Hmmmm. If a ship weighs 1000 metric tons ( wild guess) this equals 1 billion grams. The acceleration of gravity at the equator is 980.66 cm/sec/sec and the centrifugal acceleration due to earths rotation is 3.4 cm/sec/sec. The acceleration due to the Moon at a mean distance of 3.844 x 10^10 cm with a mass of 7.35 x 10^25 grams is about 0.0033 cm/sec/sec.
This means that a billion gram Tanker will have a weight that varies by as much as 3.4/980.66 x 1 billion = 3.5 million grams due to its proximity to the equator where it will be lightest! As for the Moon, its gravitational acceleration is 0.0033/980.66 x 1 billion = 3.36 kilograms lighter when the Moon is roughly above the horizon or over head.
All this is approximate, but probably not too far off in order of magnitude!
All answers are provided by Dr. Sten Odenwald (Raytheon STX)
NASA IMAGE/POETRY Education and Public Outreach program.