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

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

2. 4.3 Conservation Laws in Physics: Our goals for learning: • 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?

3. Conservation Laws • The momentum, angular momentum and energy of an ISOLATED SYSTEM are conserved • An ISOLATED SYSTEM is a system whose only interactions are internal to the system

4. Conservation of Momentum • The total momentum of interacting objects (isolated system, i.e. all interactions are internal) cannot change unless an external force is acting on them • Interacting objects exchange momentum through equal and opposite forces

5. What keeps a planet rotating and orbiting the Sun?

6. 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 (torque) is acting on it • Earth experiences no twisting force as it orbits the Sun, so its rotation and orbit will continue indefinitely

7. Angular momentum conservation also explains why objectsrotate faster as they shrink in radius:

8. Where do objects get their energy? • Energy makes matter move. • Energy is conserved, but it can: • Transfer from one object to another • Change in form

9. Basic Types of Energy • Kinetic (motion) • Radiative (light) • Stored or potential Energy can change type but cannot be destroyed.

10. 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 the average kinetic energy of the many particles in a substance.

11. Temperature Scales

12. Thermal energy is a measure of the total kinetic energy of allthe particles in a substance. It therefore depends both on temperature AND density Example:

13. Gravitational Potential Energy • On Earth, depends on: • object’s mass (m) • strength of gravity (g) • distance object could potentially fall

14. 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.

15. Mass-Energy E = mc2 • Mass itself is a form of potential energy • A small amount of mass can release a great deal of energy • Concentrated energy can spontaneously turn into particles (for example, in particle accelerators)

16. 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.

17. What have we learned? • 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 only transformed from one type to another. • Energy comes in three basic types: kinetic, potential, radiative.

18. 4.4 The Universal Law of Gravitation • Our goals for learning: • What determines the strength of gravity? • How does Newton’s law of gravity extend Kepler’s laws?

19. What determines the strength of gravity? The Universal Law of Gravitation: • Every mass attracts every other mass. • Attraction is directly proportional to the product of their masses. • Attraction is inversely proportional to the square of the distance between their centers. • NOTE: the mass is the GRAVITATIONAL mass, not the INERTIAL mass

20. IMPORTANT CONSIDERATIONS The Universal Law of Gravitation: • Is universal, i.e. works everywhere in the universe for any to pairs of objects that have mass • The force can become very large if the distance between the two objects is very small • The Law does not specify the size or the shape of the bodies. Just their mass. It works the same whether they are big or small, cubes, spheres or just small points

21. Gravitational Potential Energy • This is the energy that two body are potentially capable of, just because of their mutual gravity • This energy is real: it can be transformed into, e.g., kinetic energy, or heat or other types • It is negative, because it is energy that the system gives to us (so, it is maximum at infinity) • The gravitational potential means that two bodies are capable to produce energy, i.e. work, thanks to the fact that they can attract each other, hence cause each other to move

22. Gravitational Potential Energy • You can emphasize the body which generates the gravity field and the body subject to it by re-writing the gravitational potential energy as above. • This is useful when m<<M, i.e. Earth and a small body. • This is the “test particle” approximation. • The physical meaning is that the small body’s gravity is negligible compared to the large one, which is what sets the gravity field

23. How does Newton’s law of gravity extend Kepler’s laws? • Kepler’s first two laws apply to all orbiting objects, not just planets • Ellipses are not the only orbital paths. Orbits can be: • Bound: ellipses (circles are special ellipses) • Kinetic energy is less than gravitational potential energy • Unbound: parabola, hyperbola • Kinetic energy is less than gravitational potential energy

24. Center of Mass • Because of momentum conservation, orbiting objects orbit around their center of mass

25. Newton and Kepler’s Third Law His 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. • Examples: • 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.

26. Newton’s Version of Kepler’s Third Law p = orbital period a=average orbital distance (between centers) (M1 + M2) = sum of object masses In other words: the law is general, it does not apply only to the Solar System If M1>>M2, for a circular orbit (s=2Πa) and remembering that vorb=s/p:

27. What have we learned? • What determines the strength of gravity? • Directly proportional to the product of the masses (M x m) • Inversely proportional to the square of the separation • 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.

28. 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?

29. How do gravity and energy together allow us to understand orbits? More gravitational energy; Less kinetic energy • Total orbital energy (gravitational + kinetic) stays constant if there is no external force • Orbits cannot change spontaneously. Less gravitational energy; More kinetic energy Total orbital energy stays constant

30. Changing an Orbit • So what can make an object gain or lose orbital energy? • Friction or atmospheric drag • A gravitational encounter.

31. Escape Velocity • If an object gains enough orbital energy, it may escape (change from a bound to unbound orbit) • Escape Velocity from Earth ≈ 11 km/s from sea level (about 40,000 km/hr) • The Escape Velocity is the one required for the kinetic energy to be equal to the gravitational potential energy

32. Deriving the Escape Velocity • To derive the escape velocity remember the meaning of gravitational potential energy and the fact that it is negative. • Namely, if an object has Vorb is equal to Vesc that object is free from the gravity, i.e. must have zero total energy.

33. Escape and orbital velocities don’t depend on the mass of the test particle: Note the similarity of the two equations. Note also that for any given orbit, the escape velocity is only √2 ≈ 40% larger than the orbital velocity.

34. How does gravity cause tides? • Moon’s gravity pulls harder on near side of Earth than on far side • Difference in Moon’s gravitational pull stretches Earth (air and water stretch a lot, rock significantly less. • Earth rotates, which creates a stretch wave that propagates. Who pays the bill for this constant stretching? The Earth’s rotation energy: Earth is slowing down! • In which condition is there no propagating stretch wave? Synchronous rotation and revolution.

35. Tides and Phases Size of tides depends on phase of Moon

36. Tidal Friction • Tidal friction gradually slows Earth rotation (and makes Moon get farther from Earth). • Moon once orbited faster (or slower); tidal friction caused it to “lock” in synchronous rotation.

37. Why do all objects fall at the same rate? • The gravitational acceleration of an object like a rock does not depend on its mass because Mrock in the equation for acceleration cancels Mrock in the equation for gravitational force • This “coincidence” was not understood until Einstein’s general theory of relativity.

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