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 starts 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

The reading on the scale is N = 9W

Explanation:

Since the roller coaster drops from a height, h of 80 m, the potential energy lost equals the kinetic energy gained as it enters the loop.

So, mgh = 1/2mv² where m = mass of rider, g = acceleration due to gravity = 9.8 m/s², h = initial height of roller coaster above ground level = 80 m and v = speed of roller coaster as it enters the loop.

So, mgh = 1/2mv²

v² = 2gh

v = √(2gh)

v = √(2 × 9.8 m/s² × 80 m)

v = √(1568 m²/s²)

v = 39.6 m/s

Now, as the roller coaster gets to the top of the vertical loop, the centripetal force, F and the weight W acts downwards. For the passenger not to fall off, this must equal the normal force, N

So, F = mv²/r where v = speed of roller coaster = 39.6 m/s and r = radius of vertical loop = 20 m and m = mass of rider = W/g

F = Wv²/gr

F = W(39.6 m/s)²/(9.8 m/s² × 20 m)

F = (1568 m²/s²)W/196 m²/s²

F = 8W

Since F + W = N

8W + W = N

9W = N

So, N = 9W

So, the reading on the scale is N = 9W

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