I am being told to go to PT by the command on only 4-5 hours of sleep. As a junior enlisted, how do I solve this problem?

As a junior enlisted personnel, it shouldn't be a problem for you to explain your situation to your leader(s). If your work schedule is morning or afternoon, you may need to adjust your time to allow for PT, late evenings with friends or weekend outings, does not figure into ones personal responsibilities. Their are many places around the military, where a soldier is responsible for his\her own fitness and training for physical fitness test. Most training manuals will tell you how an exercise is to be done. The unit or organizational policy will outline the training as well as who should be in attendance of physical fitness training, all the way down to the type of PT to be conducted should a soldier be medically limited to certain exercises. It would seem, your problem can be fixed and it would most likely involve you being committed to doing what is expected of you, even if you have to do it on your own, before or after work. You can come up with your own solution, but keep in mind, it would need to be heard and agreed upon, by you supervisor or someone directly involved in your chain of command.

You have a problem with a 5:30 PT formation? That was perfectly normal when I was in. Even after all night CQ duty, 2 hours of guard duty in the middle of the night, a 12 hour overnight shift in the border ops center, a late night getting gear and uniforms squared away or just partying with buddies into the wee hours of the morning. Sounds like your not Army material. Do your time stay out of trouble and find a new career.

Manage your time better, suck it up, and realize that this too will pass.
I see this was originally posted a year ago and that you were promotable -- hopefully it all worked out for you without too much wailing and gnashing of teeth.

@SPC(P) (Verify To See) Suck it up son your in the Army now!!!

For now just go to the one that doesn't conflict with your duty shift. Ensure that when they take muster for PT you are acounted for. If the person you report to in the hospital is other than the person you report to for PT make sure that person knows. Adjust your person schedule around PT and duty. If your section rotates the time they stand duty, just suck it up. You'll be rotating out of your problem, and someone else will be rotating in. If sections don't rotate shifts you could always request a change of section. They might say no, but if you don't ask you are saying no. Either way it may be very inconvenient, may be very difficult to do you best, but I think your orders that you are complaining about are still legal, and you could be setting yourself up for a world of hurt, starting with some NJP if you don't comply.

Show up, do PT. allow your Squad/ PLT/Section Leadership try and convince the 1SG to adjust his views

As I re-read this thread, I can't help but notice... wow, there are a lot of people willing to shout "old school" as they justify not taking care of their Soldiers. Beating your chest about how hard-core you are isn't a substitute for leadership.

While stationed in Germany there were many PT formations and for PT we were in our clubbing clothes. Rolling through the hole in the gate crossing the farmers field just in time for roll call. Our first sergeant did care what you were wearing. BUT if not in PT clothes you were in the front of the pack and no falling out to puke. Indian sprints. Motto: adapt and overcome.

I think it would be best for both parties if you requested a discharge because you are not able to adapt to military discipline and regimentation. Second choice would be for you to request reclassification into the Double 0 Grunt world where you could work 24 hour shifts everyday and still make all required formations or get kicked out for the good of the army. The military force of the USA is not designed to be a democracy where you can be given all the individual pleasures you want.

Why not ask to move afternoon PT to 1430 or 1500? Barring that, I assume their are NCOs on your shift; why not suggest they run a PT formation? Seems to me this is looking for a solutions that meet both needs of the Army (your physical training) and supporting the important work you do. Presenting these options to your leadership in a respectful and constructive way shows that your on the team and developing the skillset to problem solve and become a leader yourself.
I would hope that they would listen, but you have to understand they are acting within their responsibliites and that their is no recourse but soldier up if they decide not to alter their decision. I don't know the circumstances, but try to put yourselves in shoes of the leadership. One of their obligations is to maintain the physical health of the force IOT prepare for the stress of combat (and the enormous stress casualties bring to a hospital), meet standards, mitigate having to separate, counsel, and run an overwieght program for non-complying soldiers, and most of all to look out for your individual health--as a health worker you already know that poor physical condition translates into poor mental condition and job performance decrement.