Chapter 3 - Gravity and Motion. Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 3 - Gravity and Motion Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

In 1687 Isaac Newton published the Principia in which he set out his concept of the universal nature of gravity and also his law of gravity. The line of thought leading to his mature theory of gravity started with an exchange of letters with Robert Hooke in 1679 80, but it did not become precise until after a visit from Edmund Halley in 1684. Halley, like Hooke before him, asked about the trajectory of a body under the influence of inverse-square law forces directed towards a given center.

Gravity gives the Universe its structure Gravity It is a universal force that causes all objects to pull on all other objects everywhere It holds objects together It is responsible for holding the Earth in its orbit around the Sun, the Sun in its orbit around the Milky Way, and the Milky Way in its path within the Local Group The force that pulls an upward going ball back to the Earth and keeps the Moon in its orbit.

The Problem of Astronomical Motion Galileo not only proposed this property of matter but also demonstrated it by real experiment. He put science on a course to determine laws of motion and to develop the scientific method Central to Galileo's laws of motion is the concept of inertia. Inertia is the tendency of a body at rest to remain at rest or of a body in motion to keep moving in a straight line at a constant speed.

Inertia - the property that summarizes a body's tendency to stay in the state of rest or motion Example: While riding a bicycle, if you stop pedaling you will still continue to move forward due to inertia. Galileo established the idea of inertia A body at rest tends to remain at rest A body in motion tends to remain in motion Through experiments with inclined planes, Galileo demonstrated the idea of inertia and the importance of forces (friction)

Inertia and Newton s First Law This concept was incorporated in Newton s First Law of Motion: A body continues in a state of rest or uniform motion in a straight line unless made to change that state by forces acting on it.

Newton s First Law Important ideas of Newton s First Law Force: A push or a pull The force referred to is a net force Newton's first law may not sound impressive at first, but it carries an idea that is crucial in astronomy: that if a body is not moving in a straight line at a constant speed, some net force must be acting on it. The law implies that if an object is not moving with constant velocity, then a nonzero net force must be present A body continues in a state of rest or of motion in a straight line at a constant speed unless made to change that state by forces acting on it.

Newton was preceded in stating this law by the seventeenth-century Dutch scientist Christiaan Huygens. However, Newton went on to develop additional physical laws and more important for astronomy showed how to apply them to the Universe.

Newton was not the first person to attempt to discover and define the force that holds planets in orbit around the Sun. Nearly 100 years before Newton, Kepler recognized that some force must hold the planets in their orbits and proposed that something similar to magnetism might be responsible. Newton was not even the first person to suggest that gravity was responsible. That honor belongs to Robert Hooke, another Englishman, who noted gravity's role in celestial motions several years before Newton published his law of gravity in 1687. Newton's modification of Kepler's 3rd law allows astronomers to determine the mass of celestial objects.

Newton's contribution is nevertheless special because he demonstrated the properties that gravity must have if it is to control planetary motion. Moreover, Newton went on to derive equations that describe not only gravity but also its effects on motion. The solution of these equations allowed astronomers to predict the position and motion of the planets and other astronomical bodies.

Orbital Motion and Gravity Although not the first to propose gravity as being responsible for celestial motion, Newton was the first to: Spell out the properties of gravity Write the equations of gravity-induced motion Newton deduced that: The Moon s motion could be explained by the existence of a force (to deviate the Moon from a straight inertial trajectory) and that such a force decreased with distance Orbital motion could be understood as a projectile moving parallel to the Earth s surface at such a speed that its gravitational deflection toward the surface is offset by the surface s curvature away from the projectile

Newton hypothesized that moon was simply a projectile circling Earth under the attraction of gravity 1. Newton had to test hypothesis 2. Compared fall of apple to fall of moon

Newton reasoned that gravitational attraction was diluted by distance a. Moon is 60 times farther from the center of the Earth than the apple b. Calculated difference to be 1/(60) 2

Newton s law of universal gravitation states that every object attracts every other object with a force that for any two objects is directly proportional to the mass of each object. Newton deduced that the force decreases as the square of the distance between the centers of mass of the objects increases.

Planets remain in orbit while falling around the sun due to their tangential velocities.

Newton s Law of Universal Gravitation Law states: every object attracts every other object with a force that for any two objects is directly proportional to the mass of each object 1. The greater the mass the greater the attraction 2. The farther away the objects are from each other, the less the force of attraction between them

Gravity and Distance: The Inverse Square Law A.Inverse Square Law when quantity varies as the inverse square of its distance from its source B. Also applies to light, radiation, and sound Example: If you double the distance between two bodies, the force of gravity between them would decrease by a fourth.

The strength of its force on objects follows the inversesquare law (so g weakens with increasing distance from Earth)

Astronomical Motion As seen earlier, planets move along curved (elliptical) paths, or orbits. Speed and direction is changing Must there be a force at work? Yes!

Gravity is that force! After tying a string to a rock, you twirl the rock around above your head. If the string were to break the rock will go in a straight line before falling down.

