2010 Pearson Education, Inc. Chapter 4 Making Sense of the Universe: Understanding Motion, Energy, and Gravity

Transcription

1 Chapter 4 Making Sense of the Universe: Understanding Motion, Energy, and Gravity

2 4.1 Describing Motion: Examples from Daily Life Some of the topics we will explore: How do we describe motion? (Speed, velocity, acceleration, acceleration of gravity, momentum, angular momentum, force) How is mass different from weight?

3 How do we describe motion? Precise definitions to describe motion: Speed: Rate at which object moves. It is the distance traveled per unit time speed = distance time Example: 10 m/s units of m s Velocity: The velocity includes speed and direction (In math it is called a vector) Example: 10 m/s, due east Acceleration: A change in velocity. It can be a change in speed or direction. The units are speed/time (m/s 2 ) It can be an increase of velocity (positive acceleration) or a decrease (negative acceleration)

4 All falling objects accelerate at the same rate (not counting friction of air resistance). On Earth, the acceleration of gravity g is g 10 m/s 2 This means that the speed increases 10 m/s with each second of falling. Written in an equation: v = g t The Acceleration of Gravity

5 The Acceleration of Gravity (g) Galileo showed that g, the acceleration of gravity is the same for all falling objects, regardless of their mass. This may seems to contradict our everyday experience. A feather drops slowly to the ground compared with a steel ball. The answer is the air resistance. A force exerted by the air as the object falls. In vacuum ( An example on the Moon surface) both objects will fall at the same rate Apollo 15 demonstration

6 Momentum, force, angular momentum Momentum of an object is the product of its mass and velocity Momentum = mass velocity m v This is called linear momentum A net force changes momentum, which generally means an acceleration (change in velocity). Rotational momentum of a spinning or orbiting object is known as angular momentum. Angular momentum = mass velocity radius = m v r If the mass, the velocity and the radius remain constant, the angular momentum is constant if no force is applied to the body. However, if we change the radius (without applying an external force) the velocity will change to keep the product m v r constant. We will see an example later.

7 Questions For each of the following is there a net force? Y/N 1. A car coming to a stop 2. A bus speeding up 3. An elevator moving up at constant speed 4. A bicycle going around a curve 5. A moon orbiting Jupiter

8 Questions For each of the following is there a net force? Y/N 1. A car coming to a stop: Y 2. A bus speeding up: Y 3. An elevator moving at constant speed: N 4. A bicycle going around a curve: Y 5. A moon orbiting Jupiter: Y

9 How is mass different from weight? Mass the amount of matter in an object Weight the force that acts upon the mass of an object You are weightless in free-fall!

10 Question On the Moon: A. My weight is the same, my mass is less. B. My weight is less, my mass is the same. C. My weight is more, my mass is the same. D. My weight is more, my mass is less.

11 Question On the Moon: A. My weight is the same, my mass is less. B. My weight is less, my mass is the same. C. My weight is more, my mass is the same. D. My weight is more, my mass is less.

12 You would weigh less on the Moon (less mass, weaker gravitational force) than on Jupiter or the Sun: Note: Mass weight: The person s mass never changes but the weight does.

13 Why are astronauts weightless in space? There is gravity in space. The gravitational force on the astronaut exerted by the Earth is less than at the surface but not zero. Weightlessness is due to a constant state of free-fall. The gravitational force keep pulling the astronaut toward the center of the Earth. Otherwise it will move in a straight line

14 An analogy using a ball attached to a string The inward force keep the ball moving in a circle. The magnitude of the force is F = ma In the case of the astronaut the inward force is the gravitational force.

15 Summary How do we describe motion? Speed = distance / time Speed and direction => velocity Change in velocity => acceleration Momentum = mass x velocity Force causes change in momentum, producing acceleration. Angular Momentum = product of mass x velocity x radius (An example, a sphere)

16 How is mass different from weight? Mass = quantity of matter Weight = force acting on a mass. More specifically is the force of gravity acting on the mass of a body Objects are weightless in free-fall.

17 4.2 Newton s Laws of Motion Topics we will explore: How did Newton change our view of the universe? What are Newton s three laws of motion?

18 Aristotle and Newton ideas Aristotle ( B.C.) proposed that the physical laws on Earth did not apply to heavenly motions. His ideas help him support that Earth was the center of the universe. By the time that Newton ( ) began working on his ideas and formulated the laws of motion and gravitation, the Copernican revolution and Galileo discoveries had displaced the Earth from it central position. Newton realized that gravity operated on Earth and heavens.

