noticed - planets and scientists - some of the role in - as inaccurate - as cision bodies another Tycho Brahe

Transcription

1 Lecture Notes (Planetary Motion) Intro: - for thousands of years, humans have watched the skies and noticed that the stars move in regular paths - planets and comets were also seen, but their motion seemed irregular to early scientists - it wasn't until the late 1500's that the motion of heavenly bodies began to be understood - some of the scientists that played a big role in Galileo, Kepler, Brahe, and Newton this process were Observed Motion: - Tycho Brahe was a Danish scientist ( ) who became interested in astronomy upon observing a solar eclipse when he was thirteen - as he studied science/mathematics, hee discovered that old observations were inaccurate - as a result, he started to design methods and instruments for high-prec cision measurements of positions of celestial bodies - while in college, he got into a fight with another student over who was the best mathematician Tycho Brahe

2 - this conflict ended up in a duel wheree he received a deep wound in his nose; for the rest of his life he wore a metal plate to cover his scar - Brahe was given the island of Hven by the King of Denmark to study the motion of celestial bodies; it was at Hven where he built his observatory - Brahe's lab cost an estimated 5% of Danish GNP to build - Brahe was later exiled by the King in Brahe found another supported in the Bohemian Emperor Rudolph II; Rudolph II brought Brahe to Prague (modern day capital of the Czech Republic) and funded his continuing work - it was in Prague that Brahe was introduced to Johann Kepler who became his lab assistant; Brahe died less than a year later - Kepler took the dataa Brahe accumulat ted for over twenty years and was able to draw many famous conclusions Emperor Rudolph II Kepler: - Johann Kepler ( ) was born in Germany, which was at this time part of the Holy Roman Empire - Kepler initially studied to be a Lutheran minister but was excluded from the churchh when he refused to adhere to some of the church's beliefs - Kepler also refused to convert to Catholicism; this forced Kepler to move to Prague, where he took his position as assistant to Brahe Johann Kepler

3 - after Brahe's death, Kepler used Brahe's data to formulate his three laws of planetary motion Kepler's First Law: - Kepler reported his first law of planetary motion in his first law states that planets move in ellipsess with the Sun at one focus Kepler's Second Law: - Kepler's second law was published in this law states that the radius vector describes equal area in equal times - the object in an elliptical orbit moves quickly when the radius vector is short and moves slowly when the radius vector is long Kepler's Third Law: - Kepler's third law was published in 1618

4 - this law states that the squares of the periodic times are other as the cubes of the mean distances to each - this statement can be written mathematically as: T T A 2 3 B B - Kepler's first two laws apply to each planet, moon, or satellite individually - Kepler's third law relates the motion of severall satellites about a single body Ex. this law is used to compare distances and periods of artificial satellites orbiting the Earth Universal Gravitatio on: - it was nearly forty years after Kepler'ss death, that another scientist continued work on planetary motion - that scientist was Sir Isaac Newton - Newton began his work on planetary motion by understanding that the magnitude of the force on the planet resulting from the Sun must vary inversely with the square of the distance between the center of the planet and the center of the Sun; this can be written as F 1/d 2 r r A - Newton further realized that the forcee acted in the direction of the line connecting the centers of the tow bodies

5 - Newton was uncertain, however, that the force acting between the planets and the Sun was the same force that caused objects to fall down on the Earth - upon watching an apple fall to the Earth, Newton hypothesized that the apple fell becausee the Earth attracted it - according to Newton's third law, he believed that the apple also attracted the Earth; the force of these attractions would be proportional to their masses - the name given to this attractive forcee was gravitational force - Newton was confident that the laws governingg motion on Earth would work anywhere in the universe - the same force of attraction, Newton concluded, would act between any two masses; he incorporated this idea into his Law of Universal Gravitation F mm G r in this equation, r is the distance between the centers of the masses and G is a universal constant

6 Weighing the Earth: - we can intuitively tell that the universal gravitational constant is small; you cannot feel event the slightest attraction between two massive railroad engines - it took over 100 years from the time of Newton until a device sensitive enough to measure the gravitational force was developed - in 1798, an English scientist used a device thatt could detect the force of gravity; it was called a torsion balance - Cavendish's device used a light source, a mirror, a rod supported by a cable with a mass on each end, and two large metal spheress - the force of attraction between the masses on the ends of the rod and the two large metal spheres caused the cable attached to the rod to twist - this twisting of the cable caused the light reflected by the mirror to move some distance