Ocean Tides Newton showed that the ocean tides are caused by differences in the gravitational pull of the moon on opposite sides of earth 1. Oceans bulge about 1 meter on opposite sides of Earth 2. Because Earth spins, the tides change as Earth rotates

The moon falls toward Earth in the sense that it falls beneath the straight-line path it would take without gravity.

Sun also contributes to tides 1. Sun s pull 180 times greater than moon, but contributes only half as much as the moon 2. Because difference in gravitational pull by sun on opposite sides of Earth very small (0.017% compared to 6.7% for moon's gravitation) The highest ocean tides occur when the Earth and moon are lined up with the sun.

Newton's cannon Most of Newton's work is highly mathematical, but as part of his discussion of orbital motion, he described a thought experiment to demonstrate how an object can move in orbit. Thought experiments are not actually performed; rather, they serve as a way to think about problems. In Newton's thought experiment, we imagine a cannon on a mountain peak firing a projectile. From our everyday experience, we know that whenever an object is thrown horizontally, gravity pulls it downward so that its path is an arc. Moreover, the faster we throw, the farther the object travels before striking the ground.

Orbital Motion Using Newton s First Law A cannonball fired at slow speed experiences one force gravity, pulling it downward A cannonball fired at a higher speed feels the same force, but goes farther

Orbital Motion Using Newton s First Law At a sufficiently high speed, the cannonball travels so far that the ground curves out from under it. The cannonball literally misses the ground! The ball, now in orbit, still experiences the pull of gravity! The average orbital speed of the Earth around the Sun about 30 km/s.

Notice that in the above discussion we used no formulas. All we needed was Newton's first law and the idea that gravity supplies the deflecting force. However, if we are to understand the particulars of orbital motion, we require additional laws. For example, to determine how rapidly the projectile must move to be in orbit, we need laws that have a mathematical formulation.

Newton s Second Law: Motion Motion An object is said to be in uniform motion if its speed and direction remain unchanged An object in uniform motion is said to have a constant velocity A force will cause an object to have nonuniform motion, a changing velocity Acceleration is defined as a change in velocity

Newton s 2nd Law: Acceleration Acceleration An object increasing or decreasing in speed along a straight line is accelerating An object with constant speed moving is a circle is accelerating Acceleration is produced by a force and experiments show the two are proportional

Mass Newton s Second Law: Mass Mass is the amount of matter an object contains Technically, mass is a measure of an object s inertia Mass is generally measured in kilograms Mass should not be confused with weight, which is a force related to gravity weight may change from place to place, but mass does not At the equator on Earth you would weight less.

Newton s Second Law of Motion Gravity produces a force of attraction between bodies. Gravity Produces a Force of Attraction Between Bodies F = ma Equivalently, the amount of acceleration (a) that an object undergoes is proportional to the force applied (F) and inversely proportional to the mass (m) of the object This equation applies for any force, gravitational or otherwise

F = ma

Newton s Third Law of Motion When two objects interact, they create equal and opposite forces on each other This is true for any two objects, including the Sun and the Earth! The gravitational force of the Earth on the Moon is exactly equal to the gravitational force of the Moon on the Earth.

Newton s Law of Universal Gravity Newton's second law supplies the answer. Written in the form a = F/m, we see that the acceleration an object feels is inversely proportional to its mass; that is, the more massive it is, the more force is required to accelerate it. Thus, even though the forces acting on the Earth and Moon are precisely equal, the Moon accelerates 81 times more because it is 81 times less massive than the Earth. Everything attracts everything else!!

Measuring an Object s Mass Using Orbital Motion Basic Setup of an Orbital Motion Problem Assume a small mass object orbits around a much more massive object Massive object can be assumed at rest (very little acceleration) Assume orbit shape of small mass is a circle centered on large mass Using Newton s Second Law Acceleration in a circular orbit must be: a = v 2 /r where v is the constant orbital speed and r is the radius of the orbit The force is that of gravity

Measuring an Object s Mass Using Orbital Motion Method of Solution Equate F = mv 2 /r to F = GMm/r 2 and solve for v: One can also solve for M: v = (GM/r) 1/2 M = (v 2 r)/g v can be expressed in terms of the orbital period (P) on the small mass and its orbital radius: v = 2pr/P Combining these last two equations: M = (4p 2 r 3 )/(GP 2 ) This last equation in known as Kepler s modified third law and is often used to calculate the mass of a large celestial object from the orbital period and radius of a much smaller mass

Surface Gravity One of Galileo's famous discoveries was that objects of different masses all fall at the same rate. Many people incorrectly believe that lighter objects fall slower, but that is just a result of air friction. In a vacuum all objects accelerate downward at the same rate. Astronomers call this acceleration surface gravity, which gives a measure of the gravitational attraction at a planet's or star's surface. This acceleration determines not only how fast objects fall, but since an object's weight depends on its mass and the acceleration of gravity, surface gravity also determines what a mass weighs. The Earth's radius at the equator is larger than that at the poles. The Earth's surface gravity more at the North and South Poles.