19 How did Newton change our view of the universe? Sir Isaac Newton ( ) He realized that the same physical laws that operate on Earth also operate in the heavens Heavens and earth were brought together as one universe Discovered laws of motion and gravity But he did much more than that: He experimented with light, designed the first reflecting telescope, invented calculus

20 What are Newton s three laws of motion? Newton s first law of motion: An object moves at constant velocity unless a net force acts to change its speed or direction.

21 Newton s second law of motion: Force = mass F = m a acceleration

22 Newton s third law of motion: For every force, there is always an equal and opposite reaction force.

23 Question How does the force the Earth exerts on you compare with the force you exert on it? A. Earth exerts a larger force on you. B. You exert a larger force on Earth. C. Earth and you exert equal and opposite forces on each other.

24 Thought Question How does the force the Earth exerts on you compare with the force you exert on it? A. Earth exerts a larger force on you. B. You exert a larger force on Earth. C. Earth and you exert equal and opposite forces on each other.

25 Question A compact car and a Mack truck have a head-on collision. Are the following true or false? 1. The force of the car on the truck is equal and opposite to the force of the truck on the car. 2. The momentum transferred from the truck to the car is equal and opposite to the momentum transferred from the car to the truck. Momentum : m v 3. The change of velocity of the car is the same as the change of velocity of the truck.

26 Thought Question A compact car and a Mack truck have a head-on collision. Are the following true or false? 1. The force of the car on the truck is equal and opposite to the force of the truck on the car. T 2. The momentum transferred from the truck to the car is equal and opposite to the momentum transferred from the car to the truck. T Momentum : m v 3. The change of velocity of the car is the same as the change of velocity of the truck. F

27 What have we learned? How did Newton change our view of the universe? He discovered laws of motion and gravitation. He realized these same laws of physics were identical in the universe and on Earth. What are Newton s three laws of motion? 1. Object moves at constant velocity if no net force is acting. 2. Force = mass acceleration (F = m a) 3. For every force there is an equal and opposite reaction force.

28 4.3 Conservation Laws in Astronomy Topics we will explore: Why do objects move at constant velocity if no force acts on them? What keeps a planet rotating and orbiting the Sun? Where do objects get their energy?

29 Why do objects move at constant velocity if no force acts on them? Objects continue at constant velocity because of conservation of momentum. The total momentum of interacting objects cannot change unless an external force is acting on them. Interacting objects exchange momentum through equal and opposite forces. The assumption is that there is no release of energy during the impact(elastic collision) If the mass of the two balls is the same, after the collision the second ball will move at the same velocity of the first ball. If the mass of the second ball is larger than the first ball, the velocity of the second ball will be smaller so that the product m x v is constant.

30 What keeps a planet rotating and orbiting the Sun?

31 The mutual gravitational attraction between the Sun and planets is responsible for their motion.

32 In the case of a planet, the force produced by the tension of the string is replaced by the force of the gravitational attraction between the planet and the Sun

33 Conservation of Angular Momentum Angular momentum = mass x velocity x radius = m x v x r The angular momentum of an object cannot change unless an external twisting force or torque (torque = f x r) is acting on it. Earth experiences no twisting force as it orbits the Sun, so its rotation and orbit will continue indefinitely because of conservation of angular momentum.

34 Conservation of angular momentum also explains why objects rotate faster as they shrink in radius. Left figure Angular momentum 1 = m x v1 x r1 Right figure Angular momentum 2 = m x v2 x r2 In the right figure, r2 is smaller that r1, then v2 must be larger than v1 to keep the product m x v x r constant (conservation of angular momentum)

35 Where do objects get their energy? Conservation of energy Energy cannot appear out of nowhere or cannot disappear into nothingness. But energy can change from one type to another: Transfer from one object to another. An example the collision of two balls where kinetic energy is transferred from one ball to another Energy can change in form, from kinetic to potential to radiative and vice versa Energy can be converted into mass and mass can be converted into energy.

36 Basic Types of Energy Kinetic (motion) E = ½ mv 2 Potential (stored) Gravitational potential energy Ep = m x g x h) Radiative (light) Energy can change type, but cannot be created or destroyed.

37 Thermal Energy: The collective kinetic energy of many particles (for example, in a rock, in air, in water) Thermal energy is related to temperature but it is NOT the same. Temperature is a measure of the average kinetic energy of the particles in a substance.

