A particle P of mass 8.84 kg moves on a frictionless horizontal track, and is at

Question

A particle P of mass 8.84 kg moves on a frictionless horizontal track, and is attached by two model springs to fixed supports at A and B, as shown in the figure.

The natural length of the spring AP is twice that of spring PB. Spring AP has a stiffness of 118 N/m and the stiffness of spring PB is twice that of spring AP. The distance AB between the supports is equal to the natural length of the spring AP. The particle moves so that it always remains between the supports. Take the axis to be in the direction of the track from A to B with origin at A.

Using the equation of motion of the particle, find the period, in s, of the oscillation of the particle when it is in simple harmonic motion between the supports. Give your answer to 3 decimal places.

Explanation / Answer

stiffness of AP be k
stiffness of PB=2k
force on the particle if it is at a distance x from midpoint of AB
F=k(l/2-x)-2k(x)
=kl/2-3kx

using equation of motion
F=md^x/dt^2
^2=3k/m

=(3k/m)=6.3281

T=2/=0.993s

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