto — mela e molla