!t + U " #Q. Solving Physical Problems: Pitfalls, Guidelines, and Advice

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1 Solving Physical Problems: Pitfalls, Guidelines, and Advice Some Relations in METR 420 There are a relatively small number of mathematical and physical relations that we ve used so far in METR 420 or that you learned over the course of your education and likely know. Some relations that come to mind include: (1) p =!R d T v (a version of the ideal gas law) (2) dq dt =!Q!t + c " #Q (a relation between total and partial derivatives; applies to field variables) (2.1) DQ Dt =!Q!t + U " #Q (a special case of Eq. (2), in which c = U ; that is, the observer has the same velocity as a fluid parcel and is therefore following it at that moment) (3) DQ Dt =! sources/sinks of Q (a generalized conservation law for Q; it applies to individual objects, including fluid parcels, but is not restricted to field variables) (4)!Q!t = "U # $Q + % sources/sinks of Q (a generalized tendency equation for Q; a combination of the information in Eqs.(2.1) and (3); applies to field variables only) Most of the relations above involve vectors, so we have to understand how to manipulate vectors if we are to use the relations either for theoretical purposes or for practical purposes.

2 Because of the influence that gravity has on the atmosphere, the oceans, and us we very often choose coordinate systems that have one axis oriented vertically and the other two axes horizontally. In a three- dimensional coordinate system oriented in this way, we can break any vector into two components, one of which is horizontal and the other vertical, like this:!" A =!" A + A z ˆk (where A is the horizontal part of the vector, k ˆ is the vertical unit vector, and A z is the vertical scalar component of the vector). In such a coordinate system, we can show that the dot product between two three- dimensional vectors satisfies the following relation: (5) A! B = A + A ˆ z k ( )! ( B + B ˆ z k ) = A! B + A! ˆ k B z + A ˆ z k! B + A z ˆ k! ˆ k B z = A! B + A z B z (a relation between the dot product of three- dimensional vectors and the horizontal and vertical components of the vectors involved). [Note that once a coordinate system has been selected, a vector can be broken into vector components. Each component is parallel to one of the coordinate axes and consists of a scalar multiplying a unit vector that points in the direction of the coordinate axis. The full vector is the sum of its vector components, each of which is generally perpendicular to the other(s). For example, in the example above, A z ˆk would be the vertical vector component of A!", while A!" might be called the horizontal component (though a coordinate system for the horizontal part of the 3- dimensional vector hasn t been chosen yet, so A!" has yet to be broken into its vector components). The scalar quantity multiplying the unit vector is called the scalar component.]

3 In a horizontal (and hence two- dimensional) plane, we can choose to represent the components of horizontal vectors in either a two- dimensional rectangular coordinate system or a polar coordinate system (which is also two- dimensional). If we choose a rectangular coordinate system, then we can write a vector in terms of its components as A = A xˆ i + A ˆ y j. The dot product between two such vectors can therefore be written as: (6) ( )! ( B xˆ i + B y ˆ j ) = A x B x + A y B y A! B = A xˆ i + A y ˆ j (Substituting Eq. (6) into Eq. (5) gives you the relation between the dot product and vector components in rectangular coordinates in three dimensions.) In polar coordinates a vector can be written in terms of its components as A = A R ˆ (! ), where A is the magnitude (length) of the vector, R ˆ! is a radial unit vector that points in the direction of, and is the direction of the unit vector expressed as an angle. (The unit vector, R ˆ!, is written as an explicit function of direction, which is where the ( ) ( ) A! second piece of information about the vector A that is, besides its magnitude/length resides.) The relation between the dot product of two vectors and their respective components is: (7) A! B = A B cos(" A # " B ) (Substituting Eq. (7) into Eq. (5) gives you the relation between the dot product and vector components in cylindrical coordinates, which is a three- dimensional coordinate system.) (The relation between the dot product and vector components in spherical components we ll cover later in the semester.) There are familiar trigonometric relations among the components of two- dimensional vectors, including horizontal vectors like these. For example, to convert from polar coordinates to rectangular coordinates: (8.1) A x = A cos! A (8.2) A y = A sin! A