7 - upon measuring the masses of the spheres and the distance between their centers, Cavendish found an experimental value for G - Cavendish calculated G as g 2 N m 2 /kg - Cavendish used G to determine the mass of the Earth and the Sun; the Earth's mass is kg Using Newton's Law of Universal Gravitation: - Newton was able to relate his law of universal Kepler' 's third law of planetary motion gravitation to 2 - the resulting equation: T 2 4π Gm - this equation solvess the time (T) needed for a planet to make one revolution about the Sun s r 3 m s = mass of the Sun r = radius of the planet's orbit G = Newton' s universal gravitational constant, Nm/kg 2 2 Motion of Planets & Satellites: - Newton used a drawing to illustrate a motion of satellites thought experiment on the

8 - in the figure above, a cannon is perched atop a mountain; the cannon fires a ball horizontally with a given horizontal speed - the cannonball is a projectile, therefore it has vertical and horizontal components and follows a parabolic course - in the first second of flight, the cannonball will drop 4.9 m; even if the horizontal speed were increased, the cannonball would still drop 4.9 m - because the surface of the Earth is curved, it is possible for a cannonball with just the right horizontal speed to fall 4.9 m at a point where the Earth's surface has curved 4.9 m from the horizontal - this means that after one second, the cannonball is at the same height above the Earth as it was initially - the curvature of the projectile matches precisely the curvature of the Earth; so the cannonball never gets any closer or farther away from Earth's curved surface - the cannonball is said to be in orbit - therefore, an object at the Earth's surface with a horizontal speed of 8 km/s will keep the same altitude and circle the Earth as an artificial satellite - this experiment did not take into air resistance into account; air resistance is largely removed over 150 km above the Earth's surface - a satellite in orbit that is always the same height above Earth, moves with uniform circular motion where: a c = v 2 /r F c = ma c F c = mv 2 /r

9 - solving for the speed of an object in circular orbit, we can rewrite the equation to: v Gm E r - the period, T, of an object circling the Earth is written as: T 2π 3 r Gm E - note that orbital speed and period (v and T) are independent of the mass of the satellite - satellites are accelerated by the speeds they need to achieve orbit by large rockets - due to Newton's 2nd Law, F net = ma, the more massive the satellite, the more power the rocket must be to put it into orbit Gravitational Field: - gravity is not a contact force; instead it can act on bodies that are not even close together - while studying magnetism, Michael Faraday developed the concept of fields to explain how magnets attract objects - this field concept was later applied to gravity; it was proposed that anything with mass is surrounded by a gravitational field and that it is this field which interacts with objects Einstein's Theory of Gravity: - Newton's Law of Universal Gravitation allows us to calculate the force that exists between two bodies because of their masses

10 - the gravitational field concept allows us to picture the way gravity acts on bodies far away, but does not explain the origin of gravity - Einstein proposed that gravity is not a force, but rather of space itself an effect - Einstein proposed that mass changes the space around it; it causes space to be curved - bodies are accelerating because of thee way they follow curved space this The large ball in the center represents a massive body such as a star. Its weight curves the sheet near it. The ball bearings rolling on the sheet are deflected by this curvature and go around the large ball, in the same way that planets in the gravitational field of a star can orbit it. - Einstein's theory, called the general theory of relativity, makes many prediction ns about how massive objects affect one another - to this day, virtually every experiment has confirmed Einstein's hypothesis

11 Gravitational Lensing: - one of Einstein's most interesting predictions is that gravity can act like a lens - the gravity of a massive object or objects such as a galaxy or black hole is so powerful that it distorts surrounding space; the gravitational forces of the massive object(s) deflects the light from an object lined up behind it and amplifies it, like a glass lens bending and focusing starlight in a telescope

12 Because gravity is attractive, matter always warps spacetime so that its light rays bend toward each other. - this can result in multiple images of the same object; for example, in the picture below we see four images of the same quasar being lensed by the galaxy in the middle of the images - this phenomenon is called the Einsteinn Cross

13 Black Hole: - if an object is extremely compact and massive,, then light is bent totally back to the object; as a result, it absorbs all light that hits it and reflects no light back Approaching A Black Hole: - if you watch someone else fall in, you'd see them approach the black hole normally; then 1. the light coming from the person gets redshifted; they'll start to take on a redder hue and then, eventually, will requiree infrared, microwave, and then radio "vision" to see 2. the speed at which they appear to fall in will get asymptotically slow; they will appear to fall in towards the event horizon at a slower and slower speed, never quite reaching it 3. the amount of light coming from them gets less and less; in addition to getting redder, they also will appear dimmer, even if they emit their own source of light - the person falling in notices no difference in how time passes or how light appears to them; they would continue to fall in to the black hole and cross the event horizon as thoughh nothing happened

