Mathematical routines (MR)
A sliding rod slowed by induction
- Unit 13
- 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 diagram or a graph, derive an equation using letters, calculate a numerical answer with units, and 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 and equation sheet allowed. The CED suggests 20–25 minutes.
The question
Two long, parallel, horizontal metal rails are a distance ℓ apart and are connected at their left ends by a resistor of resistance R. A metal rod of mass m lies across the rails, perpendicular to them, and can slide without friction. A uniform magnetic field of magnitude B points vertically upward everywhere. At time t = 0, the rod is moving to the right with speed v₀; nothing pushes or pulls it after that. The rails and rod have negligible resistance.
Suggested time: 22 minutes
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Part (a)
2 pointsDerive an expression for the current in the resistor when the rod's speed is v.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (b)
3 pointsStarting with Newton’s second law, derive an expression for the rod's speed v as a function of time t, in terms of v₀, B, ℓ, m, R, and t.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (c)
2 pointsThe rod has m = 0.20 kg, ℓ = 0.50 m, B = 0.80 T, R = 0.40 Ω, and v₀ = 3.0 m/s. Calculate the rod's speed at t = 1.0 s.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (d)
2 pointsCalculate the total distance the rod travels after t = 0.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (e)
1 pointCalculate the total thermal energy dissipated in the resistor after t = 0, and explain your reasoning.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
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