Theoretical Analysis of Suspended Can Oscillations · RMPH 2014 #1

Physics · mechanics

Problem

Context: You have a cylindrical tin can suspended by two wires that can perform small rotational oscillations. The can has radius a=rLa = rL for its cover, height b=pLb = pL, and wall thickness t=qLt = qL, where LL is the length of each suspension wire. An "elementary volume" is defined as v=a2bN=r2pNL3v = \frac{a^2b}{N} = \frac{r^2p}{N}L^3, where NN is five times the number of syringes needed to fill the can. Consider a coordinate system where the center of the circular upper face is at origin O when in equilibrium, with the Ox axis passing through the suspension points, Oy perpendicular to Ox, and Oz as the vertical axis of symmetry. 1.a. Write the expression for the coordinates of C" - the place where a suspension wire is caught - when the box is rotated by angle θ\theta around the vertical axis of symmetry. Express in terms of θ\theta, aa, and hh. 1.b. Deduce the expression of angle α\alpha as function of angle θ\theta, length LL, and radius aa of can cover. 1.c. Deduce the expression of angle α\alpha and elevation hh when angle θ\theta is small enough to admit that sin⁡θ≈θ\sin\theta \approx \theta, cos⁡θ≈1\cos\theta \approx 1, θ2≈0\theta^2 \approx 0. Express results as functions of θ\theta, LL, and aa. 1.d. Deduce the expression of the interior volume VV, as well as expressions for mass mm and moment of inertia JJ of the empty can. Express results as appropriate functions of LL, rr, pp, qq, and ρ0\rho_0. 1.e. Determine the expression of mass mkm_k and moment of inertia JkJ_k of the can filled with kk "elementary volumes" of water. Express results as appropriate functions of LL, rr, pp, qq, ρ0\rho_0, ρ\rho, NN, and kk. 1.f. Deduce the expression of the momentum MTM_T of the forces acting on the can when it is rotated by a small angle θ\theta. Express in terms of mkm_k, gg, aa, LL, and θ\theta. 1.g. Neglecting damping, determine the equation of motion describing the rotation of the can with a small angle θ\theta and show that it performs simple harmonic motion. 1.h. The period of oscillatory motion has the expression: Toscil=2π(q+r22p+prNk)gwL(r+r2q2p)T_{oscil} = 2\pi\sqrt{\frac{\left(q + \frac{r^2}{2p} + \frac{pr}{N}k\right)}{\frac{gw}{L}\left(r + \frac{r^2q}{2p}\right)}} Find the unknown coefficient ww in the above expression.

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