Translation between representations (TBR)
Ball on a string in a vertical circle
- Units 2 and 3
- 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 small ball of mass m = 0.20 kg is attached to a light string and moves in a vertical circle of radius R = 0.80 m around a fixed point. Only gravity and the string act on it; nothing else pushes it along. At the bottom of the circle, the ball's speed is m/s. The angle θ is measured at the center of the circle from the lowest point, so θ = 0 at the bottom and θ = π at the top. Ignore air resistance. Use g = 9.8 m/s² (answers that use g = 10 m/s² are also accepted).
Suggested time: 28 minutes
Your answers are saved in this browser as you type.
Part (a)
2 pointsDescribe the free-body diagrams of the ball at the bottom and at the top of the circle. List each force, name the object that exerts it, give its direction, and compare the lengths of the arrows in each diagram.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (b)
3 pointsDerive an expression for the ball's speed v at angle θ and for the tension T in the string at angle θ, in terms of m, R, g, , and θ. Then calculate the ball's speed at the top of the circle.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (c)
3 pointsDescribe the graph of the tension T as a function of θ for one full revolution, from θ = 0 to θ = 2π. Include the axes and labels with units, the shape, and the values at θ = 0, π/2, π, 3π/2, and 2π.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (d)
2 pointsShow that the difference between the tension at the bottom and the tension at the top does not depend on the ball's speed, and explain how this is consistent with the free-body diagrams in part (a) and the graph in part (c).
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (e)
2 pointsDerive the minimum speed at the bottom of the circle for which the string remains taut at the top. Describe how the tension graph from part (c) would change if the ball had exactly this speed at the bottom.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Checking scoring…
Scoring it yourself shows you the rubric, examples and a model answer. Try writing your answer first.