Question

Will damped oscillations occur for any values of b and k? Explain.

Short answer

No

Step-by-step solution

Types of damping

There are 4 different behaviors that depend on damping constant b, out of which in under damping$0<b<2m{\omega }_0$ Decaying oscillation. A larger value of b leads to faster decay of an oscillation.

Over damping$2m{\omega }_0<b$

Very slow monotonic decay

If $b^2>4mk$, the system is over damped

Explain the reasoning

No. If the resistive force is large compared to the restoring force of the spring (in particular, if$b^2>4mk$ , the system will be over damped and will not oscillate.)