A roller coaster car moves in a vertical circle of radius R. At the top of the circle the car has speed v1, and at the bottom of the circle it has speed v2, where v2>v1. Suppose that v2 is in the x-direction, and y-axis is directed upward.

1) When the car is at the top of its circular path, what is the direction of its radial acceleration, arad, top? The direction angle β-top of the radial acceleration arad, top is measured from the + x-direction toward the + y-direction.

2) When the car is at the bottom of its circular path, what is the direction of its radial acceleration, arad, bottom? The direction angle β-bottom of the radial acceleration arad, bottom is measured from the + x-direction toward the + y-direction.

3) In terms of v1 and v2, what is the ratio arad, bottom/arad, top?

Question

Physics

Kaylin Russo

6 November, 21:05

A roller coaster car moves in a vertical circle of radius R. At the top of the circle the car has speed v1, and at the bottom of the circle it has speed v2, where v2>v1. Suppose that v2 is in the x-direction, and y-axis is directed upward.

1) When the car is at the top of its circular path, what is the direction of its radial acceleration, arad, top? The direction angle β-top of the radial acceleration arad, top is measured from the + x-direction toward the + y-direction.

2) When the car is at the bottom of its circular path, what is the direction of its radial acceleration, arad, bottom? The direction angle β-bottom of the radial acceleration arad, bottom is measured from the + x-direction toward the + y-direction.

3) In terms of v1 and v2, what is the ratio arad, bottom/arad, top?

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Answers (1)

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Raelynn Wu · Answer 1

1) Radial acceleration is nothing but centripetal acceleration. When the car is at the top, centripetal acceleration is downwards towards the center.

In other words, its direction is at 270 degree from + x axis.

angle β-top = 270 degree

2) When the car is at the lowest position, centripetal acceleration is again

towards the center or in upward direction.

So angle β-bottom = 90 degree.

3) Velocity at the bottom v₂ = √5gr

velocity at the top v₁ = √gr

v₂ / v₁ = √5.

OR

arad (bottom) = v₂² / R

arad (top) = v₁² / R

arad (bottom) / arad (top) = v₂² / v₁²