Aim for FivePhysics 1

Mathematical routines (MR)

Roller-coaster cart in a vertical loop

  • Units 2 and 3
  • 10 points
  • About 22 minutes

You can use a calculator on this question, just like on exam day.

A multipart problem where you use math to analyze a physical situation. You draw or sketch a representation such as a free-body diagram or a graph, derive an equation using letters, calculate a numerical answer with units, and then make a claim or prediction and justify it with physics reasoning. On the exam: Question 1 of 4. Section II has 4 questions in 95 minutes (50% of the exam score), one of each type in this order; calculator allowed. The CED suggests 20–25 minutes.

The question

A roller-coaster cart of mass 450 kg is released from rest at a height H above the bottom of a vertical circular loop of radius R = 8.0 m. The cart travels on the inside of the track and goes around the loop. Treat the cart as a particle, and assume friction and air resistance are negligible. Use g = 9.8 m/s² (answers that use g = 10 m/s² are also accepted).

Suggested time: 22 minutes

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Part (a)

1 point

Describe the free-body diagram of the cart when it is at the very top of the loop. List each force, name the object that exerts it, and give its direction.

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Part (b)

3 points

Derive an expression for the minimum release height H_min for which the cart stays in contact with the track at the top of the loop. Express your answer in terms of R and physical constants, as appropriate.

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Part (c)

2 points

The cart is released from H_min for the loop with R = 8.0 m. Calculate (i) H_min and (ii) the magnitude of the normal force the track exerts on the cart at the bottom of the loop.

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Part (d)

2 points

A second cart with twice the mass is used on the same track. A student claims that the heavier cart must be released from a greater height to make it around the loop. Do you agree or disagree? Justify your answer.

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Part (e)

2 points

Suppose instead that the track exerts a small friction force on the cart. Would the minimum release height be greater than, less than, or equal to the value from part (c)? Justify your answer.

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