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Gravity Gravity is the weakest of the four fundamental forces, yet it is the dominant force in the universe for shaping the large scale structure of galaxies, stars, etc. The gravitational force between two masses m1 and m2 is given by the relationship: This is often called the "universal law of gravitation" and G the universal gravitation constant. It is an example of an inverse square law force. The force is always attractive and acts along the line joining the centers of mass of the two masses. The forces on the two masses are equal in size but opposite in direction, obeying Newton's third law. Viewed as an exchange force, the massless exchange particle is called the graviton. The gravity force has the same form as Coulomb's law for the forces between electric charges, i.e., it is an inverse square law force which depends upon the product of the two interacting sources. This led Einstein to start with the electromagnetic force and gravity as the first attempt to demonstrate the unification of the fundamental forces. It turns out that this was the wrong place to start, and that gravity will be the last of the forces to unify with the other three forces. Electroweak unification (unification of the electromagnetic and weak forces) was demonstrated in 1983, a result which could not be anticipated in the time of Einstein's search. It now appears that the common form of the gravity and electromagnetic forces arises from the fact that each of them involves an exchange particle of zero mass, not because of an inherent symmetry which would make them easy to unify. Fundamental Forces Weight The weight of an object is defined as the force of gravity on the object and may be calculated as the mass times the acceleration of gravity, w = mg. Since the weight is a force, its SI unit is the newton. For an object in free fall, so that gravity is the only force acting on it, then the expression for weight follows from Newton's second law. You might well ask, as many do, "Why do you multiply the mass times the freefall acceleration of gravity when the mass is sitting at rest on the table?". The value of g allows you to determine the net gravity force if it were in freefall, and that net gravity force is the weight. Another approach is to consider "g" to be the measure of the intensity of the gravity field in Newtons/kg at your location. You can view the weight as a measure of the mass in kg times the intensity of the gravity field, 9.8 Newtons/kg under standard conditions. Data can be entered into any of the boxes below. Then click outside the box to update the other quantities. At the Earth's surface, where g=9.8 m/s2 : The weight of mass kg is The weight of mass slugs is Newtons pounds The kilogram is the SI unit of mass and it the almost universally used standard mass unit. The associated SI unit of force and weight is the Newton, with 1 kilogram weighing 9.8 Newtons under standard conditions on the Earth's surface. However, in the US common units, the pound is the unit of force (and therefore weight).The pound is the widely used unit for commerce. The use of the pound force constrains the mass unit to an inconveniently large measuring unit called a "slug". The use of this unit is discouraged, and the use of exclusively SI units for all scientific work is strongly encouraged. What is a slug? What is a Slug? The slug is the unit of mass in the US common system of units, where the pound is the unit of force. The pound is therefore the unit of weight since weight is defined as the force of gravity on an object. While the pound force and pound weight are the widely used units for commerce in the United States, their use is strongly discouraged in scientific work. The standard units for most of scientific work are the SI units. Mass and Weight The mass of an object is a fundamental property of the object; a numerical measure of its inertia; a fundamental measure of the amount of matter in the object. Definitions of mass often seem circular because it is such a fundamental quantity that it is hard to define in terms of something else. All mechanical quantities can be defined in terms of mass, length, and time. The usual symbol for mass is m and its SI unit is the kilogram. While the mass is normally considered to be an unchanging property of an object, at speeds approaching the speed of light one must consider the increase in the relativistic mass. The weight of an object is the force of gravity on the object and may be defined as the mass times the acceleration of gravity, w = mg. Since the weight is a force, its SI unit is the newton. Density is mass/volume. Earth's Gravity The weight of an object is given by W=mg, the force of gravity, which comes from the law of gravity at the surface of the Earth in the inverse square law form: At standard sea level, the acceleration of gravity has the value g = 9.8 m/s2, but that value diminishes according to the inverse square law at greater distances from the earth. The value of g at any given height, say the height of an orbit, can be calculated from the above expression. Above the earth's surface at a height of h = corresponds to a radius r = m/s2 = m= x 106 m, which x earth radius, the acceleration of gravity is g = x g on the earth's surface. Please note that the above calculation gives the correct value for the acceleration of gravity only for positive values of h, i.e., for points outside the Earth. If you drilled a hole through the center of the Earth, the acceleration of gravity would decrease with the radius on the way to the center of the Earth. If the Earth were of uniform density (which it is not!), the acceleration of gravity