How long will it take for the rod to stop moving?

We can see from the previous plot that the vibration dissipates, and thus we infer that the rod will eventually stop moving. But, strictly speaking, the exponential term

"tends to zero asymptotically", but never reaches absolute rest. To answer the question of when the rod will stop moving, we must define our terms carefully.

We will use the total energy

of the rod as a proxy for movement.

is the sum of the rod's potential and kinetic energies. Its formula is given below $($without proof$).$

is the mass of the rod, which we assume to be $0\mathrm{.}8$ Kg$.$ We will say that the rod is motionless if its energy is less than a threshold value

which we take to be

joules $($the SI unit of energy$).$

Your task is to find the setting time Ts of the rod, which is defined as the earliest time that the rod is motionless. We will compute this in several steps.