Translation between representations (TBR)
Solid cylinder and hoop rolling down a ramp
- Units 5 and 6
- 12 points
- About 28 minutes
You can use a calculator on this question, just like on exam day.
A multipart problem about one situation shown in several ways. You draw a diagram, derive equations, sketch or draw graphs, and then explain whether your answers agree with each other or use them to predict what happens when the situation changes. On the exam: Question 2 of 4. New for May 2027: the free-response section is 95 minutes, down from 100, for all 4 questions (50% of your score), one of each type in this order. Calculator and equation sheet allowed. The CED suggests 25–30 minutes. This question type started in May 2025, so Physics C: Mechanics free-response questions from 2024 and earlier are built differently.
The question
A uniform solid cylinder and a thin-walled hollow cylinder (a hoop) have the same mass M and the same radius R. The rotational inertia of the solid cylinder about its axis is , and that of the hoop is . Each is released from rest at the top of a ramp inclined at θ = 30° above the horizontal and rolls without slipping to the bottom, a vertical drop of h = 0.90 m. Use g = 9.8 m/s² (answers that use g = 10 m/s² are also accepted).
Suggested time: 28 minutes
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Part (a)
2 pointsDescribe a force diagram for the solid cylinder as it rolls down the ramp. List each force, name the object that exerts it, give its direction, and state where on the cylinder it acts.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (b)
3 pointsStarting from Newton's second law in translational and rotational form, derive expressions for the acceleration of the center of the solid cylinder and of the center of the hoop, in terms of g and θ.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (c)
2 pointsDescribe energy bar charts for each object at the bottom of the ramp, showing the translational kinetic energy and rotational kinetic energy as fractions of the gravitational potential energy Mgh that each object had at the top (taking the bottom as zero).
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (d)
2 pointsDescribe a graph of the speed of each object's center versus time, from release until it reaches the bottom of the ramp, on the same axes. Include the shape of each graph, how the slopes compare, and the final speed and time for each.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (e)
2 pointsExplain how your energy bar charts from part (c) are consistent with your speed–time graphs from part (d).
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (f)
1 pointCalculate the minimum coefficient of static friction between the ramp and the solid cylinder needed for the cylinder to roll without slipping.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
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