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1) We are going to test out a calculation. Let's pretend we are roller coaster designers in a world where friction is essentially eradicated from modern-day roller coaster tracks. Lucky you, because the physics just got a lot easier. You are assigned to build a roller coaster where, given its starting height, a cart can barely a loop-the-loop. That is, you are to construct a loop-the-loop so that a cart at a certain height (which you will determine) would be able to go through a loop in the track without falling when it passes the highest point in the loop.
Use your knowledge of conservation of energy and rotational motion (in particular, centripetal force) to determine the difference in height between the starting point of a cart and the top of the loop-the-loop. Put your calculations and your answers below. You may include a diagram (either hand-drawn and uploaded, or drawn using a diagram. Express heights in terms of r, the radius of the circle. Let the bottom of the loop-the-loop be at a height of zero. Show your work.
Hint: start by finding the minimum speed necessary to traverse the top of a circle of radius r (discussed in the lecture slides), calculate the corresponding kinetic energy, and then use conservation of energy to find the necessary change in height.
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