The loop-the-loop

One of the most fascinating problems in mechanics: an object travelling round a vertical loop (the loop-the-loop) while staying in contact with the track. The critical moment is the top of the loop, where both the weight $\vec{P}$ and the normal reaction $\vec{R}$ point both towards the centre (downwards).

Loop-the-loop. At point B (the top), both the weight $\vec{P}$ and the reaction $\vec{R}$ point towards the centre. The condition $R \geq 0$ requires $v_B \geq \sqrt{gr}$.

At the top, the sum of weight and reaction supplies the centripetal force: $P + R = mv_B^2/r$, i.e. $mg + R = mv_B^2/r$. The physical key is that the track can only push, not pull: the normal reaction must satisfy $R \geq 0$. Imposing this condition gives the minimum speed at the top.

Principle — Condition to complete the loop

At the top of the loop the normal reaction must be $R \geq 0$ (the track can only push). This imposes a minimum speed: $\ev{v_{\min}^2 = g\,r}$ where $r$ is the radius of the loop.

Below $v_{\min}$ gravity exceeds the required centripetal force: the body leaves the track and falls inwards. To find from what height the body must start so that it reaches the top with at least $v_{\min}$, energy conservation is used (see The loop-the-loop: an energy approach), which gives the elegant result $h = \tfrac{5}{2}r$.

Real loops are not circular

In real “loops” in theme parks and roller coasters, the shape is not circular but a clothoid (Euler spiral): the radius decreases going upwards, allowing the loop to be completed at a lower speed and with less violent $g$-forces on the body.

Links

Topics: Dynamics Concepts: Centripetal force · Normal force · Conservation of mechanical energy Objects: Loop-the-loop

Related exercises: Maximum speed on a flat curve · Banking angle of the curve · Flat curve, dry and wet