1 Dimensions and Units 1.1 Physical Dimensions and Units of Measurement You've probably heard the expression that you can't compare apples and oranges. To a physicist, the key concept underlying this idea is that of physical dimension. Different types of quantities can be distinguished by their physical dimension. For example, distance and mass are two distinct physical quantities and thus correspond to different physical dimensions. Furthermore, quantities that have different physical dimension cannot be directly compared in a meaningful way-you can't compare apples and oranges. The commonly used base dimensions in mechanics are those of length (or distance), time, and mass. These can be manipulated using standard mathematical operations such as multiplication, division and exponentiatron. For example, volume is measured as a length ("length cubed"). The dimension of speed is length over time. Associated with physical dimensions are units of measurement ('units' for short). Units are standard conventions that are easily reproducible and that allow people to consistently communicate the results of their measurements. A wide variety of standard units are used in physics, but the International Standard base units of meters (for length), seconds (for time), and kilograms (for mass) are the most common . 1.2 Dimensional Analysis Engineers and scientists need to be able to identify the correct units for an expression in order to properly communicate results of measurements and calculations-a skill known as "dimensional analysis." Furthermore, if you find that the units of an expression are incorrect, this signals that an error has been made in formulating the expression or during a calculation. This fact can be used to 'troubleshoot' your work, helping you quickly spot issues on your homework and exams. Developing the ability to determine the appropriate dimensions and units for a certain quantity will help you out tremendously over the course of your science and engineering career. Example 1.1. Below is an expression that represents a relationship between physical quantities. Note that instead of variables, the expression only involves dimensionless constants or physical dimensions. Insert the missing physical dimensions into the blank spaces so that the dimensions on the left-hand-side match those on the right-hand-side of the expression. distance = (distance . time) cos (time time , Solution. The dimension on the left-hand-side is distance (or length). On the right-hand-side mathematical functions like cosine or sine are physically dimensionless, so the blank space in the denominator must have a dimension of time. Within the cosine function the full argument has to be dimensionless. There is a quantity that involves time divided by time, which is dimensionless overall. Therefore, the blank space withinthe cosine must be dimensionless pure number. So the final result is distance = distance . time cos time time time number) Question 1.1) Below is a numbered list of expressions, representing typical equations that you might find in your science and engineering classes. These expressions however focus only on the appropriate dimensions and units. To the right, are options labeled A through F. Fill the blanks in the expressions with the letter choices whose combination yield the correct quantity. Each letter choice should be used only once. (1) distance = (/_)time2 (A) 27 (2) surface area = / meters (B) s (3) distance = (distance) sin((_/_)time) (C) kg (4) mass per unit volume = _/m3 (D) m (E) m3 (F) $2 2 Vector Activity I: Concepts A scalar can be a positive or negative number, but in general, it doesn't convey any precise directional information. A geometric vector has both direction and magnitude. They have two key properties that make them quite useful: (a) Addition: Vectors can be added up in a manner that is similar to numbers from basic arithmetic. (b) Scalar multiplication: Vectors can be re-sized (rescaled) by multiplying them by scalars (numbers). Negative numbers flip the vector's orientation as well. 2.1 Vector Basics: True or False Indicate whether the statements below are true or false. If you believe the statement is false, produce a brief counterexample. Question 2.1) Vectors can only be added together. There is no such thing as vector subtraction. 2Question 2.2) Two vectors are equal A = B if all of their components are equal. Conversely, if all of the components of a pair of vectors are equal, then the vectors themselves mst be equal. Question 2.3) You cannot divide one vector by another. Question 2.4) The magnitude of a sum of two vectors cannot be smaller than the magnitude of either vector. In other words, if C - A + B, then C 2|A and C 2 B Question 2.5) A vector whose components are negative has a magnitude that is negative. Question 2.6) Three different vectors that satisfy the relationship A + B + C =0 always form a triangle. 32.2 Physical Quantities: Vectors vs. Scalars The physical world can be described and modeled using both scalar and vector quantities. For example, the change in an object's position as it moves from one location to another is captured by a vector, encoding both the magnitude of the change as well as the direction. By contrast, the number of grains of sand in a bag is best described by a scalar. Question 2.7) Indicate whether each quantity listed below is a scalar or a vector. (a) An object's mass. (b) A particle's position. (c) An object's momentum. (d) An object's speed. (e) An object's kinetic energy. (f) The negative gravitational potential energy between two planets. (g) An object's acceleration. (h) An object's surface area. (i) A single force acting on an object. (j) The net force acting on an object. 