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I know the rifling in a gun or rifle puts a spin on the bullet along the axis of trajectory. Now I don’t understand exactly what does it make the trajectory more stable and the travel grater?

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Bullet is not of perfect shape, and these imperfections will force bullet to flip. When bullet spins, these imperfections work in all directions equally, so bullet does not flip. This cause near-perfect trajectory. Flipped bullet have higher air drag -> reduced range.

Also, please note, that even bullet spin does not guarantee that bullet would not flip when going from supersonic to subsonic speed.

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Is there something about how the spin changes the aerodynamics. But what I'm relay wondering about is if the gyroscopic behavior of something so small has an influence. –  Fortunato Jul 4 '11 at 8:18
    
Spin itself does not change aerodynamics by much. But the fact that bullet flies in the proper orientation (nose - forward) all the way does improve aerodynamics (comparing to bullet flipping randomly during the flight) –  BarsMonster Jul 4 '11 at 9:09
    
@Fortunato: Actually, it is due to gyroscopic effects. It has to do with angular momentum being a vector--which requires torque to change. See the other answers. –  Manishearth Mar 30 '12 at 3:04
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One reason is due to angular momentum.

When a body is spinning around some axis, we say the angular momentum is pointing along this axis with a magnitude equal to the magnitude of the angular momentum. When a body has angular momentum is some direction, much like with regular momentum, it doesn't like to have its magnitude or direction changed.

One particular note to this is precession, where the angular momentum of a body spins about a 3rd axis, but it doesn't like to have this 3rd axis changed either.

So, when the bullet has some spin, it has some angular momentum in the direction of its motion. This spinning adds stability, because the bullet itself doesn't want to turn on some other axis, thus changing the direction of its angular momentum, so it stays pointing straight. Since it stays pointing straight, it is more aerodynamic, so it flies for longer.

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It's called precession.

The air friction and other hydrodynamical forces a bullet experiences often exert a net torque that flip the bullet, if it is not spinning. When it spins, however, it will precess and thus remain stable, like a spinning top resisting gravity.

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