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Question If we build a very tall tower with a height of one Earth radius. What would be your weight when you make a measurement with our regular spring scale on the top of the tower? It would be only one-forth of your weight on the ground. F Fground Ftower Gmearth m you Gmearth m you (2rearth ) 2 2 rearth Gmearth m you r2 100kg 9.8m / s 2 98km m s 2 1 Gmearth m you 100kg 9.8m / s 2 2 27 . 5 km m s 2 4 rearth 4 Question If you take your spring scale with you to a space tour, orbiting the Earth at a distance of 1 Earth radius above the ground. What would be your weight when you make a measurement with you spring scale in orbit? It would read zero! F Fground Ftower Gmearth m you Gmearth m you (2rearth ) 2 2 rearth Gmearth m you r2 100kg 9.8m / s 2 98km m s 2 1 Gmearth m you 100kg 9.8m / s 2 2 27 . 5 km m s 2 4 rearth 4 This calculation is still valid. That is, you still experience the force. But when you are in orbit, you and the spring scale are both in constant free fall! Centrifugal Force? If you take your spring scale with you to a space tour, orbiting the Earth at a distance of 1 Earth radius above the ground. What would be your weight when you make a measurement with you spring scale in orbit? It would read zero! A different answer is that the gravitational forces are balanced by the centrifugal forces. Centrifugal force is a fictitious force that is “experienced” by an object in circular motion. When you are in circular motion, there is a force that acts on you all the time to change your direction. Therefore, you feel there is a force that’s throwing you out. Orbital Motion Newton’s theory of gravitation explains why planets move in elliptical orbits (Kepler’s First Law of Planet Motion), Additionally, it also tells us that there are other type of orbits an object can have in gravitational fields: • Bound Orbit: Ellipses Objects in bound orbits circle around the Sun in elliptical orbits. • Unbound Orbit: Parabola, Hyperbola Objects in unbound orbits pass through the Sun only once, and never return. Capture of a comet The encounter with Jupiter may put the originally unbound comet in the right place with the right speed of a bound orbit. Escape Velocity Escape Velocity is the minimum velocity that an object is required to attain in order to escape from gravitational field of another object. • Escape velocity does not depends on the mass of the escaping object. • However, heavier object will require more energy to escape! • Vescape= 40,000 km/hr = 11 km/sec from the surface of the Earth. Click image to start movie An object with speed exceeding the escape velocity can escape from the gravitational field of Earth An object with the right velocity will stay in orbit Tides Because the gravitational force decrease quadratically with distance, the side of the Earth facing the Moon experiences stronger gravitational pull then the side facing away from the Moon. The net effect is a stretching of the entire Earth into elongated shape in the direction toward the Moon. The gravitational pull by the moon on this side is weaker The gravitational force is stronger on this side Spring and Neap Tides The Sun also causes tides on Earth. The tidal effect due to the Sun is about 3 times smaller than the tidal effect due to the Moon, because the Sun is too far away. Spring Tides When the Sun, the Earth, and the Moon as (more or less) aligned on a straight line, the tidal effect of the Sun and the Moon works together to cause much more pronounced tidal effect. Neap Tides When the tidal forces of the Sun and the Moon oppose each other, we get relatively small tides. Why do we have two tides per day? Click image to start animation Tidal Friction Gravitational pull of the Moon on the tidal bulges of the Earth generates a non-zero net torque opposite to the rotation of the Earth, causing the rotation of the Earth to slow down.=> Friction. Gravitational Pull of the Moon on Earth’s bulge Effects of Tidal Friction 1. Slowing down of Earth’s Rotation (previous page) 2. Increasing distance between the Earth and the Moon (this page) 3. Synchronous Rotation (next page) Because the tidal bulge of the Earth is always ahead of the Earth-Moon line due to Earth’s rotation, the Earth is pulling the Moon ahead of its orbit, making it rotates faster around the earth, thus moving it farther away from the Earth. This effect can also be explained by the conservation of angular momentum. The reduced angular momentum of the Earth (slower rotation) is transferred to the Moon, causing it to rotate faster around the Earth. Net force Gravitational pull due to the tidal bulge of the Earth on the Moon (Newton’s Third Law) Synchronous Rotation • • • Tidal friction also applies to the Moon’s rotation. The Moon may have being rotating much faster before than it does today, but the tidal friction effect due to the tidal force of the Earth had slowed the Moon’s rotation to the point where its rotational period is the same as its orbital period => Synchronous Rotation. Most