38 Temperature Scales The Kelvin scale is the scale used mostly in physics, astronomy and science in general The units of the Kelvin scale are Kelvin (K) At zero K, the thermal energy is zero. Molecules and particles have zero velocity at zero K Zero K is absolute Zero. There is no temperature lower than that

39 Thermal energy is a measure of the total kinetic energy of all the particles in a substance. It therefore depends on both temperature AND density. Example: A fluid (high density) and air (low density) What happens if you put your hand in: Water at 212 F Air at 400 F

40 Gravitational Potential Energy On Earth, depends on: object s mass (m) strength of gravity (g) distance object could potentially fall (h) Ep = m x g x h Ep is potential gravitational energy

41 Gravitational Potential Energy In space, an object or gas cloud has more gravitational energy when it is spread out than when it contracts. A contracting cloud converts gravitational potential energy to thermal energy. In other words, the temperature of the cloud is low before it start contracting but it can be very high after it finish contracting. The temperature in the central body (star) can be high enough to star the conversion of hydrogen into helium (10 million K).

42 Mass-Energy Mass itself is a form of potential energy. Einstein equation of energy-mass conversion: Mass can be converted into energy and energy can be converted into mass A small amount of mass can release a great deal of energy (for example, an H- bomb, a nuclear power reactor). Concentrated energy can spontaneously turn into particles (for example, in particle accelerators). E = mc 2

43 Conservation of Energy Energy can be neither created nor destroyed. It can change form or be exchanged between objects. The total energy content of the universe was determined in the Big Bang and remains the same today.

44 Summary Why do objects move at constant velocity if no force acts on them? Conservation of momentum What keeps a planet rotating and orbiting the Sun? Conservation of angular momentum Where do objects get their energy? Conservation of energy: energy cannot be created or destroyed but can only transformed from one type to another. Energy comes in three basic types: kinetic, potential, radiative.

45 4.4 The Universal Law of Gravitation Let s explore the following topics: What determines the strength of gravity? How does Newton s law of gravity extend Kepler s laws?

46 What determines the strength of gravity? Newton s universal law of gravitation: 1. Every mass attracts every other mass. 2. The force of attraction is directly proportional to the product of their masses. 3. The force of attraction is inversely proportional to the square of the distance between their centers. G is a constant called gravitational constant

47 Plot of force as function of distance Inverse square law

48 How does Newton s law of gravity extend Kepler s laws? Newton using his gravitational law, the laws of motion, trigonometry and calculus was able to deduce and generalize Kepler s law so they apply not only to planets but to any body orbiting another body. Kepler s first two laws apply to all orbiting objects, not just planets. Newton showed that Kepler s law are a consequence of the law of motion and the universal law of gravitation Ellipses are not the only orbital paths. Orbits can be: bound (ellipses). The objects goes around each other for ever if no external force act on the system Unbound (path that bring an object close to another only once; an example are some comets) Parabola hyperbola The circle, the ellipse, the parabola and the hyperbola are known as conic sections. These curves result from cutting the cone at different angles.

49 A definition of center of mass What s the Center of Mass? The equation for the center of mass: m 1 d 1 = m 2 d 2 (d 1 + d 2 = d)

50 Center of Mass

51 Orbit of bodies of different mass Two stars of about the same mass orbit around each other describing two ellipses of the same size Two stars with different mass, orbit around each other. The lowest mass star describe a larger ellipse. A star and a planet orbit each other. Because of the lower mass of the planet, the orbit of the planet is a large ellipse. Due to the large mass of the star, the center of mass is close (inside) the star. But the star also describe a small ellipse

52 Newton and Kepler s Third Law Newton s laws of gravity and motion showed that the relationship between the orbital period and average orbital distance of a system tells us the total mass of the system. Kepler s third law does not include the mass of the bodies involved. It only relates the period and the average distance Examples of the application of Newton s version of Kepler s law: Earth s orbital period (1 year) and average distance (1 AU) tell us the Sun s mass. Orbital period and distance of a satellite from Earth tell us Earth s mass. Orbital period and distance of a moon of Jupiter tell us Jupiter s mass.

53 Newton s Version of Kepler s Third Law a OR G( M M2) G p p a M M 2 p = orbital period a = average orbital distance (between centers) (M 1 + M 2 ) = sum of object masses

54 The change to Kepler s third law is small in the case of a planet orbiting the Sun, but larger in the case where the two bodies are closer in mass, e.g. Two stars orbiting each other (binary star system). Newton s version of Kepler s revised 3rd law: P 2 a 3 M total : Proportional For Planet-Sun systems: M total = M Sun + M planet M sun (M planet small compared with M Sun )

