Physics Chapter 9 Gravity
The Newtonian Synthesis Gravity is a Universal Force
The Newtonian Synthesis According to legend, Newton discovered gravity while sitting under an apple tree.
The Falling Moon If the Moon did not fall, it would move off in a straight line.
The Falling Moon Original drawing by Newton: Shows how a projectile fired fast enough would fall around Earth and become an Earth satellite.
The Falling Moon Tangential velocity is the sideways velocity the component of velocity perpendicular to the pull of gravity.
Newton s Apple-Moon Test For Newton s idea to advance from hypothesis to scientific theory, it would have to be tested.
Newton s Apple-Moon Test An apple falls 5 m during its first second of fall when it is near Earth s surface. Newton asked how far the moon would fall in the same time if it were 60 times farther from the center of Earth.
The Falling Moon If the force that pulls apples off trees also pulls the moon into orbit, the circle of the moon s orbit should fall 1.4 mm below a point along the straight line where the moon would otherwise be one second later.
The Falling Earth Newton s theory of gravity confirmed the Copernican Theory of the solar system. The tangential velocity of Earth about the Sun allows it to fall around it rather than directly into it.
Newton s Law of Universal Gravitation Every object attracts every other object with a force that for any two objects is directly proportional to the mass of each object. That the force decreases as the SQUARE of the distance between the centers of mass of the objects increases.
Newton s Law of Universal Gravitation
Newton s Law of Universal Gravitation
The Universal Gravitational Constant, G The universal gravitational constant, G, in the equation for universal gravitation describes the strength of gravity.
The Universal Gravitational Constant, G Philipp von Jolly developed a method of measuring the attraction between two masses.
The Universal Gravitational Constant, G The value of G tells us that gravity is a very weak force. It is the weakest of the presently known four fundamental forces. We sense gravitation only when masses like that of Earth are involved.
The Mass of the Earth Cavendish s first measure of G was called the Weighing the Earth experiment. Mass of Earth m 1 = 6 10 24 kilograms.
The Mass of the Earth
Understanding Gravity Gravity decreases according to the inverse-square law. The force of gravity weakens as the square of distance.
Gravity & Distance: The Inverse Square Law
Gravity & Distance: The Inverse Square Law d F R 10 N 2R ¼ (10 N) 3R (1/9) (10 N) 4R (1/16) (10 N) When a quantity varies as the inverse square of its distance from its source, it follows an inverse-square law.
Gravity & Distance: Inverse Square Law Q: At what distance will the gravitational force become zero? A: You are always affected! No matter how far away you go, Earth s gravitational influence extends through ALL space!
Gravity and Distance: Inverse Square Law
Check Question Suppose that an apple at the top of a tree is pulled by Earth s gravity with a force of 1 N. If the tree were twice as tall, would the force of gravity on the apple be only 1/4 as strong? Explain your answer.
Check Question Calculate the resulting force of gravitational attraction between two masses if one of the masses was to double and the distance between them was to triple.
Gravitational Field Newton himself couldn t understand how two objects could exert force over a distance. Moon and Earth are not in contact. It s hard to imagine a force reaching out through nothingness?
Gravitational Field Field lines can also represent the pattern of Earth s gravitational field. ΣF = F g ma = mg a = g
Finding Surface Gravity We now have two equations for the force of gravity on an object here on earth. Let s set these two equal to each other and solve for g...
Finding Surface Gravity To solve for the gravity on any planet at a point in space:
Check Question Calculate the surface gravity of the Moon. The Moon has a mass of 7.36 x 10 22 kilograms and a radius of 1.74 x 10 6 meters.
Tangential Orbital Velocity Consider a satellite of mass m orbiting a central body, M. Can you see that the gravitational force is a centripetal force? M v F g m Let s equate the two and then solve for orbital speed: F g = F c...
Tangential Orbital Velocity M v F g m M is the mass of a planet or star. Note that m is missing! The speed of an orbiting satellite does not depend on its own mass!
Check Question A satellite orbiting at a speed of v and a radius of r above the center of a planet climbs to a radius 2r. What is the satellite s new orbital speed?
Gravitational Field Inside a Planet The gravitational field of Earth exists inside Earth as well as outside. Imagine a hole drilled completely through Earth. Consider the kind of motion you would undergo if you fell into such a hole.
Gravitational Field You would oscillate back and forth, approximating simple harmonic motion. Each round trip would take nearly 90 minutes. Interestingly enough, we will see in the next chapter that an Earth satellite in close orbit about Earth also takes the same 90 minutes to make a complete round trip.
Gravity and Distance: Inverse Square Law
Gravitational Field
Weight and Weightlessness The force of gravity, like any force, causes acceleration. Objects under the influence of gravity are pulled toward each other and accelerate. We are almost always in contact with Earth, so we think of gravity as something that presses us against Earth rather than as something that accelerates us.
Weight and Weightlessness
Weight and Weightlessness Rather than define your weight as the force of gravity that acts on you, it is more practical to define weight as the force you exert against a supporting floor. According to this definition, you are as heavy as you feel. The condition of weightlessness is not the absence of gravity, but the absence of a support force.
