Homework 2 Unit 14 Problems 17, 19 Unit 15. Problems 16, 17 Unit 16. Problems 12, 17 Unit 17, Problems 10, 19 Unit 12 Problems 10, 11, 16, 17, 18 Unit 11 Problems 16, 19 Units to be covered: Units 16-18

Newton’s Universal Law of Gravitation Every mass exerts a force of attraction on every other mass. The strength of the force is proportional to the product of the masses divided by the square of the distance between them –Simply put, everything pulls on everything else –Larger masses have a greater pull –Objects close together pull more on each other than objects farther apart This is true everywhere, and for all objects –The Sun and the planets exert a gravitational force on each other –You exert a gravitational force on other people in the room!

Surface Gravity Objects on the Moon weigh less than objects on Earth This is because surface gravity is less –The Moon has less mass than the Earth, so the gravitational force is less We let the letter g represent surface gravity, or the acceleration of a body due to gravity F = mg g= (GmM/R 2 )/m=GM/R 2 On Earth, g = 9.8 m/s 2 g on the Moon is around 1/6 as much as on the Earth!

Centripetal Force If we tie a mass to a string and swing the mass around in a circle, some force is required to keep the mass from flying off in a straight line This is a centripetal force, a force directed towards the center of the system The tension in the string provides this force. Newton determined that this force can be described by the following equation:

We know that for planets, the centripetal force that keeps the planets moving on an elliptical path is the gravitational force. We can set F G and F C equal to each other, and solve for M! Now, if we know the orbital speed of a small object orbiting a much larger one, and we know the distance between the two objects, we can calculate the larger object ’ s mass! Masses from Orbital Speeds

Newton’s Modification of Kepler’s 3 rd Law Newton applied his ideas to Kepler ’ s 3 rd Law, and developed a version that works for any two massive bodies, not just the Sun and its planets! Here, M A and M B are the two object ’ s masses expressed in units of the Sun ’ s mass. This expression is useful for calculating the mass of binary star systems, and other astronomical phenomena

As we saw in Unit 17, we can find the mass of a large object by measuring the velocity of a smaller object orbiting it, and the distance between the two bodies. We can re-arrange this expression to get something very useful: Orbits We can use this expression to determine the orbital velocity (V) of a small mass orbiting a distance d from the center of a much larger mass (M)

Calculating Escape Velocity From Newton ’ s laws of motion and gravity, we can calculate the velocity necessary for an object to have in order to escape from a planet, called the escape velocity

What Escape Velocity Means If an object, say a rocket, is launched with a velocity less than the escape velocity, it will eventually return to Earth If the rocket achieves a speed higher than the escape velocity, it will leave the Earth, and will not return!

Escape Velocity is for more than just Rockets! The concept of escape velocity is useful for more than just rockets! It helps determine which planets have an atmosphere, and which don ’ t –Object with a smaller mass (such as the Moon, or Mercury) have a low escape velocity. Gas particles near the planet can escape easily, so these bodies don ’ t have much of an atmosphere. –Planets with a high mass, such as Jupiter, have very high escape velocities, so gas particles have a difficult time escaping. Massive planets tend to have thick atmospheres.

The Origin of Tides The Moon exerts a gravitational force on the Earth, stretching it! –Water responds to this pull by flowing towards the source of the force, creating tidal bulges both beneath the Moon and on the opposite side of the Earth

High and Low Tides As the Earth rotates beneath the Moon, the surface of the Earth experiences high and low tides

The Sun creates tides, too! The Sun is much more massive than the Moon, so one might think it would create far larger tides! The Sun is much farther away, so its tidal forces are smaller, but still noticeable! When the Sun and the Moon line up, higher tides, call “ spring tides ” are formed When the Sun and the Moon are at right angles to each other, their tidal forces work against each other, and smaller “ neap tides ” result.

The Conservation of Energy The energy in a closed system may change form, but the total amount of energy does not change as a result of any process

Kinetic Energy is simply the energy of motion Both mass (m) and velocity (V) contribute to kinetic energy Imagine catching a thrown ball. –If the ball is thrown gently, it hits your hand with very little pain –If the ball is thrown very hard, it hurts to catch! Kinetic Energy

Thermal Energy Thermal energy is the energy associated with heat It is the energy of the random motion of individual atoms within an object. What you perceive as heat on a stovetop is the energy of the individual atoms in the heating element striking your finger

Potential Energy You can think of potential energy as stored energy, energy ready to be converted into another form Gravitational potential energy is the energy stored as a result of an object being lifted upwards against the pull of gravity Potential energy is released when the object is put into motion, or allowed to fall.

Conversion of Potential Energy Example: –A bowling ball is lifted from the floor onto a table Converts chemical energy in your muscles into potential energy of the ball –The ball is allowed to roll off the table As the ball accelerates downward toward the floor, gravitational potential energy is converted to kinetic energy –When the ball hits the floor, it makes a sound, and the floor trembles Kinetic energy of the ball is converted into sound energy in the air and floor, as well as some heat energy as the atoms in the floor and ball get knocked around by the impact

Definition of Angular Momentum Angular momentum is the rotational equivalent of inertia Can be expressed mathematically as the product of the objects mass, rotational velocity, and radius If no external forces are acting on an object, then its angular momentum is conserved, or a constant:

Conservation of Angular Momentum Since angular momentum is conserved, if either the mass, size or speed of a spinning object changes, the other values must change to maintain the same value of momentum –As a spinning figure skater pulls her arms inward, she changes her value of r in angular momentum. –Mass cannot increase, so her rotational speed must increase to maintain a constant angular momentum Works for stars, planets orbiting the Sun, and satellites orbiting the Earth, too!