Gravity • Gravity rules the Universe • It holds objects like the Sun and Earth together Our Our Place Place in in the the Cosmos Cosmos • Sun’s gravity determines motion of the planets of the Solar System Lecture 7 • Gravity binds stars into galaxies and galaxies Gravity - Ruler of the Universe into clusters • In this lecture we will follow Newton’s lines of reasoning in arriving at his law of gravity What is Gravity? Acceleration due to Gravity • Gravity is a force between any two • Galileo observed that all freely falling objects accelerate towards the Earth at the same rate objects due to their masses regardless of their mass • It is a “force at a distance” - two • A marble and a cannonball dropped at the same time objects do not need to come into from the same height will hit the ground simultaneously contact for them to exert a • The gravitational acceleration near the Earth’s gravitational force on one another surface is usually indicated by the symbol g and has a measured value of about 10 m/s 2 • As with the law of inertia, our • An object dropped from rest will be moving at 10 m/s understanding of gravity begins with after 1 second, 20 m/s after two and so on Galileo Galilei (neglecting air resistance) Isaac Newton Weight vs Mass • Newton realised that if all objects fall with the same • Weight is the gravitational force F g acting on acceleration, then the gravitational force on an an object object must be determined by its mass • An object’s weight thus depends on its • Recall that Newton’s 2nd law says location, whereas its mass does not acceleration = Force/mass • On the Earth’s surface, weight is equal to • Since all objects have the same acceleration, then mass times g , the acceleration due to gravity the gravitational force divided by mass must be the same for all objects • It is incorrect (but common) to say that an • A larger mass feels a larger gravitational force: object “weighs 2 kg” F grav = mg • A 2 kg mass actually weighs about 2 kg x 10 • Note that gravitational mass is the same as inertial m/s 2 or 20 kg m/s 2 or 20 Newtons (20 N) mass - this equivalence is the basis for GR
Gravitational Force Gravitational Force • As with every other force, any gravitational force has • Newton realised that if doubling the mass of an an equal and opposite force (Newton’s 3rd law) object doubles the gravitational force between it and the Earth, then doubling the mass of the Earth would • Drop a 20 kg cannonball and it falls towards the do the same Earth • Thus the gravitational force experienced by an object • At the same time Earth falls towards the cannonball! is proportional to the product of the mass of the • We do not notice the Earth’s motion in this case object times the mass of the Earth: because the Earth is so much more massive than the F g = something x mass of Earth x mass of object cannonball • Since objects fall towards the centre of the Earth, • Each object feels an equal and opposite force but F g is an attractive force acting along a line between acceleration equals force divided by mass the two masses Gravitational Force Inverse Square Law • But why, reasoned Newton, should this law of • Kepler had already reasoned that since the gravity apply only to the Earth? Sun is at the focus of planetary orbits, then it must be exerting some influence over the • Surely the gravitational force between any planets’ motion two masses m 1 and m 2 should be given by the product of the masses: • He also reasoned that this influence weakens F g = something x m 1 x m 2 with distance - why else does mercury orbit so much faster than Jupiter or Saturn? • Above reasoning follows from Galileo’s observations of falling objects and Newton’s • The area of a sphere increases with the laws of motion square of its radius ( A = 4 � r 2 ) • But what is the “something” in the above • Thus Kepler reasoned that the Sun’s influence equation? should decrease with the square of distance Newton’s Universal Law of Inverse Square Law Gravitation • Kepler’s proposal was interesting but not a Gravity is a force between any two objects, • scientific theory as he lacked a good idea as and has the following properties to the true source of the influence and also 1. It is an attractive force acting along a straight lacked the mathematical tools to predict how line between the objects an object should move under such an influence 2. It is proportional to the product of the masses • Newton had both - he realised that gravity of the objects m 1 x m 2 should act between the Sun and the planets, 3. It decreases with the square of the separation r and that the gravitational force was probably between the objects Kepler’s “influence” F g = G x m 1 x m 2 / r 2 • • In this case, the “something” in Newton’s expression for gravity should diminish with the universal gravitational constant square of the separation between two objects
Weakness of Gravity • It is now possible to measure the value of the gravitational constant G using sensitive equipment: G = 6.67 x 10 -11 N m 2 / kg 2 • The force between two bowling balls placed 1 foot apart is F g � 4 x 10 -8 N, about the same as the weight of a single bacterium! • Gravity is only noticeable in everyday life because the Earth is so massive Acceleration due to Gravity Mass of the Earth • For an object of mass m , Newton’s 2nd law of motion • Rearranging the last expression for g , we find says F g = mg M � = g R 2 � / G • Everything on RHS may be measured • Universal law of gravitation says F g = G M � m / R 2 • g by acceleration of falling objects � • R � by altitude of celestial pole with latitude • Equating these two expressions gives mg = G M � m / R 2 • G via lab experiments � • The mass m appears on both sides and so may be • We find M � � 6 x 10 24 kg divided out to give • Newton inverted this argument to estimate a value g = G M � / R 2 for G by assuming that Earth has the same density as � • Thus the acceleration g due to gravity is independent typical rocks of the mass of the object - as observed by Galileo! Gravity and Orbits Predicting Orbits • Newton speculated that Kepler’s solar “influence” on • A full prediction of planets’ orbits requires the planets’ orbits is gravity, but a good physical use of the branch of mathematics known as theory should be testable calculus that Newton invented for the purpose • He lacked the sensitive apparatus to measure • However, we can still gain a conceptual gravitational forces directly, but he was able to show understanding of how orbits come about by a that his law of gravity predicted that the planets series of thought experiments should orbit the Sun just as Kepler’s empirical laws described • These are experiments that are not • Newton was thus able to explain Kepler’s laws executable in practice, but that still give us a • Gravity is just one example of a physical law that good conceptual grasp of a physical problem was first tested by astronomical observations
Falling around the Earth • In this thought experiment we fire a cannonball horizontally from a height of a few metres and neglect air resistance • As cannonball travels horizontally, it also falls towards the ground • The faster we fire the ball, the further it travels before hitting the ground • As the ball travels further and further, the ground starts to curve away from underneath it • If we fire the ball fast enough, it will maintain a constant height above the ground and complete an orbit of the Earth Captions Falling around the Earth Astronaut falling freely around the Earth An object orbiting the Earth is literally • “falling around it” - it is always falling towards the Earth’s centre Shuttle and astronaut First man-made satellite to orbit the Earth • experience same was the Sputnik I satellite launched in 1957 gravitational acceleration Astronauts float around the cabin of an • orbiting spacecraft for the same reason: both the spacecraft and the astronaut are in Both are independent free fall - according to Newton’s law of satellites sharing the gravity both accelerate towards the Earth same orbit at the same rate Orbital Velocity • How fast must Newton’s cannonball move to orbit the Earth? • An object moving round in a circle requires a centripetal force to prevent it from flying off in a straight line (Newton’s 1st law) • For a ball on a string, the string provides the force, for an object in orbit it is gravity • For a satellite on a circular orbit, force required for uniform circular motion = force provided by gravity
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