PHYA10H3 Lecture 1: CM2

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Department
Physics and Astrophysics
Course
PHYA10H3
Professor
Benj Hellie
Semester
Fall

Description
Classical Mechanics 2 Isaac Newton discovered that his Law of Gravitati on could be applied to a falling apple and to the Moon in orbit around the Earth. The same force acts in both situations. In the absence of air resistance, an apple released above the surface of the Earth accelerates to the ground. The Moon also accelerates towards the Earths surface. Ho wever, since the Moon has a large sideways motion, it avoids hitting the Earth. Elliptical orbit of the Moon around the Earth (not to scale). The arrow indicates the direction the Moon would travel if the Earth was not present. Due to the gravitational attraction of the Earth, the Moon fa lls below the path of the arrow and follows the curved orbit. The successful application of Newtons Laws in determining the motion of the planets in the Solar System led Pierre Simon de Laplace (1749 1827) to conclude that the universe behaves in a deterministic manner. If we knew the masses of, forces on, and initial positions and velocities of every object, then we could predict their future trajectories or paths exactly. Although this is true in many situations, it is not always possible. Henri Po incare (18541912) calculated the evolution of three bodies inte racting via Newtons Law of Gravitation and found that some configurations were highly unstable. It was possible under certain conditions for one of the objects to be ejected from the sy stem. Poincare discovered an ex ample of what is now called chaos. Chaos is a state of a system in which the initial c onditions strongly influe nce the trajectories followed by objects in the system. There is a strong sensitivity to th e initial conditions. Computer simulations of the Solar System indicat e that some of the asteroids and planets may exhibit chaos over thousands or millions of years. It is hard to make accurate forecasts of the weather because of the chaotic behaviour of the Earths atmosphere. Here is a comparison between a simple and a do uble pendulum. Suppose that the initial velocity of the objects is zero in all cases. A simple pendulum that is allowed to move freely is non chaotic. Whether its initial position is x=20.0 cm or x=19.9 cm, the pendulum will swing in a regular manner between 20.0 cm and 20.0 cm or 19.9 cm and 19.9 cm, respectively. x=20.0 cm 0 x=20.0 cm
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