Vibrations and Waves · Problem 3-19

Forced vibrations and resonance — A.P. French

Physics — Waves · two-spring-mass-oscillation

Problem

A mass

m

rests on a frictionless horizontal table and is connected to rigid supports via two identical springs each of relaxed length

l_0

and spring constant

k

(see figure). Each spring is stretched to a length

l

considerably greater than

l_0

. Horizontal displacements of

m

from its equilibrium position are labeled

x

(along

AB

) and

y

(perpendicular to

AB

). (a) Write down the differential equation of motion (i.e., Newton's law) governing small oscillations in the

x

direction. (b) Write down the differential equation of motion governing small oscillations in the

y

direction (assume

y \ll l

). (c) In terms of

l

and

l_0

, calculate the ratio of the periods of oscillation along

x

and

y

. (d) If at

t = 0

the mass

m

is released from the point

x = y = A_0

with zero velocity, what are its

x

and

y

coordinates at any later time

t

? (e) Draw a picture of the resulting path of

m

under the conditions of part (d) if

l = 9l_0/5