Social Security Disability is not an area I am particularly familiar with. As to rent and late fees, there is no Utah law which would prohibit late fees of this sort. Quite simply, if you agree on a due date and do not pay on time, Utah law permits late fees and the landlord is entitled to demand them.
You have various options which do not really involve the application of the law. I understand that your finances are very tight. Do not misunderstand or feel that I do not sympathize--I am just trying to suggest possible ways of dealing with the situation.
1. You could rearrange your finances so that you have enough money available when the rent is due. If you are a member of any local religion which engages in charitable activities, they might pay your rent for one month to permit you to achieve this goal. Perhaps a family member would be willing to make a gift to you to help you in this regard.
2. If your lease is nearly over and you are willing to move, you could consider seeking another place to rent which will permit you to pay rent on a more convenient date.
3. I suspect that in reality the apartment manager probably could modify your rent due date if he wanted--perhaps you could offer to pay another ten days rent in advance in exchange for having the due date modified.
Providing this answer does not form an attorney-client relationship. Most legal questions are exceedingly fact-sensitive and therefore this answer is a best-guess based on the information you provide. You should consult an attorney licensed in your state to further discuss your matter.
Along with what Mr. Koyle stated, you may also find that if you contact the owner of the property, that he or she may be willing to adjust the payment dates so that no late fees are incurred. I represent various landlords that are happy to make adjustments for good tenants. It has been my experience that property owners (not necessarily the manager) want the income stream and in this economy good tenants are hard to come by. The owner probably doesn't want you to leave. They would then have to spend the time and money to fix up the property and try to find a new tenant.
** LEGAL DISCLAIMER I am licensed to practice law in the State of Utah. My practice includes the areas of landlord/tenant litigation. I am ethically required to state that the above answer does not create an attorney/client relationship. My responses should be considered general legal education and are intended to provide general information about the question asked. Many times, the question does not include important facts that, if known, could significantly change my answer. Information provided on this site should not be used as a substitute for competent legal advice from a licensed attorney that practices in your state. The law changes frequently and varies from state to state. You should verify and confirm any information provided with an attorney licensed in your state.