Rollercoaster Loop

question physics

Question

A rollercoaster starts at the top of a ramp, at a height $h$. It then moves from rest, into a circular loop-the-loop of radius $R$. What is the minimum value of $h$, in terms of $R$, such that the rollercoaster makes it all the way around the loop.


Hints

Consider the conditions at the top of the loop. Balance forces - what condition is satisfied at the minimum speed required to pass through the loop.


Answer

As the cart moves around the circle, the net force it experiences is $\frac{mv^2}{R}$. At the top, the reaction force from the track, and the force from gravity act in the same direction, such that $\frac{mv^2}{R}=N+mg$. At the minimum required speed, $N=0$, as if the cart were to move any slower then it would fall off the track. This gives $v_{\text{min}}^2=gR$.

Thus, assuming no energy loss, the initial energy can be equated to the energy at the top of the loop: $$mgh=\frac{1}{2}mv_{\text{min}}^2 + mg(2R)$$ $$h=\frac{1}{2}R + 2R$$ $$h=\frac{5}{2}R$$