# Can the minimum kinetic energy of a simple harmonic oscillator be non zero?

I was practicing physics problems when I faced one which said that minimum kinetic energy of a simple harmonic oscillator (in the question it was a spring block system) can be non zero.

However, it occurred to me that it cannot be possible since in such a motion a time will come when the spring will be stretched to its maximum and the velocity of the block will change direction. So, at a time between the change in direction of the velocity there will be an instant of zero velocity where the kinetic energy will be zero(the instant of minimum Kinetic Energy).

So who is right here, and if the minimum kinetic energy of a simple harmonic oscillator can be non zero then cite an example.

Edit: Sorry the question is actually about "minimum Kinetic Energy"

• The kinetic energy is $\frac12 m\dot{x}^2.$ Plug in $x(t)$ for the standard harmonic oscillator and you'll see it's non-zero for certain $t$, as expected... Jan 22 '17 at 19:51
• There are times when it is zero and times when it is not. The total energy, however, is always non-zero and positive. Jan 22 '17 at 22:54
• I am sorry for the inconvenience caused. The question is about the minimum KE not just KE. I have updated it, and if you find time then do upvote the question, if you feel it correct this time. Thank You. Jan 23 '17 at 16:03

So who is right here, and if the kinetic energy of a simple harmonic oscillator can be non zero then cite an example.

A simple harmonic oscillator has to have a non-zero velocity at some point by definition.

The point of a simple harmonic oscillator is that it oscillates, or moves back and forth. If there were no kinetic energy (velocity), it wouldn't move.

Can you site any examples of a harmonic oscillator that has zero velocity?

• Well, a classical oscillator with zero total energy would be permanently at rest at the equilibrium point. But even that special case is not possible in QM. Jan 23 '17 at 0:55
• Actually the question is about minimum Kinetic Energy. Sorry for the inconvenience caused. Jan 23 '17 at 5:04
• I am sorry for the inconvenience caused. The question is about the minimum KE not just KE. I have updated it, and if you find time then do upvote the question, if you feel it correct this time. Thank You. Jan 23 '17 at 16:04

There are times where there is zero kinetic energy in a harmonic motion. However, there are times where it is non-zero. And the question is just whether it can be non-zero. And it certainly can be non-zero.

Update to reflect the updated question:

The minimum kinetic energy can be non-zero if the oscillator is two dimensional. Then for every trajectory that is circular or elliptical, the pendulum would never completely stop.

• Please refer to the recent edit. Jan 22 '17 at 19:33
• What's to refer to? His answer remains correct. Keep in mind that while total energy is conserved the kinetic and potential energies can (and do) both change as a function of time: it is only their sum which remains constant. Jan 22 '17 at 21:49
• Actually the question is about minimum Kinetic Energy. Sorry for the inconvenience caused. Jan 23 '17 at 5:03
• I am sorry for the inconvenience caused. The question is about the minimum KE not just KE. I have updated it, and if you find time then do upvote the question, if you feel it correct this time. Thank You. Jan 23 '17 at 16:04