A mass weighing 32 pounds stretches a spring to 6 inches. The mass moves through a medium offering a dampering force that is numerically equal to Beta times the instantaneous velocity. Determine the values of Beta>0 for which the spring/mass system will exhibit oscillatory motion.
I got a mass of 1 slug (32/32), and converted the 6 inches to .5 feet. Furthermore, I got k=64 lb/ft
So my equation d^2x/dt^2=-64x+beta*dx/dt
->d^2x/dt^2+64x-beta*dx/dt=0
so m^2-beta*m+64=0. To be oscillatory, sqrt(beta^2-256) has to =0, so Beta=16?
I got a mass of 1 slug (32/32), and converted the 6 inches to .5 feet. Furthermore, I got k=64 lb/ft
So my equation d^2x/dt^2=-64x+beta*dx/dt
->d^2x/dt^2+64x-beta*dx/dt=0
so m^2-beta*m+64=0. To be oscillatory, sqrt(beta^2-256) has to =0, so Beta=16?
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From what you have written, you need the discriminant of the quadratic equation
β^2 - 256 < 0 <==> |β| < 16 for oscillatory behavior.
Since β > 0 for this problem, the possible values of β are given by 0 < β < 16.
I hope this helps!
β^2 - 256 < 0 <==> |β| < 16 for oscillatory behavior.
Since β > 0 for this problem, the possible values of β are given by 0 < β < 16.
I hope this helps!