Weight and Weightlessness Both people are without a support force and therefore experience weightlessness.
Weight and Weightlessness Exactly how much less gravity does an astronaut experience than we do on the surface?
Ocean Tides Newton showed that the ocean tides are caused by differences in the gravitational pull of the moon on opposite sides of Earth.
Ocean Tides This difference in pulls across Earth slightly elongates it. The oceans bulge out about 1 meter on average, on opposite sides of Earth. Because Earth spins once per day, a fixed point on Earth passes beneath both of these bulges each day, producing two sets of ocean tides per day two high tides and two low tides.
Factors Affecting Ocean Tides The sun also contributes to ocean tides. Its pull on Earth is 180 times greater than the moon s pull on Earth, so why aren t solar tides 180 times greater than lunar tides? The difference in gravitational pulls by the sun on opposite sides of Earth is very small.
Spring Tides When the sun, the moon, and Earth are aligned, spring tides occur.
Neap Tides When the attractions of the sun and the moon are at right angles to each other (at the time of a half moon), neap tides occur.
Ocean Tides The tilt of Earth s axis, interfering landmasses, friction with the ocean bottom, and other factors complicate tidal motions.
Ocean Tides - Question Why does the gravitation pull of the moon create almost no tide in a lake? The moon produces scarcely any tides in a lake because no part of the lake is significantly closer to the moon than any other part this means there is no significant difference in the moon s pull on different parts of the lake.
Einstein s Theory of Gravitation In the early 20 th century, Einstein realized something else about gravity: Gravity is a dent, a bend or a curve in space-time.
Black Holes Two main processes go on continuously in stars like our sun. Gravitation tends to crush all solar material toward the center. Thermonuclear fusion, consisting of reactions similar to those in a hydrogen bomb, tends to blow solar material outward. When the processes of gravitation and thermonuclear fusion balance each other, the result is the sun of a given size.
Black Holes The size of the sun is the result of a tug of war between two opposing processes: nuclear fusion and gravitational contraction.
Black Holes If the fusion rate increases, the sun will get hotter and bigger. If the fusion rate decreases, the sun will get cooler and smaller. When the sun runs out of fusion fuel (hydrogen), gravitation will dominate and the sun will start to collapse.
Black Holes For a star that is at least two to three times more massive than our sun, once the flame of thermonuclear fusion is extinguished, gravitational collapse takes over and it doesn t stop! The star caves in on itself and the atoms that compose the star cave in on themselves until there are no empty spaces. The density becomes infinite near these black holes. Even light cannot escape a black hole.
Black Holes Suppose you are on a star and you are indestructible: F = 10,000 N F = 40,000 N R ½ R
Gravitational Field Around Black Holes A black hole is no more massive than the star from which it collapsed. The gravitational field near the black hole may be enormous but the field beyond the original radius of the star is no different after collapse than before. The amount of mass has not changed, so there is no change in the field at any point beyond this distance.
Gravitational Field Near Black Holes The gravitational field strength near a giant star that collapses to become a black hole is the same before collapse (left) and after collapse (right).
Black Holes The gravitational field around a black hole is usually represented as a warped two-dimensional surface.
Universal Gravitation The Earth is round because of gravitation. Since everything attracts everything else, Earth had attracted itself together before it became solid. The sun, the moon, and Earth are all fairly spherical because they have to be.
Universal Gravitation Gravity played a role in the formation of the solar system.
Universal Gravitation If everything pulls on everything else, then the planets must pull on each other. The net force that controls Jupiter, for example, is not just from the sun, but from the planets also. Their effect is small compared with the pull of the more massive sun, but it still shows. The deviation of an orbiting object from its path caused by the action of an additional center of force is called a perturbation.
Universal Gravitation The shapes of distant galaxies provide further evidence that the law of gravity applies to larger distances. According to current scientific understanding, the universe originated and grew from a tiny singularity, inflating rapidly over the next 13.7 billion years to become the cosmos we know today. This is the Big Bang theory of the origin of the universe.
Problems Earth, with its mass of 5.98 x 1024 kg pulls a 60-kilogram person downward with a force of 600 N. With what force does the person pull on the earth?
Problems A satellite of mass m orbits Earth at a height of h and a speed of v. What would the speed be for a satellite of mass 3m at a height of h?
Problems Determine the acceleration of gravity, in terms of g, at a point in space that is located a distance equal to two Earth radii (2r E ) above the surface of Earth, shown in the figure above.
Problems A very large star of mass M and radius r undergoes a transformation into a neutron star. Assume that the mass remains constant while the radius of the star becomes 1/1000 its original size, (1x10-3 )r. How does the change in the radius of the star affect the force of gravity, Fg, acting between the star and a planet that is orbiting the star?
Problems What would the surface gravity be on a planet with three times the mass of Earth and a radius twice that of Earth?
Problems The gravitational force of attraction between two masses is 16 newtons. If the distance between the masses is quadrupled, what is the resulting force of attraction?