Measuring the tidal effect of the moon on the earth

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I'm an undergraduate physics major and I want to create an experiment where I measure the difference in gravity between sunrise, noon, sunset, and midnight (and everything in between for that matter) due to the tidal effect from the moon and sun. I did some calculations and I expect the difference in gravity due to the sun to be on the order of 10^-6 m/s^2 and due to the moon to be about 3 times that.

My dilemma is how to measure that tiny difference in gravity.

I have yet to come up with a good way. Any scale I could get that's accurate enough couldn't handle the large mass needed to generate a tiny difference in force. I've thought a little about using interferometry to try to measure small variations in the deformation of a wire or a bar or something, but that still seems complicated because I'd have to know the exact properties of the material I am using.

Any ideas?
 

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