Math Review Science and The Universe Math Review

Math Review; Science and The Universe

Math Review • Excellent math reviews are available online several places • http: //glencoe. mcgrawhill. com/sites/0078740428/student_view 0/m ath_review. html • Also, Google search “Math Review” for more sites

Sample Problems

Formulas • 1. What is the formula for: – area of a circle – circumference of a circle • 2. What is the formula for: – hypotenuse of a right triangle? – Area of a triangle? • 3. What is the formula for the: – volume of a sphere? – surface area of a sphere?

Algebra

Trigonometry

Sources of Knowledge • Authority – Reliable if source is reliable • Intuition – Can often be wrong • Reason – Basis of math – Relies on correct postulates • Sensory Data – Can be mistaken – Machines can compensate

Four Stages of Science • Hypothesis – Reasonable starting guess • Theory – A well-tested hypothesis • Law – Hypothesis so tested as to be virtually unquestioned • Model – Schematic of system (usually mathematical) used to derive properties

Self-Evident Truths • Existence – Cogito ergo sum • Causality – Probably true – Neutrinos faster than light would lead to violation • Position Symmetry – Leads to conservation of momentum, angular momentum • Time symmetry – Leads to conservation of energy • Principle of noncontradiction – Probably true – No way to prove/disprove • Occam’s Razor – Assumed true – Magical elves in the sun?

Fundamental Forces • http: //hyperphysics. phyastr. gsu. edu/hbase/forces/funfo r. html • Gravity – Graviton particle (? ) – Infinite in range • Electromagnetism – Photon – Infinite in range • Weak – Z bosons – 10^-18 meters • Strong – W boson – 10^-15 meters

Laws of Motion

Position •

Change in Position •

Velocity •

Velocity Problem •

Acceleration •

Acceleration Notes • Acceleration is any change in the speed or direction of an object’s motion • Which of the following would involve acceleration? – A ball rolling down a hill – A ball rolling up a hill – A ball being twirled on the end of a string at a constant speed

Acceleration Problem •

Acceleration Problem #2 • A car is at rest on a freeway. • What does this statement tell us about the acceleration of the car?

The First Law of Motion • “Every object at rest, or in uniform motion, will remain in that state of motion unless compelled to do otherwise by forces acting upon it. ” • http: //www. grc. nasa. gov/ WWW/k 12/airplane/newton 1 g. html • In order to understand this mathematically and physically, lets introduce some notation

Examples of the First Law • Car sliding on an icy road • Package sliding across the seat of a car during a sharp turn • Car coasting along a level road, in neutral, with the engine turned off

Mass • An intrinsic property of matter • It is still an open question why objects in the universe have mass! – It is thought that mass comes from interactions particles with the Higgs field via the Higgs boson • Mass is not the same as weight • The SI unit of mass is the kilogram = 2. 2 pounds

Mass Problem • A ball is weighed on the surface of the earth and on the surface of the moon (where the gravitational field is approximately 1/6 as strong as that of the Earth). • What is the percent change in the mass of the ball in the two locations?

Second Law of Motion •

Force Problem #1 • What is the direction of the resulting acceleration?

Force Problem #2 • Assume that the object from the last slide has a mass of 2 kg and the forces act uniformly on the object for 5 seconds • What is the final velocity of the object?

Pythagorean Theorem • Equation relating the lengths of sides of a right triangle • A right triangle has one angle equal to 90 degrees • All interior angles of a triangle must sum to 180 degrees

Net Force • The net force acting on an object is the vector sum of all forces acting on the object • A vector has both a direction and a magnitude • What is the length of the vector in the picture? • Use Pythagorean theorem

Newton’s Third Law of Motion • “All forces result from interactions between pairs of objects, each object exerting a force on the other. The two resulting forces have the same strength and act in exactly the opposite directions. ” • The force of you on the earth is the same as the force of the earth on you.

Homework • Matching: 1 -9 • Analysis: 1 -3, 5 -7, 9 • Synthesis: 2, 5 -7

Gravity

Falling Objects •

Converting Units •

Uniform Acceleration Due to Gravity •

Orbit of the Moon • The moon orbits the Earth because it is falling towards the earth at the same rate that the surface of the Earth falls away under it • What would happen if the earth was flat? • The force of gravity between the Earth and moon are the same as if all the mass was concentrated at the center of the Earth/Moon • What would happen to the orbit of the Moon if the Earth collapsed into a black hole?

Gravity and Third Law of Motion • What is the reason that you don’t fall through the surface of the Earth due to gravity? • How does this relate to Newton’s Third Law of Motion? • Hypothesize as to why you might lose bone density during prolonged space travel.

Universal Law of Gravity •

Gravity of Curved Space-Time • Einstein’s Theory of General Relativity says that an object traveling on a curved path through a straight space-time is equivalent to an object traveling on a straight path through a curved space-time • There is also no difference between an object being uniformly accelerated and being in a uniform gravitational field

Movie on General Relativity • Check out: http: //www. youtube. com/watch? v=0 roc. Ntn. D -y. I • Clip from a Brian Greene documentary

Homework • • Comprehension: 1 -3 matching 1 -5 true/false Analysis: 1, 3 -9 Synthesis: 3 -5 Read about Cavendish experiment