Mass/Spring System Damping Constant

[Theriot2]

A mass weighing 32 pounds stretches a spring 6 inches. The spring constant is equal to 64 lb/ft.The mass moves through a medium offering a damping 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.



2*$$\lambda$$=$$\frac{\beta}{m}$$

$$\omega$$$$^{2}$$=$$\frac{k}{m}$$

$$\lambda$$$$^{2}$$ - $$\omega$$$$^{2}$$>0 is overdamped

$$\lambda$$$$^{2}$$ - $$\omega$$$$^{2}$$=0 is critically damped

$$\lambda$$$$^{2}$$ - $$\omega$$$$^{2}$$<0 is underdamped

1 slug = 32 pounds

I've solved that $$\beta$$ is equal/less than/greater than 2*$$\sqrt{k*m}$$=32, but I don't understand when it will or will not have oscillatory motion.

 

[jack action]

Underdamped --> There is an oscillatory motion
Overdamped --> No oscillatory motion (gradually goes to final position)

So the limit that separates the two is the critically damped situation.

 

[Theriot2]

Thank you!