Our machines are usually powered by the energy of electricity or heat; light is another form of energy, converted into electricity by the solar cells which power most artificial satellites. Gravity can also provide energy. The wheels of grandfather clocks are turned by weights which gradually descend to the bottom of the clock, at which point they must be cranked up again, or else the clock stops. Thomas Jefferson, at his home near Charlottesville, Virginia, had a clock whose weights hanging on the side of a room were cannonballs strung on a rope, and to give the clock a 7-day range, a hole was cut in the floor allowing the balls to descend to the basement.
When a weight or cannonball is raised against the force of gravity, it gains potential energy--energy by virtue of its position, proportional to the height to which it was raised. If the weight is dropped, it loses height and potential energy, but gains speed and kinetic energy, energy due to the speed of motion.
Kinetic energy can be converted back to potential energy, as happens to a roller coaster after it passes the bottom of a dip and climbs up again. A similar change occurs when a stone is thrown upwards with some velocity v. If its mass is m mass will be defined in a later section; for now let it be a property of the stone proportional to its weight , its kinetic energy can be shown to be.
As the stone rises, v and the kinetic energy decrease, while the potential energy grows. It is given by h m g. The sum of the two is the total energy E 1 and by a fundamental law of mechanics it stays constant:. On the downward trip, the opposite changes take place. A later section of "Stargazers" comes back to this formula and makes it more meaningful. From that. Optional Earlier it was stated that a third orbital element is needed to mark where the satellite is located in its orbit.
Since the equation of the orbital ellipse is. One could in principle use the true anomaly as a third orbital element.
Unfortunately, the speed of the satellite varies unevenly around its orbit, it grows larger near perigee and drops again near apogee. The way M is defined assures that it grows at a constant rate as time t advances:. The mean anomaly is regarded as the third orbital element. Copernicus to Galileo Average Distance m.
Period yr. Average Distance au. Kepler's third law provides an accurate description of the period and distance for a planet's orbits about the sun. In the next part of Lesson 4 , these principles will be investigated as we draw a connection between the circular motion principles discussed in Lesson 1 and the motion of a satellite.
Newton's comparison of the acceleration of the moon to the acceleration of objects on earth allowed him to establish that the moon is held in a circular orbit by the force of gravity - a force that is inversely dependent upon the distance between the two objects' centers.
Establishing gravity as the cause of the moon's orbit does not necessarily establish that gravity is the cause of the planet's orbits. How then did Newton provide credible evidence that the force of gravity is meets the centripetal force requirement for the elliptical motion of planets?
Recall from earlier in Lesson 3 that Johannes Kepler proposed three laws of planetary motion. His Law of Harmonies suggested that the ratio of the period of orbit squared T 2 to the mean radius of orbit cubed R 3 is the same value k for all the planets that orbit the sun.
Known data for the orbiting planets suggested the following average ratio:. Newton was able to combine the law of universal gravitation with circular motion principles to show that if the force of gravity provides the centripetal force for the planets' nearly circular orbits, then a value of 2.
Here is the reasoning employed by Newton:. Consider a planet with mass M planet to orbit in nearly circular motion about the sun of mass M Sun. The net centripetal force acting upon this orbiting planet is given by the relationship. This net centripetal force is the result of the gravitational force that attracts the planet towards the sun, and can be represented as.
Substitution of the expression for v 2 into the equation above yields,. By cross-multiplication and simplification, the equation can be transformed into.
The mass of the planet can then be canceled from the numerator and the denominator of the equation's right-side, yielding. The right side of the above equation will be the same value for every planet regardless of the planet's mass.
Newton's universal law of gravitation predicts results that were consistent with known planetary data and provided a theoretical explanation for Kepler's Law of Harmonies. Our understanding of the elliptical motion of planets about the Sun spanned several years and included contributions from many scientists.
Which scientist is credited with the collection of the data necessary to support the planet's elliptical motion? See Answer Tycho Brahe gathered the data. Johannes Kepler analyzed the data. Isaac Newton explained the data - and that's what the next part of Lesson 4 is all about. Galileo is often credited with the early discovery of four of Jupiter's many moons.
But these other options come with an additional cost in energy and danger to the astronauts. Visit this site for more details about planning a trip to Mars.
Consider Figure The time it takes a planet to move from position A to B , sweeping out area A 1 A 1 , is exactly the time taken to move from position C to D , sweeping area A 2 A 2 , and to move from E to F , sweeping out area A 3 A 3. Comparing the areas in the figure and the distance traveled along the ellipse in each case, we can see that in order for the areas to be equal, the planet must speed up as it gets closer to the Sun and slow down as it moves away. This behavior is completely consistent with our conservation equation, Equation The cross product for angular momentum can then be written as.
Note that the angular momentum does not depend upon p rad p rad. Since the gravitational force is only in the radial direction, it can change only p rad p rad and not p perp p perp ; hence, the angular momentum must remain constant. Now consider Figure The areal velocity is simply the rate of change of area with time, so we have. Since the angular momentum is constant, the areal velocity must also be constant. You can view an animated version of Figure Equation For an ellipse, recall that the semi-major axis is one-half the sum of the perihelion and the aphelion.
For a circular orbit, the semi-major axis a is the same as the radius for the orbit. In fact, Equation We have changed the mass of Earth to the more general M , since this equation applies to satellites orbiting any large mass.
This yields a value of 2. The semi-major axis is one-half the sum of the aphelion and perihelion, so we have. Substituting for the values, we found for the semi-major axis and the value given for the perihelion, we find the value of the aphelion to be The nearly circular orbit of Saturn has an average radius of about 9.
As an Amazon Associate we earn from qualifying purchases. Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4. Skip to Content Go to accessibility page. University Physics Volume 1 My highlights. Table of contents. Chapter Review. Waves and Acoustics. Answer Key. By the end of this section, you will be able to: Describe the conic sections and how they relate to orbital motion Describe how orbital velocity is related to conservation of angular momentum Determine the period of an elliptical orbit from its major axis.
From this definition, you can see that an ellipse can be created in the following way.
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