You are designing a new roller-coaster. The main feature of this particular desi

Question

You are designing a new roller-coaster. The main feature of this particular design is to be a vertical circular loop-the-loop where riders will feel like they are being squished into their seats even when they are in fact upside-down (at the top of the loop). The coaster start at rest a height of 80m above the ground, speeds up as it descends to ground level, and then enters the loop which has a radius of 20m. Suppose a rider is sitting on a bathroom scale that initially reads W (when the coaster is horizontal and at rest). What will the scale read when the coaster is moving past the top of the loop? (You can assume that the coaster rolls on the track without friction)

Explanation / Answer

From the energy conservation:

mgH=mg(2R)+mv^2/2

mv^2/2 = mgH - 2mgR

mv^2/R = 2mg(H/R -2)

The speed at the top of the loop enters into the calculation of the centripetal force

Fcp =mv^2/R = 2mg(H/R -2)

W = mg

Fcp = 2W(H/R -2)

Since the rider pushes against the scale while moving upwards, the scale pushes him back: the normal force is directed towards the centerof the loop..

At the top of the loop, both the weight and the normal are directed vertically down.

. The net force there is Fnet = W+FN

Here FN = normal reaction force from the scale.

By imposing that the net force gives the necessary centripetal force, we get

W+F = 2W(H/R -2)

FN = W(2H/R -5)

From given data H = 80 m ,R = 20 m

FN = W (2(80/20) -5)

FN = 3W

Scale reading is 3 times of W.

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