The two-cart system of blocks and described previously has

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Oscillations and normal modes · Oscillations and normal modes

Action and Symmetry, Ch Normal modes as a generalised eigenvalue problem

Problem

The two-cart system of blocks m=1.0 kgm=1.0\ \mathrm{kg} and 2m=2.0 kg2m=2.0\ \mathrm{kg} described previously has T=diag(1.0,2.0) kg\mathbf T=\mathrm{diag}(1.0,2.0)\ \mathrm{kg}, normal modes a(1)=(1,1)\mathbf a^{(1)}=(1,1) at ω12=100 rad2 s−2\omega_1^2=100\ \mathrm{rad^2\,s^{-2}} and a(2)=(2,−1)\mathbf a^{(2)}=(2,-1) at ω22=250 rad2 s−2\omega_2^2=250\ \mathrm{rad^2\,s^{-2}}. Block 1 is pulled aside by 5.0 cm5.0\ \mathrm{cm} and block 2 by 2.0 cm2.0\ \mathrm{cm}, both in the same direction, and the system is released from rest at t=0t=0. ((a)) Find the normal coordinates η1(0)\eta_1(0) and η2(0)\eta_2(0), using mode vectors normalised so that a^(r)TTa^(s)=δrs\hat{\mathbf a}^{(r)\mathrm T}\mathbf T\hat{\mathbf a}^{(s)}=\delta_{rs}. ((b)) Calculate the energy stored in each mode and the total energy, and give the two numbers (in joules, and as a percentage of the total) that a bar chart of mode energies would show.

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