Universal Gravitation

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 Nuclear Force – The force that is involved in holding the nucleus of an atom together Electromagnetic Force – The force that exists between charged particles Weak Nuclear 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.

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, 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.

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.

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) 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.

Newton’s Law of Universal Gravitation Example 6-2: Spacecraft at 2rE. What is the force of gravity acting on a 2000-kg spacecraft when it orbits two Earth radii from the Earth’s center (one earth radius is: rE = 6380 km)? The mass of the Earth is mE = 5.98 x 1024 kg. Figure 6-4. Answer: Since the force of gravity goes down like the square of the distance, it will be ¼ as much as on the surface, or 4900 N.

Newton’s Law of Universal Gravitation Example 6-3: Force on the Moon. Find the net force on the Moon (mM = 7.35 x 1022 kg) due to the gravitational attraction of both the Earth (mE = 5.98 x 1024 kg) and the Sun (mS = 1.99 x 1030 kg), assuming they are at right angles to each other. Figure 6-5. Caption: Example 6–3. Orientation of Sun (S), Earth (E), and Moon (M) at right angles to each other (not to scale). Solution: Find the individual forces – the Moon is 3.84 x 108 m from the Earth and 1.50 x 1011 m from the Sun – and then add them as vectors. F = 4.77 x 1020N, θ = 24.6°.

So… If gravity is always present and goes on for ever and ever, why hasn’t the Moon crashed into the Earth? Since the Moon is not only falling toward Earth, but also moving “tangentially,” the tangential velocity keeps the Moon from crashing into Earth.

Newton’s Thought Experiment To make something orbit the Earth, all you need to do is shoot it at a tangential 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 in circular motion that is directed towards the center – “centripetal” This is the force that causes centripetal acceleration Fc= mac

Try This: What is the centripetal acceleration on the moon if it travels around Earth with a tangential velocity of 1023m/s and its average distance from the Earth is 384,000,000m? What is the centripetal force on the moon? What is the force due to gravity on the moon from the Earth?

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 it’s rotation axis, the greater the “moment of inertia” - The greater the moment of inertia, the greater the torque required to alter the angular motion. Angular Momentum is the product of the moment of inertia and the angular velocity.

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

Precession The motion of the axis of a spinning body, such as the wobble of a spinning top, when there is an external force acting on the axis.

Conservation of Angular Momentum and the seasons of Earth