would decrease linearly to half the surface value of g at half the radius of the Earth and approach zero as you approached the center of the Earth. Hole through center of Earth Journey through the center of the Earth Suppose you could drill a hole through the Earth and then drop into it. How long would it take you to pop up on the other side of the Earth? Your initial acceleration would be the surface acceleration of gravity but the acceleration would be progressively smaller as you approached the center. Your weight would be zero as you flew through the center of the Earth. For our hypothetical journey we will assume the Earth to be of uniform density and neglect air friction and the high temperature of this trip. For a spherically symmetric mass, the net gravity force on an object from that mass would be only that due to the mass inside its radius, and that would act as if it were a point mass located at the center. When this is analyzed in detail, you find that the gravity at any radius r less than REarth will be linearly proportional to the distance from the center. Gravity force of spherical shell On mass outside the shell On mass inside the shell Taking positive r as outward from the center of the Earth: This is the same form as Hooke's Law for a mass on a spring. It would cause the transEarth traveler to oscillate back and forth through the center of the Earth like a mass bobbing up and down on a spring. The angular frequency and period for this oscillation are For this case the period of oscillation is The traveler accelerates toward the center of the Earth and is momentarily weightless when passing through the geometric center at about 7900 m/s or almost 17,700 miles/hr. The traveler would pop up on the opposite side of the Earth after a little more than 42 minutes. But unless he or she grabs something to hold on, they will fall back for a return journey and continue to oscillate with a round-trip time of 84.5 minutes. As a further feature of this fanciful journey, suppose a satellite could be put in a circular orbit about the Earth right above the surface. Ignoring air drag and the terrific sonic boom that would accompany such an orbit, suppose it passed overhead just above the falling person as they popped up out of the hole. The period of such an orbit would be such that it would be passing overhead every time the oscillating person popped up on either side of the Earth. The period of the orbit is calculated from which is the same as the period of the oscillating body. Other examples of weightlessness Weightlessness While the actual weight of a person is determined by his mass and the acceleration of gravity, one's "perceived weight" or "effective weight" comes from the fact that he is supported by floor, chair, etc. If all support is removed suddenly and the person begins to fall freely, he feels suddenly "weightless" - so weightlessness refers to a state of being in free fall in which there is no perceived support. The state of weightlessness can be achieved in several ways, all of which involve significant physical principles. Click on any of the examples for further details. Inverse Square Law, Gravity As one of the fields which obey the general inverse square law, the gravity field can be put in the form shown below, showing that the acceleration of gravity, g, is an expression of the intensity of the gravity field. Tides The Earth experiences two high tides per day because of the difference in the Moon's gravitational field at the Earth's surface and at its center. You could say that there is a high tide on the side nearest the Moon because the Moon pulls the water away from the Earth, and a high tide on the opposite side because the Moon pulls the Earth away from the water on the far side. The tidal effects are greatly exaggerated in the sketches. Why is the moon's tidal effect greater than the Sun's? How large is the Sun's tidal effect compared to the moon's influence? Moon as Dominant Tidal Source The Moon is the dominant tidal influence because the fractional difference in its force across the Earth is greater than the fractional difference seen from the Sun. This difference in force follows the inverse square law. Sun's tidal influence Sun's Tidal Effect Even though the Sun is 391 times as far away from the Earth as the Moon, its force on the Earth is about 175 times as large. Yet its tidal effect is smaller than that of the Moon because tides are caused by the difference in gravity field across the Earth. The Earth's diameter is such a small fraction of the Sun-Earth distance that the gravity field changes by only a factor of 1.00017 across the Earth. The actual force differential across the Earth is 0.00017 x 174.5 = 0.03 times the Moon's force, compared to 0.068 difference across the Earth for the Moon's force. The actual tidal influence then is then 44% of that of the Moon. Why is the moon the dominant tidal influence? Moon as Dominant Tidal Source The Moon is the dominant tidal influence because the fractional difference in its force across the Earth is greater than the fractional difference seen from the Sun. This difference in force follows the inverse square law. Sun's tidal influence Jupiter Effect A popular book warned of disastrous gravitational effects from the special alignment of the planets Jupiter and Mars along with the Moon. The forces of attraction compared to that of the Earth are illustrated for a person weighing 160 lb on the Earth's surface. Don't lose much sleep worrying about the Jupiter effect. You change the gravity force on yourself by taking one step up a stairway more than the combined gravitational effects of both Jupiter and Mars if they were perfectly aligned!