14 Falling Into A Black Hole: - once in the black hole, the differencess in gravitational force, known as tidal forces, would start to stretch your body; this processs is called spaghettification * * * * * * * tearing your extremities (head, arms, legs) from your torso, tearing the individual muscles, tendons, ligaments, etc., apart from your body tearing individual cells apart from one another, tearing the organelles inside each cell apart, destroying cells themselves tearing the individual molecules apart into atoms, tearing your atoms apart into nucleii and electrons, and finally tearing the individual nuclei apart into, eventually, quarks and gluons

15 Lecture Notes (Impulse & Momentum) Intro: - earlier in the year, we introduced Galileo's Principle of Inertia and we talked how Newton used this idea to formulate his first law of motion - Newton discussed inertia in terms of objects both at rest and in motion; today we will discuss the concept of inertia in motion only - when you are dealing with both momentum and motion you are talking about momentum - momentum is inertia in motion; specifically, it is the product of mass and velocity Momentum = mass velocity p = mv - the unit of momentum is kg m/s - it is intuitive to say that an aircraft carrier is harder to stop than a plastic toy boat moving at the same speed; the carrier has more momentum than the toy boat - by looking at the definition of momentum you can see that a moving object can have a large momentum either by having a large mass or a large velocity or both - the aircraft carrier has more momentum traveling at the same speed as the toy boat by virtue of its larger mass - the toy boat could have as much or even more momentum as the aircraft carrier, but its velocity would have to be extremely fast

16 Impulse: - a change in momentum can occur if there is a change in mass, a change in velocity, or both - if the momentum of an object changes while the mass remains unchanged, which is usually the case, then it is due to a change in velocity - changes in velocity are due to accelerations; accelerations are due to net forces acting on objects - the greater the net force on an object, the greater the acceleration, the greater the velocity change, the greater the change in momentum - there is another important factor in changing momentum; time - apply a force to a crate on the ground for a brief amount of time and you get a small change in momentum; if you apply that same force for a longer amount of time, then you get a larger change in momentum - a long sustained force will produce a greater change in momentum than the same force applied briefly - in terms of momentum, both the amount of force and the amount of time during which the force act are important; the product of these two terms is called the impulse Impulse = force time interval Impulse = FΔt - whenever you exert a force on an object you also exert an impulse

17 - the resulting acceleration depends on the force; the resulting change in momentum depends on the force and the time in which the force acts Impulse-Momentum Equation: - the relationship of impulse and momentum is derived from Newton's 2nd law of motion; F = ma; then time interval of impulse is hidden in the acceleration variable (Δv/Δt) - if you rearrange Newton's 2nd law you get: Force time interval = change in mass velocity (momentum) FΔt = mδv Increasing Momentum: - if you want to increase the momentum of something as much as possible, you not only apply the greatest force possible, you also apply that force for as long as possible - for example, long range cannons have long barrels; the longer the barrel the greater the velocity of the shell - this occurs because the longer barrel allows more time for the force of the exploding gunpowder to act on the shell (increasing Δt) - the force that is exerted on most objects varies over time; for example, in the cannon the gunpowder applies a greater force initially and the begins to lessen as the shell goes down the barrel - so, in order to simplify things, we will use the average force acting on an object

18 Decreasing Momentu um (Over a Long Time): - one intuitive way to discuss decreasin ng momentum is to imagine you are in a car out of control and you can either slam into a concrete wall or a haystack - it' 's obvious which solution is the better choice; the more important reason in physics is why this is the better choice - in either case, hitting the wall or the haystack, your momentum will be decreased by the same impulse (FΔt); remember impulse is the product of force and time, but does not mean the same force or same time - you have a choice; if you hit the haystack your time will be extended (the time during which your momentum is brought to zero) - this longer time corresponds to a lesser force; for example, if you extend the time by 100 times, then the force will decrease by if you hit the wall, the time will be reduced, so the force will increase

