Lesson 20 Angular Momentum

Kepler’s second law – an imaginary line from the sun to a planet sweeps out equal areas equal times   - applies to the vortexes of hurricanes, firestorms, and bathtubs, the center of this swirling motion is the law of conservation of angular momentum.

During the 16^th and 17^th century the Renaissance came to an end and Europe turned to Christian theology. This period was known as the Reformation, and afterward the Counter Reformation developed during the Thirty Years War. During this time, Galileo, Kepler, and Tycho Brahe all contributed to the study of the universe and physics. Kepler was a professional astrologist that caste horoscopes in order to make ends meet. Kepler almost discovered integral calculus over the dispute of the shape of a wine barrel, and eventually developed three laws that describe planetary motion.

Kepler’s second law in particular explains the shape of all galaxies as a factor of the conservation of momentum. Astronomy was the first physical science before medicine. In fact, early astronomers attempted to make a model of heaven and earth.

The Greeks made many models, including Aristotle, who in 300 B.C. placed the Earth at the center of the universe, as the sun and all other planets rotated around it in uniform circular motion. Astronomers continued to develop complex models with epicycles, however they were not correct. Copernicus later placed the planets around the sun, but kept the uniform circular motion of the bodies. This belief was scientifically accepted until Kepler found the three laws of planetary motion.

One law in particular (Kepler’s second law) states that a planets orbit sweeps out equal area in equal time.

The area of the parallelogram formed by the two vectors r and (delta) r is the cross product.

(delta) A = ½ r x (delta) r because half the area of the parallelogram would be equivalent to the triangular area swept out by the planet’s orbit.

(delta)A/(delta)t = ½ r x (delta)r/(delta)t

dA/dt = ½ r x v is equivalent to the rate of change of the area vector, which is constant in all parts of a planet’s orbit. This means that as the distance r increases, velocity decreases and as the distance r decreases, velocity increases.

When momentum is angular there is a twist known as torque. Even if force is applied to a body, the torque may be zero. If the twist is zero than vector r x F = 0

Gravity and fluid pressure are the main influence on vortexes.

F = 0, therefore integral was constant p

dL/dt = r x F

d( r x v)/dt = dr/dt xv + rx dv/dt

= r x dv/dt

= r x a

F = ma

d(mr x v)/dt = r x ma

= r x F = T or torque

L = mr x v

r x F = 0 so mr x v is constant as angular momentum constant L

A planet in orbit has no torque so it conserves angular momentum. That’s why Kepler’s second law is true. According to the law of inertia, a body moving in a straight line will continue at constant speed unless a force is applied to it. In a similar way, a spinning body rotates in the same direction unless a twist or torque is applied. This is the essence of the law of the conservation of momentum.

Angular momentum L is a vector following the right hand rule

L = mr x v

L = mrv when v and r are perpendicular

If orbit is smaller speed must increase to maintain L as a constant.

Bathtubs, tornadoes, and hurricanes all exhibit angular momentum.


Water lacks enough viscosity to apply torque. In tornadoes calm air towards the center moves upward forcing circulating air upward to replace it. Jupiter’s Red Spot is an enormous hurricane that has lasted for at least three hundred years since it was first observed, due to the principles of inertia. Furthermore, the Solar System, the planets, and the galaxies exhibit flat disc shaped planes with globular masses at the center due to the conservation of angular momentum.

Below is a video that explains the concept of angular momentum in depth:

Lesson 19 Torques and Gyroscopes

When a torque acts on a spinning object the angular momentum changes. The rate of change of angular momentum is equal to the applied torque. Under the influence of a torque, a spinning object precesses.

All conservation laws come from F = m a

Newton's Mechanics based on three laws, however everything we do is based off of F = m a

Newton's third law implies conservation, thus a torque on one body implies a reverse torque on another. Angular momentum flows from one body to another and angular momentum is conserved.

The advancement of cultures over the course of history can be traced by advancements in transportation. Transportation dramatically changed with the creation of the wheel.

The force of gravity naturally tips the wheel down. However,  this is different with a spinning wheel.

When there is angular momentum there is a twisting force that changes the angular momentum. When angular momentum is conserved r x F (the radius vector and the force vector) is conserved.

A spinning wheel already has angular momentum, and torque can change that.

T = dl/dt

Below is a video that explains the concept of torque:


Gravity plays a strange trick on a spinning wheel. The force of gravity is downward, however the force of torque is sideways. This is why a spinning wheel will precess when stationed on an axis. When gravity tries to make a non-spinning wheel fall, the spokes push the top of the wheel outward rather than downward, which causes the weel to fall. This creates a component of velocity in the outward direction. This same thing happens at the bottom of the wheel but the force of gravity pulls the wheel inward.

This same idea also occurs when the wheel spins, however, when the outward velocity builds up it is at the side of the wheel. The force is outward on one side and inward on the other -- each spoke exerts its own push at the top -- causing a jog at the side. The net result is a downward but sideways motion when the wheel spins (precessing).


A top and bicycle wheel are a lot alike. A top acts as if all its mass is concentrated at one point, known as the center of mass. When the top is tilted there is a torque created. When this happens to a spinning top the torque nudges the anguar momentumof the top around in a little circle. Heat is produced at the tip of the top where energy is transfered into a random motion of molecules.

If a gyroscope is balanced at its own center of mass, the force of gravity will not make it precess. Even if it supports swivel in all directions the axis will remain stable and keep the gyroscope pointed in the same fixed direction.


Gyroscopes are used in planes in order to fly over magnetic poles using computer aide. Gyroscopes are improved by removing torque.

:C or capital omega is the rate of precession

U = wr

T = :C L

:C = Rg/wr^2

:C is just torque divided by angular momentum.

An object will precess least if wheel is large, spinning fast, and carefully balanced.

The best gyroscope on earth is the earth itself. The plane of the equator where day and night are equal length intersects the earth's orbit at the equinoxes. These points are used to denote seasonal changes and slowly dift to the west each year in the ever expanding universe.