Workshop Week 1 2021 PDF

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Workshop questions for Classical Mechanics Week 1...

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Workshop 1: PHYS2201 Suppose you have a string of length L with linear density 𝜇 and tension 𝑇 that is clamped at one end (left, 𝑥 = 0) and free to move on a frictionless rod at the other (right, 𝑥 = 𝐿). For an initial condition, 𝑦(𝑥, 0) = 𝑓(𝑥), 𝑦󰇗 (𝑥, 0) = 0, find a solution to the motion of this string, 𝑦(𝑥, 𝑡), as a sum of the normal modes and find an equation for the amplitude of each mode. A hint. The clamped end has a boundary condition with which we are familiar, since it is easy to implement physically. A clamp means that the position of the end of the wire is fixed so 𝑦(0, 𝑡) = 0. What about the other end - what is the boundary condition here? A key assumption* here is that the ring is massless. That means that any vertical force on this ring will result in unphysical, infinite acceleration. Think about what this means about the slope of the string as it meets the ring.

For the specific case of L=4, 𝑦(𝑥, 0) = 𝑥/4, 𝑦󰇗 (𝑥, 0) = 0, what is the function 𝑦(𝑥, 𝑡)?

* Physics is riddled with details and assumptions, sometimes they matter, sometimes they have their own Wikipedia entry, e.g. spherical cows....