19 - people often use the knowledge of impulse to their advantage; for example, a wrestler thrown to the floor will relax their muscles in orderr to extend the crash into a series of smaller impacts - the wrestler will have his/her foot, knee, hip, ribs, and shoulder hit the mat in turn; this will increase the time of impact whichh reduces the amount of force on thee body - a boxer will move his face backwardss in order time of impact of an opponent's blow to increase the - ballet dancers preferr to dance on a wooden floor rather than a concrete floor, because the wooden floor has "give" Decreasing Momentu um (Over a Shortt Time): - in certain situations it is advantageouss to decrease your momentum m quickly - when you decrease the time interval, the size of the force will increase; for example, a karate master will endeavor to hit a stack of bricks with a quick blow; this will maximize the force of the blow

20 Bouncing: - impulses become even greater if bouncing occurs - bouncing will amplify the impulse because it takes an additional impulse to throw an object back after collision - for example, take the case of a flowerr pot hitting you head; if the pot breaks upon colliding with your head itt does some damage; if the pot bounces off your head without breaking, it would do even more damage to your body - your head has to provide an additiona al impulse to send back off your head the pot - the effect was known and used for mining in Californiaa during the gold rush; a man named Pelton created a water wheel with curved blades to maximize the impulse of the water on the wheel

21 Lecture Notes (Conservation of Momentum) Intro: - when you throw a ball, shoot a bullet or give someone a push you tend to move backward - Newton s third law of motion explained that action and reaction were equal and opposite forces - a study of momentum can describe the motion of interacting bodies mathematically - the two most common interactions we can study are explosions and collisions; we ll start with explosions because they are a bit simpler Explosions: - an explosion can be thought of as a single object separating into two or more fragments - the word explode was first used to mean burst with destructive force in the nineteenth century when a mathematical treatment of explosions became necessary; prior to that, the Latin verb explodere meant to drive off the theatre stage with hisses, boos, loud noises and claps - it came from ex- meaning out and plaudere meaning clap ; many scientific words started off meaning something else

22 - consider a 10 kg bombb at rest that explodes into two fragments - if a 4 kg piecee (m 1 ) travels west at 15 m/ /s (v 1 ), then the 6 kg piece (m 2 ) would have moved in the opposite direction (at a speed v 2 ) - as there was no external unbalanced forces acting on the bomb (all forces were internal), we have a closed system and there would be no change in the total momentum of the system - this is called the law of conservation of momentum; in a closed system, the change in momentum is zero - relationships such as this can be applied to all sorts of explosions, a cannon or rifle being fired, a bomb exploding, a heart pumping a pulse of blood, a hose squirting waterr and even a nucleus givingg off radioactive particles

23 - the recoiling rifle has just as much momentum as the bullet - both the gun and the bullet have gained momentum, but since they are in opposite directions, they cancel out leaving the total momentum of the system at zero - this is exactly the same momentum the system started out as; no momentum was gained or lost - this example shows us two important qualities of momentum: 1. momentum is a vector quantity 2. momentum is conserved (only in a closed, isolated system) - the forces that are applied to an object, in order to accelerate it, must be external forces, not internal ones - external forces originate from outside the object you are studying; internal forces occur within the object itself - for example, the molecular forces within a rock do not change the momentum of the rock; similarly, if you are sitting in a car and push on the dashboard, you cannot change the momentum of the car; these are internal forces - cases in which the bodies explode in a straight line are not that common, however; explosions in two dimensions will be dealt with later Airplanes, Balloons, And Rockets: - conservation of momentum also applies to flight - with propeller-driven airplanes, the interaction occurs when the propeller pushes against the surrounding air molecules, increasing their momenta in the backward direction

24 - this is accompanied by an equal change of the airplane's momentum in the forward direction - releasing an inflated balloon is not like the airplane, because the molecules in the atmosphere are not necessary - the air molecules in the balloon rush out, acquiring a change in momentum toward the rear; this is accompanied by an equal change in momentum of the balloon in the forward direction - the air molecules do not need to push on anything; the balloon can fly through a vacuum - this is also true of rockets and explains why they can be used in space flight Collisions: - a collision occurs when objects crash into each other - some familiar collisions are: meteorite craters a billiard ball being struck by a cue a boxer punching a body bag hammering a nail into a piece of wood gas molecules bouncing off each other - collisions can be grouped into two types: Elastic (rebound), where objects bounce off each other (ex. gas molecules or billiard balls) Inelastic (coupled), where objects remain locked together (ex. a bullet in a target)