4 and to convert from rectangular coordinates to polar coordinates: (9.1) A = A x2 + A y 2 ( ) (9.2)! A = arctan A y / A x Although vectors are compact, efficient notational devices for use in abstract relations such as many of the relations above, to apply them quantitatively you must first choose a coordinate system, decide where to put its origin, decide how to orient its axes, then break vectors into components in that coordinate system so that ultimately you can assign values and units to each scalar component. Choosing a coordinate system, placing its origin, and orienting its axes so that your particular problem is as easy to solve as possible is a problem- solving skill that you have to learn. For example, if you want to compute the dot product between two horizontal vectors, you must decide whether to use Eq. (6) or Eq. (7), which in turn requires that you choose a coordinate system. The coordinate system you choose depends on which information about the vectors you have available to you most directly with the least amount of conversion from one coordinate system to another needed. If you are given the speed and direction of a velocity vector, you should recognize immediately that you ve been given the components of the velocity vector in polar coordinates. Whether you use Eq. (7) or Eq. (6) to compute the dot product between the horizontal velocity and another horizontal vector depends on whether you know the components of the second vector in polar coordinates or in rectangular coordinates. If it s the latter, then you have to decide whether to convert its components into polar coordinates and use Eq. (7) or convert the velocity components into rectangular coordinates and use Eq. (6). Otherwise you d just use Eq. (6).

5 Common Mistakes with Vectors There are a number of mistakes that students are prone to make when learning to solve physical problems that involve vectors. ere are some common ones: (1) Assigning a vector a single (scalar) value, as in U = 10 m/s. Since a vector is a package of two or more pieces of information bundled together in some way, this sort of statement doesn t make much sense. To assign values to a vector, you must first choose a coordinate system (implicitly or explicitly), identify the vector s components in that coordinate system, and assign scalar values to the scalar components. (2) Writing two vectors side by side without any operation between them specified: AB. This is not defined and hence is meaningless. You must write either the dot or the cross between the vectors to indicate which operation you mean, the dot product or the cross product. (3) Dividing by a vector: A B. There is no such operation. You can multiply or divide a vector by a scalar; the result is still a vector, though dividing by a scalar will typically change it s length and possibly its units, and if the scalar is negative its direction will reverse. owever, you can t divide a scalar or a vector by a vector that s an operation that isn t defined and is therefore meaningless. (4) Substituting scalar values directly into a dot product without writing the dot product in terms of its components in one coordinate system or another first. (This is analogous to (1) above.)

6 Common Mistakes Implementing a Structured Problem Solving Approach When implementing the structured problem solving approach * that I ve asked you to follow, it is not uncommon for students to take shortcuts or depart from the structured approach in several respects, often to the detriment of the integrity of the solution. For example: (1) Substituting numerical values and units before developing a symbolic solution. You get a symbolic solution when you ve identified the relations you need to connect (a) the information given or otherwise known to you in the problem, with (b) the information you want to determine, and then manipulated the relations into a single equation that has the information you want to know on the left- hand side and has only things that you know on the right- hand side. Only then should you insert values and units and start manipulating those (first units, then powers of ten, and lastly the arithmetic). [An exception is when the value of some quantity is given or known to be zero. You might choose to invoke that information earlier in your derivation of the symbolic solution to help you get rid of a quantity multiplying it that you don t know.] (2) Failing to list, label (with numbers, for handy reference subsequently), and name the relations that you need to solve the problem, including mathematical/geometrical/trigonometric relations such as Eqs. (5)- (9) above. If you don t list and label them, it s hard to invoke them when you need them without making their appearance look simply like magic, in which your equations transform themselves into something else without explanation. Not a good idea! (3) Changing notation in the middle of the problem without defining the new notation. This can only confuse the reader and, probably, you. (4) Ignoring without justification parts of an equation that might not apply to the problem at hand but do apply to other problems. A common (mal)practice is to ignore without comment the vertical part of a vector (such as a gradient of a scalar) in a problem that seems to involve otherwise horizontal vector quantities. Make sure you account in some way or another for all components of a vector that appears in an equation, including gradients and dot products involving gradients!

7 (5) Failing to keep careful track of signs. Signs carry physically meaningful information; you can check the sign of a term at any time to see if it seems reasonable or not. If it does seem wrong, that might be telling you that you lost a sign somewhere and need to track it down. It is not uncommon for students to make two sign errors that cancel each other out. Although the final solution might well be right, the procedure that produced the solution would have had two major errors in it! It s the solution procedure and the quality and clarity with which you execute it and communicate it that I pay attention to when I evaluate problem assignments. (6) Failing to check the final numerical solution for physical reasonableness. Sometimes this is very trivial, but you can often at least ask questions such as, does it have a reasonable sign? Is the magnitude physically reasonable for the problem at hand? Doing this does take some intuition into what is reasonable and what isn t, but it s worth practicing. *See the handouts Solving Physical Problems and Understanding a Relation at and ng.html, respectively.

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