55 An application of the modified Kepler 3 rd Law to Jupiter and its satellites We want to obtain the mass of the Jupiter using the mean distance (a) of one of its moon (For example Io) and its orbital period (P) M total = M Jupiter + M Io But the mass of Io is much smaller than the mass of Jupiter The mass of Io can be neglected If a is in AU and P is in years, we can substitute the proportional sign for an equal sign M Jupiter =a 3 / P 2 The mass of Jupiter will be in solar mass

56 Summary What determines the strength of gravity? Force is directly proportional to the product of the masses (M 1 M 2 ) Force is inversely proportional to the square of the separation d How does Newton s law of gravity allow us to extend Kepler s laws? Applies to other objects, not just planets Includes unbound orbit shapes: parabola, hyperbola Can be used to measure mass of orbiting systems

57 4.5 Orbits, Tides, and the Acceleration of Gravity Our goals for learning: How do gravity and energy together allow us to understand orbits? How does gravity cause tides? Why do all objects fall at the same rate?

58 How do gravity and energy together allow us to understand orbits? Total orbital energy (gravitational potential energy+ kinetic energy) stays constant if there is no external force. Gravitational potential energy proportional to r Kinetic energy proportional to v 2 Total orbital energy stays constant. When a planet is closer to the Sun, it has more kinetic energy (larger v) and less gravitational potential energy (smaller r) When the planet moves to a point farther from the Sun, the kinetic energy is converted into gravitational potential energy. Orbits cannot change spontaneously. To change an orbit it is necessary to apply an external force

59 Changing an Orbit If an object can gain or lose orbital energy, it can change its orbit. So what can make an object gain or lose orbital energy? Friction or atmospheric drag Examples: The ISS (International Space Station) A gravitational encounter Example: Comets

60 Escape Velocity If an object gains enough orbital energy, it may change its orbit and escape (change from a bound to unbound orbit). Escape velocity from Earth 11 km/s from sea level (about 40,000 km/hr) G is gravitational constant M is mass of the object R is the distance above the object center To calculate the escape velocity from the surface, R is the radius of the object

61 How does gravity cause tides? Gravitational attraction weaker here Gravitational attraction stronger here Moon s gravity pulls harder on near side of Earth than on far side. Difference in Moon s gravitational pull stretches Earth. What causes the tidal effect is the differential gravitational attraction

62 Inverse square law How can we understand the differential gravitational attraction

63 The differential gravitational force and the tidal effect

64 Does the Sun has a tidal effect on the Earth? The Sun has much more mass than the Moon. The Sun gravitational attraction on the Earth is larger than the Moon gravitational attraction. But the Sun is much farther away than the Moon. The differential gravitational attraction of the Sun on the near and far side of the Earth is smaller than the effect of the Moon Both the Sun and the Moon cause tidal effect on Earth The moon has a larger differential gravitational attraction and causes larger tides.

65 Tides and Phases Size of tides depends on the phase of the Moon. Spring tides occur at new and full moon. The tidal forces of the Sun and the Moon add together Spring tides Neap tides occur at first quarter and third quarter moon when the Sun and the Moon tidal forces act at 90degrees from each other Neap tides The Earth develop two bulges. A consequence of this is that every 24 hours there are two high tides and two low tides.

66 Tidal Friction If the Earth didn t rotate, the tidal bulges would be oriented along the Earth-Moon line. Since the Earth rotate faster (24 hours) than the Moon orbital period (27 days), the tidal bulge is displaced a few degrees ahead from the line connecting the Earth-Moon. The slight misalignment of the bulges has two important effects: It slow down the Earth rotation and pulls the Moon ahead increasing the orbital distance Tidal friction dissipate rotational energy. This gradually slows Earth s rotation (and makes the Moon get farther from Earth). Tidal friction caused the Moon to lock in synchronous rotation. The result is that the Moon rotational period coincides with the orbital period.

67 Why do all objects fall at the same rate? a rock F g F g G M M Earth rock 2 M rock R Earth a rock G M EarthM rock 2 R Earth M rock G M Earth 2 R Earth The gravitational acceleration of an object like a rock does not depend on its mass because M rock in the equation for acceleration cancels M rock in the equation for gravitational force. This coincidence was not understood by Newton, even if his equations show it and it was not understood for 240 years. Until 1915 when Einstein discovered that it not a coincidence, it is a consequence of his General theory of relativity. This is something deeper about the nature of gravity.

68 What have we learned? How do gravity and energy together allow us to understand orbits? Change in total energy is needed to change orbit Adding enough energy (escape velocity) and the object leaves. How does gravity cause tides? The Moon s gravity stretches Earth and its oceans (Differential gravitational force). Why do all objects fall at the same rate? The mass of object in Newton s second law exactly cancels the mass in law of gravitation.