Almost sounds like the Cavendish experiment.
If the moon pulls on the tides, does it pull on anything else?
Cavendish did a pendulum for constant mass interactions.
My experiment idea focuses on the difference of masses. The ideal experiment would be to teleport an object of large mass into a field. That sudden appearance of mass would create a new force on the object.
Think of teleporting the earth next to a space craft moving in a straight line. Or even just floating in space and then suddenly falling because earth appeared below you.
The spring retracts the second large mass a huge distance relative to the point where the force was measured, making it essentially "disappear", and the second mass would then feel a sudden (and hopefully strong) recoil effect because of the now missing mass.
The vacuum would try to make sure no air currents pushed on the first mass that would make it sway or whatever.
Can you think of an experiment without teleporting? Or a vacuum?
Something you can do at home...
Depends on how much you want to spend.
There are a couple ways to something homemade - your tide idea is pretty good.
If you have an accurate scale, you could set something on it and look up the "tide charts" for your area (not sure if there are such things for like Kansas, but there might be) and see when the moon is overhead and directly away.
The tradeoff is that a very accurate scale, think 0.001grams, can't measure a lot of mass, and only a little mass probably won't change by too much.
In any case, put something on the scale, I would suggest whatever the maximum measurable mass is. If it is a 50.000g scale, put 49g on it, etc.
Then take 8 readings, one for each high and low tide time (4) and then inbetween those.
the 4 inbetween times should give you something of a "neutral" tide reading, and then the highs and lows should give you measurable changes from the inbetween value.
To be most accurate, you should set up your mass at an 'inbetween' time, and then measure to make sure that number reappears at the other 3 'inbetween' times.
The problem with a low reading scale, is that the mass of the scale is probably the same or higher than the mass you are trying to measure, meaning the tide would pull the scale upwards stronger than the mass itself, giving weird readings.
But any difference in readings is a step in the right direction.
It pulls on everything else. We can't see air, otherwise we would most likely see small air tides up to 20 miles or so.
Nothing else has as much freedom to rearrange as liquids or gas. Dirt wants to move but can't.
I guess you could suspend a weight on a scale and measure the difference as the moon goes by.
We can see clouds. Are they attracted to the moon?
Physics says yes, but with wind patterns and other temperature effects, the moon's attraction is definitely minor.
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