25 Elastic Collisions: - momentum is conserved in collisions; that is the total momentum of the system of colliding objects is unchanged before, during, and after the collision - if you have one billiard ball hitting another ball att rest, head on, then the original moving ball will come to rest and the second ball which was at rest will move with the speed of the colliding ball - this type of collision is called an elastic collision; without deformation or the generatio on of heat the objects collide Inelastic Collisions: - when objects stick together or are joined together they are coupled (Latin copula = to bond ) said to be - in a collision where the objects become coupled, the law of conservati ion of momentum still holds but the mass of the combined body after the collision is equal to the sum of the individual masses of the colliding bodies

26 - some examples of coupled collisions aree arrows sticking into their targets and two cars colliding head-on - in inelastic collisions, the colliding objects are severely deformed and they generate heat

27 Lecture Notes (Work & Energy) Intro: - one of the most central concepts in science is energy; the combination energy and matter makes up our universe - matter is the substance of the universe, while energy is what moves the substance - matter is what we can see, touch and feel; energy is somewhat more elusive to categorize; energy is not typically seen or felt - it was so abstract that the idea of energy was largely unknown until the mid 1800's - energy is both a thing and a process; people, places and things all have energy but we only see it when it is being transformed - before we talk further about energy it is customary to talk about a related concept; work Work: - when discussing momentum and impulse in the previous lecture, we discussed how time played a factor in the motion of an object - when discussing impulse, "How long" meant time; with work, however, "How long" means what distance - work is the product of the force and distance Work = Force Distance W = Fd - the unit of work is the joule (J); where 1 J = 1 N m

28 - when you lift an object off the ground you are doing work; the heavier the load ( F) or the higher you lift the object ( d), the more work you are performing - there are two things that must occur if work is being done: 1. a force must be exerted 2. the object must move - sometimes there are situations that appear to be work, but are actually not; for example, if you hold a barbell steady over your head, you are not doing any work (remember, the object has to move in order to do work) - lifting the same barbell up over your head is different however; in this case, work is being done - no work is done if you push on a wall but it does not move

29 Power: - the definition of work does not say anything about the time it takes to do the work - we know from experience, however, that the time it takes to perform work makes a difference; for example, say you went to Sam's and bought 25 cases of soft drinks for a party; your job was to unload them from your mom's car - you would obviously be much more tired (and unhappy) if you had to perform this task in 5 minutes rather than 30 minutes - this difference in how fast the work is done is called power - power is equal to the amount of work done per unit time Power = Work/time interval P = W/t - the unit of power is the watt (W); 1 W = 1 J/s - engines are valued because they can perform a lot of work very quickly Mechanical Energy: - energy is that which enables objects to do work - there are many types of energy; like work, energy is measure in joules (J) - we will focus of mechanical energy; mechanical energy is separated into two categories; potential and kinetic

30 Potential Energy: - an object may store energy because of its position; this is called potential energy (PE) - a book on a shelf, a rock tossed into the sky, a compressed spring, and a drawn bow all have potential energy; work was done to all of these objects which stored the energy - work is required to elevate objects against Earth's gravity; the amount of gravitational potential energy possessed by an object is equal to the amount of work done against gravity in lifting it Gravitational Potential Energy = Weight Height U g = mgd - the unit of potential energy (as with any type of energy) is the joule (J) - in the diagram below, we see three situations where a ball is raised three meters off the ground; in each case, the ball takes different paths to get to the three meter position off the ground

31 - in each case, the same amount of work is being done Kinetic Energy: - if we perform work on a moving object, then we can change its energy of motion - if an object is in motion, then by virtue of that motion, it is capable of performing work - kinetic energy (KE) is energy of motion; the kinetic energy of an object depends on its mass and speed Kinetic Energy = ½ mass speed 2 KE = ½mv 2 - notice that the speed variable is squared, so that if the speed is doubled, the KE is quadrupled; the change speed has a more dramatic effect on KE than a change in mass - when a ball is thrown, work is done on it, giving it KE; that moving ball can hit something and push against it and cause it to move; the ball can perform work equal to the amount of work put into the ball

32 - as a result: Conservation of Energy: Net force distance = kinetic energy Fd = ½mv 2 Work = ΔKE - an important concept about energy is to understand how it can change from one form into another

33 - for example, as you stretch a slingshot we perform work in stretching the rubber band; we give the slingshot potential energy - when the rubber band is released, the potential energy of the slingshot is transformed into kinetic energy equal to the potential energy of the rubber band - the stone then transfers this energy to the object it hits; in each case the amount of energy being transformed remains the same - the study of energy transformations led to another conservation law; the law of conservation of energy

34 - the conservation of energy law states that energy cannot be created nor destroyed; it may be transformed from one form into another, but the total amount remains the same

35 Summary: