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# 9.3: Conservation of Angular Momentum

Newton masterfully summarized the science of motion in three laws. He had an additional insight, based on the concept of momentum. Momentum is simply defined as the product of mass and velocity:

Momentum = M v

Normally we call this linear momentum because it refers to motion in a straight line. Momentum incorporates mass and motion. A stationary object has no momentum. The faster an object moves, the larger its momentum, and the more massive an object is, the larger its momentum. The idea is familiar to you: a car moving at 20 mph has much more momentum than a bicyclist moving at 20 mph, and can do a corresponding amount of damage! Newton knew that momentum could change when objects collided or interacted. But he realized that the total amount of momentum was the same before and after the interaction. This is the law of conservation of momentum.

If you hold a gun it has no momentum. After it is fired, the bullet travels at very high speed so its momentum is the product of a very small mass and a very large velocity. The recoil of the gun is exactly the same amount of momentum in the opposite direction — it has the opposite sign so it cancels out the gun’s momentum. To keep the recoil small, the gun is made heavy so that its momentum is the product of a large mass and a small velocity. The key point is that the total amount of momentum has not changed. Zero momentum before has become the sum of two equal and opposite amounts of momentum after, which is also zero. You would see the same effect if you stood on the ice wearing skates and threw a heavy rock away from you. The momentum you gave the rock would be balanced an equal momentum propelling you backward. But since your mass is much larger, you backward velocity would be smaller than the rock’s forward velocity.

Many people wonder what makes a rocket fly. How does it stay up in the air? What does it "push" against when it is in space? The answer is clear if you think of momentum. To create upward motion and momentum of the rocket, fuel is burned and ejected at high speed backward. In fact, rocket and jet nozzles are specifically designed to make the exit speed of the hot gas as high as possible. At every point in its flight, the forward momentum of a rocket is exactly balanced by the backward momentum of the fuel vapor. The rocket has large mass and moderate velocity and the gas has small mass and extremely high velocity. It makes no difference whether the rocket is in air or in the vacuum of space!

The Space Shuttle Atlantis taking off, using the momentum of the fuel ejected from the rocket engines. Click here for original source URL.

Angular momentum is just like linear momentum but it applies to rotating or orbiting systems. It is defined as the product of mass, velocity, and radius:

Angular momentum = M v r

In this case the radius is the size of the rotating object or the distance of an orbiting body from the center of gravity. The law of conservation of angular momentum says that angular momentum will stay constant as a system changes its configuration.

If the Solar System really collapsed from a gas cloud that extended at least to the orbits of Neptune and Pluto, then the rotation speed must have increased greatly. By how much? The mean distance of Neptune from the Sun is 30 A.U. or 30 × 1.5 × 108 = 4.5 × 109 kilometers. If we assume that all the material within this orbital radius ended up in the Sun, then after collapse the size of the cloud is just the radius of the Sun, or 700,000 kilometers. In the product Mvr, the mass has not changed. So if the radius r has decreased by a factor 4.5 × 109 / 700,000 = 6500, then the rotation velocity must have increased by the same factor. It turns out that the Sun does not spin at the high rate we would expect, leaving us with a mystery in our understanding of the formation of the solar system.

Composite image of the Crab pulsar in optical (red) and x-ray (blue). Click here for original source URL.

Another example of changing angular momentum is the Earth in its orbit of the Sun. The Earth’s orbit deviates from a circle by 3.4%. This means it varies from 1.017 times the mean Earth-Sun distance to 0.983 times the mean Earth-Sun distance. Since angular momentum is conserved, when the distance goes up the velocity must go down, and when the distance goes down the velocity must go up. The mean orbital velocity of the Earth is 2πr / (Time in a year) = (2 × 3.14 × 1.5 × 108) / (365 × 24 × 3600) = 14.94 kilometers per second. So when the Earth is closest to the Sun its speed is 14.94 × 1.017 = 15.18 kilometers per second. When it is farthest from the Sun its speed is 14.93 × 0.983 = 14.69 kilometers per second. This slight variation in orbital speed affects timekeeping based on the stars.

Diagram, exaggerating the elliptical orbit of the Earth. Click here for original source URL.

There is a concrete and physical explanation for Kepler’s second law of planetary motion. This law says that an imaginary line that connects a planet (or a comet or any other orbiting body) to the Sun sweeps out equal areas in equal intervals of time. Kepler’s law is no more than a statement of the law of conservation of angular momentum. Orbital velocity depends on distance to the Sun, but both vary so as to keep the angular momentum constant.

Animation of Kepler's second law of planetary motion. The same (blue) area is swept out in a fixed time period. The green arrow is velocity. The purple arrow directed towards the Sun is the acceleration. The other two purple arrows are acceleration components parallel and perpendicular to the velocity. Image used with permission (CC BY-SA 3.0; Gonfer).