Two carts, each of mass , run on a frictionless track. Each cart is joined to a wall…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Linear algebra and vector spaces

Mathematical Methods for Physics, Ch Vector spaces, bases, linear maps and dual vectors

Problem

Two carts, each of mass m=0.20 kgm = 0.20\,\mathrm{kg}, run on a frictionless track. Each cart is joined to a wall by an ideal spring of stiffness k=50 N m−1k = 50\,\mathrm{N\,m^{-1}}, and the two carts are joined to each other by a third ideal spring of stiffness k12=10 N m−1k_{12} = 10\,\mathrm{N\,m^{-1}}. Let x1,x2x_1, x_2 be the displacements of the carts from equilibrium, collected as x=(x1,x2)\mathbf{x} = (x_1, x_2). The equations of motion are mx¨1=−kx1−k12(x1−x2),mx¨2=−kx2−k12(x2−x1),m\ddot{x}_1 = -kx_1 - k_{12}(x_1 - x_2), \qquad m\ddot{x}_2 = -kx_2 - k_{12}(x_2 - x_1), i.e. x¨=−Ax\ddot{\mathbf{x}} = -\mathsf{A}\mathbf{x} for a linear map A\mathsf{A} on x\mathbf{x}. The potential energy stored in the three springs is V(x1,x2)=12kx12+12kx22+12k12(x1−x2)2V(x_1, x_2) = \tfrac12kx_1^2 + \tfrac12kx_2^2 + \tfrac12k_{12}(x_1 - x_2)^2. ((a)) Write the matrix A\mathsf{A} (with x1,x2x_1, x_2 in metres) in the standard basis {(1,0),(0,1)}\{(1,0),(0,1)\}, stating its units. ((b)) The normalised eigenvectors of A\mathsf{A} are e1=12(1,1)\mathbf{e}_1 = \tfrac{1}{\sqrt2}(1,1) and e2=12(1,−1)\mathbf{e}_2 = \tfrac{1}{\sqrt2}(1,-1) (the symmetric and antisymmetric normal modes). Using these as the columns of a change-of-basis matrix P\mathsf{P}, compute A′=P−1AP\mathsf{A}' = \mathsf{P}^{-1}\mathsf{A}\mathsf{P} and state the two normal-mode angular frequencies ω1,ω2\omega_1, \omega_2 read off its diagonal. ((c)) Confirm that tr A=tr A′\mathrm{tr}\,\mathsf{A} = \mathrm{tr}\,\mathsf{A}' and det⁡A=det⁡A′\det\mathsf{A} = \det\mathsf{A}'. ((d)) At the instant when (x1,x2)=(0.020 m,−0.010 m)(x_1, x_2) = (0.020\,\mathrm{m}, -0.010\,\mathrm{m}), compute the gradient ∇V=(∂V/∂x1,∂V/∂x2)\nabla V = (\partial V/\partial x_1, \partial V/\partial x_2), a dual vector whose pairing with a displacement δx\delta\mathbf{x} gives δV=∇V⋅δx\delta V = \nabla V\cdot\delta\mathbf{x}. ((e)) A further small displacement is δx=(1.0 mm,0.50 mm)\delta\mathbf{x} = (1.0\,\mathrm{mm}, 0.50\,\mathrm{mm}). Compute δV\delta V using the metre-based components from part (d). Then rescale the coordinates to xi′=1000xix_i' = 1000x_i (so xi′x_i' is the numerical value of xix_i in millimetres). Find the gradient components ∂V/∂xi′\partial V/\partial x_i' and the displacement components δxi′\delta x_i' in the new coordinates, and recompute δV\delta V from them. ((f)) Name one quantity from parts (b)-(c) that is the same in the standard and normal-mode bases despite the matrix entries changing, and state what is unchanged in parts (d)-(e) despite the numbers of the gradient and the displacement both changing.

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