NATS From the Cosmos to Earth Billiard Balls

NATS From the Cosmos to Earth Angular Momentum Momentum associated with rotational or orbital motion angular momentum = mass x velocity x radius

NATS From the Cosmos to Earth Torque and Conservation of Angular Momentum Conservation of angular momentum - like conservation of momentum - in the absence of a net torque (twisting force), the total angular momentum of a system remains constant Torque - twisting force

NATS From the Cosmos to Earth A spinning skater speeds up as she brings her arms in and slows down as she spreads her arms because of conservation of angular momentum

NATS From the Cosmos to Earth The law of universal gravitation.

NATS From the Cosmos to Earth The force on a body of mass m 1 is: (Newton’s Second Law) If this force is due to gravity, then: m 1 cancels out, and: Newton’s 2nd Law and the Acceleration Due to Gravity

NATS From the Cosmos to Earth The acceleration due to the force of gravity is called g, so: Mass of the Earth (m 2 ) = 5.97 X kg Radius of Earth (d) = X 10 6 m G= 6.67 x Nm 2 / kg 2 g= (6.67 x Nm 2 / kg 2 ) X (5.97 X kg)/(6.378 X 10 6 m) 2 g= 9.79 m/s 2 g does not depend on the mass of the body m 1 - so the feather falls at the same speed as the steel ball - Galileo learned this by experimentation (the Leaning Tower of Pisa experiment) - Newton showed why. Weight is the result of the force of gravity on a body of mass m 1 : Therefore all objects on earth having the same mass have the same weight.

NATS From the Cosmos to Earth The acceleration of gravity and therefore a person’s weight is dependent on a planet’s mass and radius. Planetary Mass, Radius and Weight

NATS From the Cosmos to Earth Newton’s Formulation of Kepler’s Laws As a planet moves around its orbit, it sweeps out equal areas in equal times - a planet moves slower when it is farther from the Sun and faster when it is closer Kepler’s Laws were based on observation (experimentation). Newton’s laws explained Kepler’s Laws Kepler’s Second Law

NATS From the Cosmos to Earth For a circular orbit: (r = radius of orbit) Substitute (2) into (1): F is the force of gravity: Cancel m 1 and r; then (1) (2) (3) (4) The smaller the radius, the greater the speed.The orbital speed is independent of the mass of the orbiting body (m 1 ). As the radius (the distance to the orbiting body) increases, the orbital speed decreases. When you swing a ball around, the string exerts a force that pulls the ball inward (gravity for orbiting body). The acceleration is also inward.

NATS From the Cosmos to Earth The square of any planet's period of orbital revolution, P, is proportional to the cube of its mean distance, r, from the sun. Kepler’s 3rd Law Orbital Period vs Distance Animation

NATS From the Cosmos to Earth From Kepler’s Second Law (previous slide): Speed around orbit: Circumference (2  r)/ time P=period, time of 1 orbit (1) (2) (3) (4) (5) Combine (1) and (3): Rearrange terms: Square both sides:

NATS From the Cosmos to Earth A more complex derivation of this equation yields: From this equation, if one knows the mass of the orbiting body (m 1 ), the mass of the central body (m 2 ) may be calculated. What is the mass of the Sun? M Sun (m 1 ) >> M Earth (m 2 ) so: m 1 + m 2  m 1 M 1 = 4  2 r 3 /GP 2 G = 6.67 x Nm 2 / kg 2 r = 1.5 x m P = 3.15 x s So: M sun = 2 x kg

NATS From the Cosmos to Earth Geosynchronous/Geostationary Orbits A geosynchronous orbit has a period the same as the rotational speed of the Earth - e.g., it orbits in the same amount of time that the Earth rotates - 1 sidereal day. A geostationary orbit is a geosynchronous orbit at the equator - it always stays above the same place on the Earth - communications satellites, satellite TV, etc… What is the altitude of a geostationary orbit? From Newton’s formulation of Kepler’s 3rd Law: M Earth (m 1 ) >> M Satellite (m 2 ) so: r = (GM Earth P 2 /4  2 ) 1/3 G = 6.67 x Nm 2 / kg 2 P = 3.15 x s M Earth = 5.97 X kg So: R = 42,000 km above the center of the Earth and the altitude is about 35,600 km

