Two blocks of unequal mass and sit in a line between two fixed walls

PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Oscillations, Waves & Acoustics · Scientific programming for physicists

Scientific Computing for Physicists, Ch Letting the computer do advanced algebra: operators, series and differentiated programs

Problem

Two blocks of unequal mass

m

and

2m

sit in a line between two fixed walls, joined by three identical springs of stiffness

k

: a wall, a spring, block

m

, a spring, block

2m

, a spring and a wall. The blocks move only along the line joining them, with no damping and no gravity. Declare

k

and

m

positive before any square root is simplified. ((a)) Write the two equations of motion and form the stiffness matrix

K

and mass matrix

M

such that the normal-mode condition is

\det(K-\omega^2M)=0

. ((b)) Solve for the two values of

\omega^2

, exact in surd form, in units of

k/m

. ((c)) For each mode, give the ratio

u_2/u_1

of the second block's amplitude to the first's. ((d)) In one sentence, say what a computer algebra system would return for

\sqrt{k/m}

-type quantities here if

k

and

m

were left with no declared sign, and why declaring them positive is not a cosmetic step.