Circular Motion Chapter 9

Uniform Circular Motion The movement of an object at a constant speed around a circle with a fixed radius is called uniform circular motion. 𝒗 = ∆𝒓 ∆𝑡 Velocity is always tangent to the circle. v1 v2 r1 𝒂 = ∆𝒗 ∆𝑡 Acceleration is constant and directed toward the center of the circle. r2

Centripetal Acceleration - 1 For uniform circular motion the acceleration is called centripetal acceleration and is constant, directed toward the center of the circle. The magnitude of the centripetal acceleration is 𝑎 𝑐𝑒𝑛𝑡𝑟𝑖𝑝𝑒𝑡𝑎𝑙 = 𝑎 𝑐 = 𝑣 2 𝑟 (directed towards center of circle) Newton’s 2nd Law for Circular Motion: The net centripetal force on an object moving in a circle is equal to the object’s mass times the centripetal acceleration. 𝐹 𝑐𝑒𝑛𝑡𝑟𝑖𝑝𝑒𝑡𝑎𝑙 = 𝑚 𝑣 2 𝑟 (directed towards center of circle)

Centripetal Acceleration - 2 We can measure the speed of an object moving in a circle by measuring its period (𝑇), which is the time needed for the object to make one revolution. In one revolution the object travels a distance 2𝜋𝑟 so its speed is 𝑣= 2𝜋𝑟 𝑇 𝑎 𝑐 = 𝑣 2 𝑟 = 2𝜋𝑟 𝑇 2 𝑟 = 4 𝜋 2 𝑟 𝑇 2

Examples of Uniform Circular Motion Hammer Throw Satellites Roller Coaster

Circular Motion Circular motion vignettes Science of the Winter Olympics Science of Golf

Circular Motion Activities Uniform Circular Motion Interactive Complete Activity Race Track Interactive

Universal Gravitation Newton realized that the Moon doesn’t follow a straight line path, but instead circles the Earth. He knew that circular motion is accelerated motion, which requires a force ( 𝑭 𝑛𝑒𝑡 =𝑚𝒂). He then reasoned that the Moon must be falling towards Earth for the same reason that other objects fall—because of the pull of gravity. By this reasoning process Newton arrived at his law of universal gravitation that describes the force between any two objects with mass m1 and m2 separated by a distance r: 𝐹 𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 = 𝐺 𝑚 1 𝑚 2 𝑟 2 where G is the universal gravitational constant: 𝐺=6.67 × 10 −11 𝑁∙ 𝑚 2 𝑘𝑔 2

Satellite Motion For a satellite orbiting the Earth, the net centripetal force is 𝐹 𝑛𝑒𝑡 = 𝑚 𝑠𝑎𝑡 𝑎 𝑐 = 𝑚 𝑠𝑎𝑡 𝑣 2 𝑅 where msat is the mass of the satellite and R is its orbital radius (measured from the center of the Earth. But since the net centripetal force is supplied by the gravitational force we can set the gravity force equal to the net centripetal force; 𝐹 𝑛𝑒𝑡 = 𝐹 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 𝐺 𝑚 𝑠𝑎𝑡 𝑀 𝐸𝑎𝑟𝑡ℎ 𝑅 2 = 𝑚 𝑠𝑎𝑡 𝑣 2 𝑅 𝑣= 𝐺 𝑀 𝐸𝑎𝑟𝑡ℎ 𝑅

The Relation Between G and g We previously used the equation 𝐹 𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 =𝑚𝑔 for the gravitational force, where g is the acceleration due to gravity on Earth. Setting that equation equal to the force in Newton’s law of universal gravitation gives us: 𝑚𝑔= 𝐺 𝑀 𝐸𝑎𝑟𝑡ℎ 𝑚 𝑅 2 𝑔= 𝐺 𝑀 𝐸𝑎𝑟𝑡ℎ 𝑅 2 So we can see that the acceleration due to gravity is dependent upon the mass of the Earth (approx. 5.98x1024 kg) and the distance (R) that an object is from the center of the Earth.

Gravitation Videos How to Think About Gravity Gravitation Tutorial

Kepler’s Laws

Kepler’s Laws In the 1600s Johannes Kepler proposed three laws of planetary motion based on observational data. The laws are: The path of the planets about the sun is elliptical in shape, with the center of the sun being located at one focus. (The Law of Ellipses) An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. (The Law of Equal Areas) The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. (The Law of Harmonies)

Law of Ellipses The path of the planets about the sun is elliptical in shape, with the center of the sun being located at one focus. (The Law of Ellipses)

Law of Equal Areas

Law of Harmonies Law proposes that the ratio 𝑇 2 𝑅 3 is the same for all planets Data for our solar system: Planet Period (years) Avg. Distance (AU) 𝑻 𝟐 𝑹 𝟑 Mercury 0.241 0.39 0.98 Venus 0.615 0.72 1.01 Earth 1.00 Mars 1.88 1.52 Jupiter 11.8 5.20 0.99 Saturn 29.5 9.54 Uranus 84.0 19.18 Neptune 165 30.06 Pluto 248 39.44

Kepler’s Laws Activities Planetary Orbit Simulator Which planet has the most eccentric orbit? Which planet’s orbit is closest to circular (e = 0)? At which point is each planet going fastest? Slowest? Kepler’s 2nd Law Interactive Orbits and Kepler's Laws 

Quiz on Wed. 11/4/15 Momentum (Chapter 7) Energy, Work and Power (Chapter 8) Circular Motion (Chapter 9) Universal Gravitation (Chapter 12)

Today’s Lab Read the tutorial Definition and Mathematics of Work Do the interactive Stopping Distance and complete the worksheet