1937-1954 "rear ends" are basically the same. The most common parts to fail are
#1-front pinion bearing. Will "whine" during acceleration. Other wise pinion shaft would need to be removed to inspect.
#1-rear pinion bearing. When "bad" will make noise while backing off.
#3-side carier bearings. Makes a grumbling at steady speed. Rear axle bearings makes a similar noise.
#4-worn or improperly adjusted ring gear and pinion. Check wear pattern on gear teeth.
#5-Condition of side and pinion gear and the shat the pinion ears ride on.
Replacement or inspection of most of the above parts require disasembly of differential. Assembly requires a dial indicator and shop manual.
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Chevgene