of the moons of the jovian planets rotate synchronously. Pluto and its companion (not its moon any more!) Charon both rotate synchronously. Why do we always see the same face of the Moon? • The tidal force of the Earth stretches the Moon, just like the tidal force of the Moon causes the tide on Earth. • If the Moon is trying to rotate faster or slower, the gravitational pull of the Earth on the bulge A is stronger than on bulge B (because of shorter distance, Newton’s law of gravity), it will be pulled back. A Moon Earth B Momentum and Energy Momentum Momentum • A quantity describing the motion of an object that depends on both the mass and the velocity of the object • An object with mass m moving with a velocity v has momentum P defined by P=mv Example of Momentum • You can stop a rolling shopping cart in a slopped parking lot, but you cannot easily stop a rolling car (with the same speed) in the same parking lot… The momentum of the heavy car is much larger than that of the shopping cart moving at the same speed, and much larger force is needed to stop the car. • Consider a baseball (heavy object) and a bullet (light object)… • The baseball thrown by a person cannot easily break a wooden board. • The bullet fired by a gun can easily penetrate the wooden board. The momentum of the fast moving light object is much higher than that of the slow moving heavy object. Momentum is the product of mass and velocity! 2m v P = 2mv m 2v P = 2mv Force and Momentum • Force and momentum are related by Force = rate of change in momentum or F = dP/dt • This means that to change the momentum of a moving object, we need to apply a force to it (assuming that the mass of the object remains constant)… Linear and Angular Motions Correspondence Between Linear and Angular Motions Linear Motion Angular Motion Linear Velocity Angular Velocity Force Torque Linear Momentum = m v Angular Momentum = m r v Examples of Angular Momentum Orbital motions are more easily described as angular motion. For example, an object in circular motion has constant angular momentum. But its linear momentum is constantly changing. The 24-hour rotation of the Earth possesses angular momentum also. v r Basic Types of Energy • Energy of Motion, or kinetic energy – Energy associated with motion, E = ½ mv2. – Thermal Energy is associated with the collective kinetic energy of a system of many particles. • Energy carried by light, or radiative energy • Stored energy, or potential energy – Gravitational Potential Energy – Chemical potential energy is stored in gasoline and battery – A person on the top floor of a tall building has more gravitational potential energy than one that sits at the ground floor of the building. – A compressed spring has energy stored in it. – Mass energy: Matter can be converted into energy (Einstein’s famous equation). Thermal Energy • Thermal energy is the total kinetic energy of a system of many particles in random motion… • Temperature is a measure of how much thermal energy a system has. • Thermal energy does not include the kinetic energy of the whole system moving as a whole. v Gravitational Potential Energy The gravitational potential energy between two bodies with mass m1 and m2 separated by a distance r is given by Gm1m2 Eg r m2 r m1 Mass Energy • Mass can be converted into energy (Einstein)… E = mc2 • In nuclear fission and fusion reactions, a small amount of the mass is converted into energy according to Einstein’s formula, generating a very large amount of energy. Energy Comparison • Table 4.1 of Textbook. Conservation Laws Conservation law states that certain properties of a physical system remain the same unless something is done to change it. Conservation of Momentum – The momentum of a moving object will remain unchanged unless a force is acted upon it. This is true regardless of how far the object has moved. – the total momentum of all interacting objects always stays the same. Conservation of energy means that – the total energy of a system remains constant unless more energy is added into the system, or some energy is removed from the system. – Within a closed system, the energy can change from one form into another, but the total energy is always the same. Conservation of momentum and conservation of energy are the two types of conservation laws that we encounter most frequently in astronomy. Conservation of Linear Momentum Consider two balls each with mass m, initially at rest placed on the two ends of a compressed spring, as depicted in (a). Then, the spring is released, pushing the two balls moving with speed –v and v in opposite direction, as depicted in (b)… • The total momentum of the two balls in (a) is zero. • The total momentum of the two balls after the spring is released (b) is still zero, although the two balls are now moving. The total linear momentum is conserved. It is the same before and after the compressed spring is released. Spring is compressed (a) v=0 v=0 m P=0 m Spring is released (b) -v v m P = -mv + mv = 0 m Conservation of Angular Momentum • The