I'm going to ask to share gym equipment
Why do you need to ask someone to share their gym equipment?
The work is a lot faster when you share it with someone else.
It's good that you want to be efficient, just make sure that the other person is comfortable with sharing with you. When you pressure someone into sharing with you, you may make them uncomfortable. This can lead to them avoiding you, so be sure that everyone is on the same page.
Oh well, I can wait. It's only because I thought that today is their turn.
Well if it is their turn with the gym equipment I think you need to be patient and wait for them to finish before using it, or at the at least try to just use a different equipment as it will be unsanitary if you use it right away after their turn.