Mathematical routines (MR)
Swinging a rod whose density increases along it
- Units 2, 5 and 6
- 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, including calculus, 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. 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 20–25 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 thin rod of length L = 1.2 m has a linear mass density that increases along its length, , where x is the distance from end A and . The rod is mounted on a frictionless horizontal axle through end A, so it can swing in a vertical plane. It is held horizontal and released from rest. 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)
2 pointsDerive expressions for the mass M of the rod and the distance of its center of mass from end A, in terms of β and L. Then calculate their values.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (b)
2 pointsDerive an expression for the rotational inertia of the rod about the axle at end A, in terms of M and L.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (c)
2 pointsCalculate the magnitude of the angular acceleration of the rod just after it is released.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (d)
2 pointsCalculate the angular speed of the rod when it swings through the vertical position.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (e)
2 pointsA student claims that a uniform rod with the same mass M and length L, mounted and released the same way, would have a greater angular acceleration just after release. Do you agree or disagree with the student? Justify your answer.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
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