I am locked in a room with no windows and I need to tell if the room is moving ...
The actual problem is completely different than you think:
What exactly does "moving" mean?
According to special relativity theory, there is no "fixed" coordinate system in space, but you always have to specify some coordinate system relative to some object or similar when talking about motion.
You sit on your chair in your garden. How fast do you "move"?
If you are using a coordinate system which is relative to the earth, you are not "moving".
If you are using another coordinate system, you may be moving at the speed of about 1000 km/h (depending on where you live) due to the rotation of the earth around its own axis or even with 100000 km/h due to the rotation of the earth around the sun.
So the answer to your question is completely different than you think:
Can I determinate if my room is moving ...
If you use the coordinate system which is relative to your room, your room does not move. So you don't need to do any measurement to determine if your room moves: It does not move!
If you use a coordinate system which is relative to some other object (for example the earth), the question "Can I determinate if my room is moving?" is the same question as "Can I determine if the other object is moving relative to my room?"...