Surface Gravity Surface gravity is the acceleration a mass undergoes at the surface of a celestial object (e.g., an asteroid, planet, or star) Surface gravity: Determines the weight of a mass at a celestial object s surface Influences the shape of celestial objects Influences whether or not a celestial object has an atmosphere

Surface Gravity Calculations Surface gravity is determined from Newton s Second Law and the Law of Gravity: ma = GMm/R 2 where M and R are the mass and radius of the celestial object, and m is the mass of the object whose acceleration a we wish to know The surface gravity, denoted by g, is then: g = GM/R 2 Notice dependence of g on M and R, but not m g Earth = 9.8 m/s 2 g Earth /g Moon = 5.6 and g Jupiter /g Earth = 3

Escape Velocity To overcome a celestial object s gravitational force and escape into space, a mass must obtain a critical speed called the escape velocity Escape velocity: Determines if a spacecraft can move from one planet to another Influences whether or not a celestial object has an atmosphere Relates to the nature of black holes

The speed an object needs to move away from the gravitational pull of the Earth. Escape Velocity Vesc = Escape velocity G = Gravitational constant M = Mass of the body to be escaped from R = Radius of the body to be escaped from

Escape Velocity Calculation The escape velocity, V esc, is determined from Newton s laws of motion and the Law of Gravity and is given by: V esc = (2GM/R) 1/2 where M and R are the mass and radius of the celestial object from which the mass wishes to escape Notice dependence of V esc on M and R, but not m V esc,earth is about 11 km/s, V esc,moon = 2.4 km/s

Escape Velocity Escape velocity is the speed an object must have to overcome the gravitational force of a body such as a planet and not fall back. Notice in the equation for Vesc that if two objects of the same radius are compared, the larger mass will have the larger escape velocity. Likewise, if two objects of the same mass are compared, the one with the smaller radius will have the greater escape velocity. Escape velocity is particularly important in understanding the nature of black holes. In chapter 15, we will see that a black hole is an object whose escape velocity exceeds the speed of light. Thus, light cannot escape from it, thereby making it black. A black hole has such a huge escape velocity, not necessarily because it has an unusually large mass, but because it has an abnormally small radius.

Assessment Questions 1. Newton determined that the pull of Earth s gravity caused both apples and the moon to fall toward Earth. the moon to move away from Earth. the sun to move away from Earth. stars to fall toward Earth.

Assessment Questions 2. The moon falls toward Earth in the sense that it falls with an acceleration of 10 m/s 2, as apples fall on Earth. with an acceleration greater than 10 m/s 2. beneath the straight-line path it would take without gravity. above the straight-line path it would take without gravity.

Assessment Questions 3. Planets remain in orbit while falling around the sun due to their tangential velocities. zero tangential velocities. accelerations of about 10 m/s 2. centrifugal forces that keep them up.

Assessment Questions 4. Newton did not discover gravity, for early humans discovered it whenever they fell. What Newton did discover is that gravity tells us about why the universe expands. tells us how to discover new planets. accounts for the existence of black holes. extends throughout the universe.

Assessment Questions 5. Consider a space probe three times as far from Earth s center. Compared at Earth s surface, its gravitational attraction to Earth at this distance is about one third as much. one half as much. one ninth as much. zero.

Assessment Questions 6. Compared to the gravitational field of Earth at its surface, Earth s gravitational field at a distance three times as far from Earth s center is about one third as much. one half as much. one ninth as much. zero.

Assessment Questions 7. Compared to the gravitational field of Earth at its surface, Earth s gravitational field at Earth s center is zero. half as much. twice as much. three times as much.

Assessment Questions 7. Compared to the gravitational field of Earth at its surface, Earth s gravitational field at Earth s center is zero. half as much. twice as much. three times as much. Answer: A

Assessment Questions 8. When an astronaut in orbit is weightless, he or she is beyond the pull of Earth s gravity. still in the pull of Earth s gravity. in the pull of interstellar gravity. beyond the pull of the sun s gravity.

Assessment Questions 9. The highest ocean tides occur when the Earth and moon are lined up with the sun. at right angles to the sun. at any angle to the sun. lined up during spring.

Assessment Questions 9. The highest ocean tides occur when the Earth and moon are lined up with the sun. at right angles to the sun. at any angle to the sun. lined up during spring. Answer: A

Summary A gravitational force exists between any two objects in the Universe. The strength of this force depends on the masses of the bodies and their separation. Gravitational forces hold astronomical bodies together and in orbit about one another.

Summary An object's inertia makes it remain at rest or move in a straight line at a constant speed unless the object is acted on by a net force. Thus, for a planet to orbit the Sun, the Sun's force of gravity must act on it. The law of inertia and Newton's other laws of motion, when combined with the law of gravity, allow us to relate the size and speed of orbital motion to the mass of the central object.

Summary The gravitational force exerted by a planet determines its surface gravity and escape velocity. The former determines your weight on a planet. The latter is the speed necessary to leave the surface and escape without falling back.