Physics 1A Lecture 1B
Angles: a Tricky Unit θ Angles are formally defined as a ratio of lengths; e.g. θ = Arclength/Radius [θ] = L/L = 1 This makes the angle unitless! The fundamental unit of angle is called the radian 2π radians = 360º 1 radian 57.3º
Estimation Techniques In this term you should learn how to estimate basic physical quantities using simple relations and order of magnitude numbers (closest factor of 10) Important for: Checking solution (does my answer make sense?) Experimental planning (can I test X specimens in a given timeframe?) Cost/benefit analysis (how safe does a plane have to be?) Large scale problems (how large of a human population can the Earth support?)
Review of Last Lecture Physical quantities are different from numbers in that you must include units, uncertainty/significant digits and dimension. Fundamental quantities are length, time and mass; derived quantities are combinations of these. In your calculations, be sure to carry your units throughout and check for agreement. Dimensional analysis is a useful trick for guessing a solution or checking your answer. Order of magnitude estimation allows for quick calculations and checks on solutions practice this!
Types of coordinate systems X 1D Linear Y X r θ 2D Rectangular 2D Polar
Polar to rectangular coordinates y r θ x
What is the best coordinate system to use? (A) 1D (B) 2D rectangular (C) 2D polar Ball falling close to Earth s surface
What is the best coordinate system to use? (A) 1D (B) 2D rectangular (C) 2D polar Block sliding down an inclined plane
Order 0: Order 1: Order 2: Scalars = magnitude only (example: mass, temperature) Vectors = magnitude and direction (example: displacement, velocity) Tensors = matrix operations (example: moment of inertia tensor) Scalar Vector Tensor Never ever!
Examples: scalar or vector? Mass Gravity Map directions Current speed Speed and heading Time scalar vector vector scalar vector scalar (for this class)
Vector: notation & properties y θ x A vector is usually denoted with an arrow over it; (e.g., ) or in bold (e.g., A). Vectors can be defined by a magnitude (scalar) and an angle θ relative to some axis.
Vector: notation & properties y θ Magnitude: Angle: x projection of vector along X and Y axes unit vectors
Equality of two vectors Two vectors are equal if they have the same magnitude and the same direction if A = B and they point along parallel lines All of the vectors shown here are equal
Adding vectors graphically Continue drawing the vectors tip-to-tail The resultant is drawn from the origin of end of the last vector Measure the length of and its angle Use the scale factor to convert length to actual magnitude A to the R
Adding vector components To derive the components of the new vector, add each component separately e.g.: R X = A X + B X R Y = A Y + B Y etc. Y X Units must still agree!!
Adding vectors graphically When you have many vectors, just keep repeating the process until all are included e.g.: R X = A X + B X + C X + D X R Y = A Y + B Y + C Y + D Y etc. Units must still agree!!
Subtracting vectors Special case of vector addition Continue with standard vector addition procedure Units must still agree!!
Multiplying vectors The only multiplication functions you can perform on a vector are (contd. next slide): Multiply vector by scalar result is a vector with same direction, different length Scalar (dot) product of two vectors result is a scalar; zero if vectors are perpendicular
Multiplying Vectors (contd.) Vector (cross) product of two vectors: result is a vector pointing perpendicular to A and B
Question... Which of the following vector operations is allowed? A. B. C. D.
Question... In which case are the two vectors shown perpendicular? A. θ = 90º B. C. θ D. All of the above E. None of the above
Position, Distance & Displacement Position: where you are right now (vector) Distance: the length of the path you have followed (scalar) Displacement: the net separation between two points (vector) Y finish now start X
North East 3 m 4 m 5 m 36.9º Total distance travelled = 7 m Total displacement = 5 m at angle of 36.9º East of North
North East 3 m 4 m 5 m Total distance travelled = 12 m Total displacement = 0 m
Question... Can you travel a large distance and zero displacement? A. Yes B. No C. Don t Know
Question... Can you travel zero distance and have a large displacement? A. Yes B. No C. Don t Know
Question... Can the magnitude of the displacement be larger than the distance? A. Yes B. No C. Don t Know
For Next Time (FNT) Start Reading Chapter 2 Finish the homework for Chapter 1 Bring your clicker to the next lecture