Universal Gravitation Chapter Opener. Caption: The astronauts in the upper left of this photo are working on the Space Shuttle. As they orbit the Earth—at a rather high speed—they experience apparent weightlessness. The Moon, in the background, also is orbiting the Earth at high speed. What keeps the Moon and the space shuttle (and its astronauts) from moving off in a straight line away from Earth? It is the force of gravity. Newton’s law of universal gravitation states that all objects attract all other objects with a force proportional to their masses and inversely proportional to the square of the distance between them.

A space station revolves around the earth as a satellite, 100 km above Earth’s surface. What is the net force on an astronaut at rest inside the space station? Equal to her weight on Earth. Zero (she is weightless). Less than her weight on Earth. Somewhat larger than her weight on Earth.

What is Gravity? Fundamental Forces: Strong Force – The force that is involved in holding the nucleus of an atom together Electromagnetic Force – The force that exists between charged particles Weak Force – The force involved in nuclear decay Gravity – The force that exists between any two objects that have mass. - Always attractive

Newton’s Law of Universal Gravitation If the force of gravity is being exerted on objects on Earth, what is the origin of that force? Newton’s realization was that the force must come from the Earth. He further realized that this force must be what keeps the Moon in its orbit. Figure 6-1. Caption: Anywhere on Earth, whether in Alaska, Australia, or Peru, the force of gravity acts downward toward the center of the Earth.

Newton’s Law of Universal Gravitation The gravitational force on you is one-half of a Newton’s Third Law pair (the action force): The Earth exerts a downward force on you, and you exert an upward force on the Earth. When there is such a difference in masses, the reaction force is undetectable, Figure 6-2. Caption: The gravitational force one object exerts on a second object is directed toward the first object, and is equal and opposite to the force exerted by the second object on the first. but for bodies more equal in mass it can be significant.

Newton’s Law of Universal Gravitation Therefore, the gravitational force must be proportional to both masses. By observing planetary orbits, Newton also concluded that the gravitational force must decrease as the inverse of the square of the distance between the masses. (This is called the inverse square law) In its final form, the law of universal gravitation reads: Where:

Inverse Square Law for universal gravitation: As the distance decreases, the strength of Fg increases by the square of the number the distance went up by. If d is halved (1/2) the force will be 4 times what it was. Review of Scientific Notation.

Newton’s Law of Universal Gravitation The magnitude of the gravitational constant G can be measured in the laboratory. This is the Cavendish experiment. Figure 6-3. Caption: Schematic diagram of Cavendish’s apparatus. Two spheres are attached to a light horizontal rod, which is suspended at its center by a thin fiber. When a third sphere (labeled A) is brought close to one of the suspended spheres, the gravitational force causes the latter to move, and this twists the fiber slightly. The tiny movement is magnified by the use of a narrow light beam directed at a mirror mounted on the fiber. The beam reflects onto a scale. Previous determination of how large a force will twist the fiber a given amount then allows the experimenter to determine the magnitude of the gravitational force between two objects.

Gravitational Attraction of Spherical Bodies Even though the gravitational force is very small, the mirror allows measurement of tiny deflections. Measuring G also allowed the mass of the Earth to be calculated, as the local acceleration of gravity and the radius of the Earth were known.

Gravitational Attraction of Spherical Bodies Gravitational force between a point mass and a sphere: the force is the same as if all the mass of the sphere were concentrated at its center.

Newton’s Law of Universal Gravitation (ignore) Newton’s Law of Universal Gravitation Using calculus, you can show: Particle outside a thin spherical shell: gravitational force is the same as if all mass were at center of shell Particle inside a thin spherical shell: gravitational force is zero Can model a sphere as a series of thin shells; outside any spherically symmetric mass, gravitational force acts as though all mass is at center of sphere

Gravitational Attraction of Spherical Bodies The acceleration of gravity decreases slowly with altitude:

Gravitational Attraction of Spherical Bodies Once the altitude becomes comparable to the radius of the Earth, the decrease in the acceleration of gravity is much larger:

Tides Usually we can treat planets, moons, and stars as though they were point objects, but in fact they are not. When two large objects exert gravitational forces on each other, the force on the near side is larger than the force on the far side, because the near side is closer to the other object. This difference in gravitational force across an object due to its size is called a tidal force.

Tides This figure illustrates a general tidal force on the left, and the result of lunar tidal forces on the Earth on the right. Second high tide (smaller then the other) because this side is pulled the least Gravity of moon Moon’s pull is greatest here because it is closer. Highest tide

Tides Tidal forces can result in orbital locking, where the moon always has the same face towards the planet – as does Earth’s Moon. If a moon gets too close to a large planet, the tidal forces can be strong enough to tear the moon apart. This occurs inside the Roche limit; closer to the planet we have rings, not moons.

Tidal locking This is what allows only one side of the moon to face the Earth It can even keep moons from forming!

Newton’s Law of Universal Gravitation Example 6-1: Can you attract another person gravitationally? A 50-kg person and a 70-kg person are sitting on a bench close to each other. Estimate the magnitude of the gravitational force each exerts on the other. Answer: Assume the distance between the two people is about ½ m; then the gravitational force between them is about 10-6 N.

So… If gravity is always present and goes on for ever and ever, why hasn’t the Moon crashed into the Earth? It is true that the moon is technically falling towards Earth because of gravity. BUT, it is also speeding along with a velocity. It is this velocity that keeps it from falling into Earth!

Newton’s Thought Experiment To make something orbit the Earth, all you need to do is shoot it with a horizontal velocity that will make it fall along with the curve of the Earth. As a satellite or the Moon falls towards Earth, the earth also falls (or curves) away from the satellite at the same rate!

Gravity and Orbits Centripetal acceleration Centripetal Force The acceleration towards the center if something going in a circle ac= v2/r Centripetal Force The force that causes this acceleration (also towards the center Fc= mac

Lets try one! If the distance between the Earth and the Moon is 384,405,000m and the moon travels around Earth with an orbital velocity of 1023 m/s, what is the centripetal acceleration on the moon?

All of the planets as a system. The deviation of a planet from its normal orbit is called a perturbation. It is caused by other planets.

Some more facts about angles and planetary motion

Angular Velocity Just like how velocity is the rate at which an object covers a distance, Angular Velocity is the rate at which an angle (or an amount of degrees) is covered.

Moment of Inertia and Angular Momentum Inertia is_____________________? Moment of Inertia The further mass is from its spinning axis, the harder it is to make it spin. - Just like with regular inertia, things that are spinning also don’t like to change.

Gyroscopic Motion A device used to both measure and maintain its orientation while it is spinning. Works based on the principles of conservation of angular momentum. Bicycles Airplanes Hubble space telescope Yo Yos Frisbees

Conservation of Angular Momentum and the seasons of Earth