You have a roller coaster with height h and a single circular loop of radius R. After the loop there is a frictional portion to the track with a coefficient of kinetic friction ?k. What is the mathematical condition such that the cart stays on the track at the highest point of the loop? Derive the height h such that this condition is met (algebraically in terms of R). If h = 50 m, and the coefficient of friction at the end of the track is 0.1 how long does the frictional part have to be so that the cart stops? Calculate the length both using energy methods and with standard kinematics (like we would have prior to Exam 1).
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To determine the mathematical condition for the cart to stay on the track at the highest point of the loop, we need to consider the forces acting on the cart at that point. The two main forces are the gravitational force (mg) and the normal force (N) exerted by Show more…
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The diagram below shows a roller-coaster ride which contains a circular loop of radius (r). A car of mass (m) begins at rest from point A and moves down the frictionless track from A to B, where it then enters the vertical loop which is also frictionless, traveling once around the circle from B to C to D to E and back to B, after which it travels along the flat portion of the track from B to F which is not frictionless. a) Assume that H = 4r. Use the conservation of energy to find the velocity of the car when it reaches point C. b) Determine the minimum speed at point D for the cart to maintain centripetal motion and not fall from the track. c) Suppose H = 4R. Determine if the cart successfully traverses the loop. yes no Justify your answer mathematically.
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Here's a classic physics problem. A roller coaster has a loop of radius, R. If the roller coaster car starts from rest a height, h, above the ground, what is the minimum value of h such that car stays on the track and the riders don't fall out at the top of the loop? This is a variation on the water-in-pail demonstration in lecture: as long as the centripetal acceleration required is at least as large as the acceleration due to gravity, all will be well. The situation is shown in the figure below. Start by finding the minimum velocity needed when the car is at the top of the loop, then use the conservation of energy (you can assume there is no friction, nor are there other significant losses of energy) to find h. Your answer should be a number times R.
Ethan D.
In an amusement park ride a cart of mass $300 \mathrm{kg}$ and carrying four passengers each of mass $60 \mathrm{kg}$ is dropped from a vertical height of $120 \mathrm{m}$ along a frictionless path that leads into a loop-the-loop machine of radius $30 \mathrm{m}$ The cart then enters a straight stretch from $A$ to C where friction brings it to rest after a distance of $40 \mathrm{m} \text { . (See Figure } 8.19 .)$ (a) Find the velocity of the cart at A. (b) Find the reaction force from the seat of the cart onto a passenger at $\mathrm{B}$. (c) What is the acceleration experienced by the cart from A to C (assumed constant)?
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