angular momentum of a rotating body is constant unless an torque is applied to it. • When a net torque is applied to an object, the object will change its rotational speed. • Like mass, which determines how fast an object can react to applied force, the moment of inertia determines how fast an object can respond to an applied torque. • The ice skater is in fact changing her moment of inertia with respect to her rotation axis. It is larger when she extends her arms, thus in order to satisfy angular momentum Conservation of Angular Momentum in Astronomy—Orbital Motion • Although the orbital speed of Earth around the Sun is changing according to its distance to the Sun, its angular momentum is constant regardless of its orbital speed… Conservation of Angular Momentum—Earth-Moon System • The total angular momentum of the Earth-Moon system is also conserved—Earth probably use to rotate much faster, with the Moon much closer to Earth. Because of the tidal effect, the rotation of the Earth slowed while the Moon move farther away from Earth (and thus rotate around Earth faster, Kepler’s third law). In terms of angular momentum conservation, the angular momentum of the Earth decreases while the angular momentum of the Moon increase, but the total angular momentum remains constant. Conservation of Angular Momentum—The Solar System • All the planets of the solar system rotate in the same direction around the Sun, and their orbital planes are pretty much the same…Why? – Angular Momentum Conservation: The orbital rotation follows the original rotation of the planetary nebular that forms the planets. Examples of Angular Momentun Conservation in Daily Life Riding Bicycle: • When you ride a bicycle, you don’t fall off the bike easily. But when you stop, is it very difficult to maintain your balance. Why? • When you make a turn, you tip the bicycle to the side. Why doesn’t the bicycle just fall to the ground? • If you are riding a motorcycle real fast, then you can tip over much more when you make the turn. Why? These are consequences of Angular Momentum Conservation! Conservation of Energy in the Ball-Spring Sytem Consider two balls each with mass m, initially at rest placed on the two ends of a compressed spring, as depicted in (a). Then, the spring is released, pushing the two balls moving with speed –v and v in opposite direction, as depicted in (b)… • The total kinetic energy E of the two balls is zero in (a) • There are potential energy V stored in the compressed spring in (a) • The potential energy stored in the compressed spring is released in (b), and transfomed into the kinetic energy of the two balls. The total energy U of the balls and spring system is conserved. The potential energy of the spring is converted into kinetic energy of the balls. Spring is compressed (a) v=0 v=0 m U = V, E=0 m Spring is released (b) -v v m U=E = ½ m(-v)2 m + ½ mv2 = mv2, V=0 Conservation of Energy of a Falling Ball When a ball is thrown up and then falls down to the ground, the total energy of the ball is conserved… – Chemical potential energy stored in our muscle is converted in kinetic energy of the ball going up. – The kinetic energy is converted into gravitational energy as the ball gains height but loss speed. – The gravitational potential energy is converted back into kinetic energy again as the ball falls and gains speed. – When the ball falls to a level lower than where it started, its original gravitational energy is converted into kinetic energy, making it falls at a higher speed. Conversion of Gravitation Energy into Thermal Energy • A cloud of interstellar gas has more gravitational potential energy when it is more spread out. • As the cloud collapse under its own gravitational pull, the gravitational potential energy is converted into thermal energy of the system. • If a star is formed in this process, then some of the mass energy will be converted into radiative energy. Weighing the Earth We can use Newton’s Theory of Gravity to derive 2 4 p2 a3 G(m1 m2 ) Where m1 and m2 are the masses of two objects (in kg) orbiting each other, p is the orbital period (in second), and a is the average radius of the orbit (in m). (Note that this is exactly Kepler’s Third Law of Planet Motion). Therefore, we can use it to determine the mass of the Sun, the planets, or even black holes as long as we can measure 1. The period p of the orbital motion, 2. The average radius a of the orbit, And that one of the object is much more massive than the other… We know that • The average orbital period of the Moon around the Earth is about 27.3 days p = 2.35 106 sec • The average distance between the Earth and the Moon is 384,000 km. a = 3.84 108 m If we assume that mearth >> mmoon, so that we can write 2 4 p2 ~ a3 , Gmearth or mearth 4 2 3 4 2 (2.35106 ) 3 24 ~ a [ kg ] 6 10 [kg ] 2 11 8 2 Gp 6.67 10 (3.8410 ) Using mearth derived from the observing Moon’s orbital motion around Earth, we can calculate the gravitational acceleration on the surface of Earth: Gmearth 6.671011 6 1024 2 a 9 . 8 m / sec r2 (6.4 106 ) 2