NATS From the Cosmos to Earth Center of Mass Newton also showed that two objects attracted to each other by gravity actually orbit about their center of mass - the point at which the objects would balance if the were connected. This idea is used to find planets orbiting other stars - massive planets cause star to move against background stars Center of Mass - Binary Star

NATS From the Cosmos to Earth Tides The gravitational attraction of the Moon varies as the square of the distance (Newton’s Law of Gravitation) - gravity stronger on side facing the Moon than on opposite side. The Moon pulls the ocean water towards it on facing side - creates tide - and pulls the Earth away from the ocean water on the other side - reason for tides twice a day. Time of tides varies by 50 min per day - Moon at its highest point every 24 hrs 50 min because Moon orbits Earth while Earth rotates.

NATS From the Cosmos to Earth The Sun also causes tides - why are they weaker than the Moons’? Neap tides - when Moon’s and Sun’s gravitational forces oppose each other Spring tides - when Moon’s and Sun’s gravitational forces add up

NATS From the Cosmos to Earth Tides

NATS From the Cosmos to Earth Tidal Bulge Because the Earth rotates, friction drags the tidal bulges off of the Earth- Moon line. This tidal friction causes the Earth’s rotation to slow and the Moon to move farther out.

NATS From the Cosmos to Earth The Moon pulls on tidal bulge - slows Earth’s rotation The excess mass in Earth’s tidal bulge exerts a gravitational attraction on the Moon that pulls the Moon ahead in its orbit - Moon moves farther away - Conservation of Angular Momentum

NATS From the Cosmos to Earth Matter and Energy

NATS From the Cosmos to Earth DEFINITION: Anything that occupies space and has mass PROPERTIES OF MATTER: Mass - a measure of a body’s resistance to a change in its state of motion - its inertia Density - mass per unit volume Dimensions - height, length, width Electric charge - positive/negative/neutral Heat content - everything above absolute 0 ( º F) has heat - no such quantity as cold - only absence of heat Resistance to flow of electric current - flow of charged particles - electrons Pressure - exerted by moving molecules in all directions - resists compression Matter

NATS From the Cosmos to Earth Energy Definition of Energy: Anything that can change the condition of matter Ability to do work – the mover of substance (matter) Work is a force acting over a distance Force: The agent of change – push or pull on a body Hence: Work is the change in the energy of a system resulting from the application of a force acting over a distance.

NATS From the Cosmos to Earth Four Types of Forces: Gravitational – holds the world together Electromagnetic – accounts for many observed forces - Push or pull on a body Strong Nuclear – hold nucleus together Weak Nuclear – involved in nuclear reactions Power: Rate of change of energy

NATS From the Cosmos to Earth Energy Units Energy: Joule = 1 kg m 2 /s 2 1 Joule = 1/4184 Calorie, so 2500 Cal = 1 x 10 7 J (average daily requirement for a human) Power: 1 watt = 1J/s Thus for every second a 100 W light bulb is on, the electric company charges for 100 J of energy. The average daily power requirement for a human is about the same as for a 100-W light bulb.

NATS From the Cosmos to Earth Solar energy striking Earth’s surface per second = 2.5 x J. Energy released by burning 1 liter of oil = solar energy striking square 100 m on a side in 1 second Energy Comparisons

NATS From the Cosmos to Earth Energy Three basic categories: Kinetic energy = energy of motion Potential energy = stored energy Radiative - energy carried by light

NATS From the Cosmos to Earth Types of Energy Energy cannot be created or destroyed, only changed –Mechanical – Potential- energy of position P.E.= mgh Kinetic- energy of motion K.E.=1/2mv 2 –Electrical –Chemical –Elastic –Gravitational –Thermal –Radiant –Nuclear