Question

Determine the exact height h from which a roller coaster cart must be released in order for the normal force exerted on said cart at the top of the loop to equal twice the force due to gravity; i.e. Fx = 2mg (at the top of the loop). Note that the roller coaster cart is made to traverse a vertical (circular) loop with radius R. Assume, for the purposes of this exercise, that the roller coaster cart is released from rest AND ignore frictional forces See Figure 2. 101 Figure 2

Answer 1

As the roller coaster cart descends from the top from heighit ', åt grins some Kinetic energy as it reaches the lowermost point (A) of the loop. Therefore we can calculate the velocity of the roller coaster at the lowest echanical énergy .. 0 sen IVit Ki .Ufit Kft Beint using conservation of m = mgh.to Oto L.me? sino . = v= Jagh . As the roller coaster reaches the top of the loog, suppose its velocity becomes v'. The forces lacting on the roller coaster are centripetal force lacting towards the centre) and weight of the roller coaster (acting downwards). comu

T. At the Tacting on top the most point (B), the roller coaster is het - force R Ning = mult; where N is the normal force 1. R . on the cart. I here N=zing (given) I... ang mg=mu'2 = v'= √3Rg . I Now, applying conservation of mechanical energy for the cart's journey from A to B, we have Kit Vi = K4+Uf. T lmu²+0 = 1 m012+ mg (28) m (szga)?= 1 m (Ekg) + 2 mgr. - mgh = 3 mg R+ 2mgR. = h = 7 ... The value of height h from u le the cart must be released is ch=7R