3 Vector Activity II: Vector Algebra and Components 3.1 Vector Algebra Practice Figure 3.1: Vectors A, B, C, and D used in the problems below. Question 3.1) Consider the vectors A, B, C, D diagrammed above. (a) What is the algebraic relationship between vectors A and C? (example question) Answer: C = -A(b) What is the algebraic relationship between vectors B and D? (c) What is A + B + C + D equal to? Question 3.2) The vectors i and j below are called unit vectors since they have magnitude 1. Figure 3.2: The Cartesian horizontal and vertical unit vectors. (a) Use vector algebra to express i in terms of some combination of the vectors A, B, C, D from Figure 3.1. Example solution: i = = (A + B) Your solution: (b) Use vector algebra to express j in terms of some combination of the vectors A, B, C, D from Figure 3.1. (c) Express the vectors A, B, C, D in terms of the unit vectors i and j. Are they themselves unit (magnitude 1) vectors? If not, what factor must you scale them by to get unit vectors corresponding to them?3.2 Vector Decomposition into Components In a Cartesian coordinate system in two dimensions (i.e. x, y coordinates), two unit vectors can be defined that point along the positive r, and y directions. These are traditionally denotedV i = unit vector in positive x direction 3 = unit vector in positive y direction and are depicted in Figure 3.2. Using vector addition, any vector in two dimensions can be decomposed into two component numbers multiplying each of the unit vectors. A = Azi + Ay] This can be seen graphically in the diagram below: A Ayb Figure 3.3: Decomposition of a vector A into its component vectors. Question 3.3) Suppose that you know the angle 0 that the vector A makes with the positive horizontal direction. What is the relationship between that angle and the components A, and Ay? Assume that the angle 0 is acute. Question 3.4) If you are only told the angle that a vector makes with a given axis (say, the positive r-axis), what numerical information can you determine about the components of that vector? 6Question 3.5) What is the significance in terms of vector A's components of assuming 0 is acute? If it was obtuse, how might that relate to the signs of A, and Ay?Question 3.6) In each of the following, use the given data to draw a decomposition diagram similar to Figure 3.3 above. Find the unknown sides and angles and depict them on your diagram: (a) Ax = 4, Ay = 3. ( b ) |A = 5 , A y = 4 , A * > 0 (c) A = 10, 0 = 30 above negative I-axis (d) Ax = 6, 0 = 60 right of the positive y-direction 8Question 3.7) Spider-Man isn't feeling so amazing at the moment. He managed to get tangled in his own web and is hanging precariously over New York City's Times Square as people gawk at him from below. Looking at the scene from further downtown (so the street numbers are running up and you are looking North), Spider-Man's web is attached to two buildings on the East and West sides of Times Square. The thread attached to the Eastern building is pitched at an angle of 30 degrees above the horizontal. The thread attached to the Western building is angled at 450 degrees above the horizontal. Each thread can support a maximum tension of 500 N before snapping. Assume that Spider-Man weighs 700 N. (a) Draw a free body diagram depicting the forces acting on Spider-Man. Don't worry too much about the overall scales of the unknown forces, but try to get their relative directions some- what accurately. (b) Decompose each tension force vector into its components using the grids provided below as an aid. (c) Write down the equations of balance (Newton's 2nd Law) in the east/west and north/south directions. Use your formulas to find the unknown tension magnitudes. Does Spidey make it? 9Vector Activity Part 3 Vector Valued Functions So far we have dealt with vectors that have non-variable components. In physics, there are many vectorial quantities that depend on time. Consider for example the position of a particle T ( t ) = I (t)ity(t ); (3.1) The position vector is actually a vector-valued function of time. The three components are the position functions along a, y, and z directions. Question 3.8) What operation do you apply to the position function of a particle to compute the particle's velocity? Write out explicitly how that operation affects the left and right sides of equation (3.1) above. A vector-valued function can also assign a vector to each location in space. For example the function F (x, y ) = rity) (3.2) assigns a particular vector to each location (x, y) in 2D space. The vector's tail or start is fixed to the given point (x, y) and the vector stretches out from that point. Its components are defined by the formula above. The function above is called a constant vector field. The word "constant" here refers to the fact that the function does not depend on time. Question 3.9) The position function (3.1) and the function defined in formula (3.2) look quite similar. What are the differences between them? What happens if you take an z-derivative of the function in formula (3.1)? What about taking a t-derivative of the function in formula (3.2)? Question 3.10) Draw a graph that visualizes the function defined in formula (3.2). Let the middle of the grid below correspond to (0, 0) and draw the vectors associated with the origin, the points (1,0), (-1,0), (0,1), (0,-1), (1,1), (1,-1), (-1,1), and (-1,-1). Each vector you draw should start at the given point and then extend away in the direction determined by its components. 10A constant and uniform vector field is a vector field that does not depend on either position or time. For example: S( x, y) = 1 -2j (3.3) is a constant and uniform vector field. Question 3.11) What is the difference between the vector field defined in (3.3) and the earlier vector field defined by (3.2)? Question 3.12) Draw a graph that visualizes the function defined in formula (3.3). Let the middle of the grid below correspond to (0, 0) and draw the vectors associated with the origin, the points (1,0), (-1,0), (0,1), (0,-1), (1,1), (1,-1), (-1,1), and (-1,-1). Each vector you draw should start at the given point and then extend away in the direction determined by its components. 11