The U.S. Army was formed in 1775 as the Continental Army. The army's job is to protect the country from land attacks. As with most military forces, the army has its own logo. The Department of the Army Seal and the Department of the Army Emblem serve as this logo. The seal was created in 1778 and the emblem in 1947. While both the seal and the emblem are still in use today, few people actually know what they look like.
Most people are actually more familiar with the image used on this US Army Patch. It features the green background with the circle and star on it. The eagle on the front of the star is actually not associated with the army at all. Instead, it is from the Great Seal of the United States.
So what does the actual army emblem look like? It features a roman cuirass (a type of armor) surrounded by a number of different symbols. There is an unsheathed sword, a Phrygian cap, an espontoon (a type of pole arm), a U.S. flag, a musket with bayonet, several cannonballs, a drum, two drumsticks, and the flag of the Department of the Army. Overall of this is a rattlesnake with a banner that reads, 'This we'll defend'. It's a busy design that few people, especially those who aren't in the army, would recognize.
Of course, each of these items on the emblem represents something, but it would be difficult to create a small patch that has all of these different items done in any detail. Since few people would even know what the symbol was for, many patches instead opt to use the eagle holding an olive branch and arrows.
