Two blocks slide on a frictionless horizontal track

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

Two blocks slide on a frictionless horizontal track. Block 1 has mass

m=1.0\ \mathrm{kg}

and is joined to a wall by a spring of stiffness

k=100\ \mathrm{N\,m^{-1}}

. Block 1 is joined to block 2 (mass

2m=2.0\ \mathrm{kg}

) by a second spring of stiffness

k

, and block 2 is joined to a second wall by a spring of stiffness

2k

. Let

x_1,x_2

be the displacements of the two blocks from equilibrium. ((a)) Write down the kinetic and potential matrices

\mathbf T

and

\mathbf V

for

(x_1,x_2)

. ((b)) Find the two normal-mode angular frequencies and the ratio

x_1:x_2

for each mode. ((c)) A horizontal force

F(t)=10\cos\omega_{\rm d}t\ \mathrm{N}

with

\omega_{\rm d}=10\sqrt2\ \mathrm{rad\,s^{-1}}

is applied to block 1 only. Find the steady-state amplitude of each block.