Definition Problems
These are straightforward problems that take you between two closely related concepts. Definition problems may be strictly mathematical (e.g. components of a vector), may involve rates (e.g. acceleration is the rate at which velocity changes), or they may simply be definitions (e.g. pressure is defined as force/area).
How to work with vectors
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1. Identify the Problem
Displacement, velocity, acceleration, force, torque, angular displacement, angular velocity, angular acceleration, momentum, electric and magnetic fields are all vectors. Whenever you work with vectors that are not all along the same straight line, you need to divide the vectors into their x- and y- (and sometimes z-) components. In some cases, you will have explicit practice problems but most of the time you will work with vector components as a preliminary step to solving other problems. In the later case, it is easiest to divide the vectors into components at the "Draw a Picture" stage so that you don’t have to worry about it as you get into the problem itself.
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2. Draw a Picture
The most visual way to see the components of a vector is to draw that vector alone on your coordinate system. (It doesn’t matter if the vector quantity actually acts at the origin—you can move your coordinate system around as long as you point the axes in the same direction.) Then draw a right triangle with the vector as the hypotenuse and the sides on or parallel to the axes.
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3. Select the Relation
There are the four relations which describe a right triangle:
- a2 + b2 = c2 (a and b are the sides of the triangle, c is the hypotenuse)
- Sinθ = (opposite side)/(hypotenuse)
- Cosθ = (adjacent side)/(hypotenuse)
- Tanθ = (opposite side)/(adjacent side)
The last three relations are often remembered as soh-cah-toa.
If you are given a vector (the hypotenuse) and asked for components (the sides,) use the definitions of sine and cosine. The Pythagorean Theorem (Equation 1) is useful if you know the components and are asked for the vector itself.
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4. Solve the Problem
Solving vector problems and sub-problems is merely a matter of doing the math. There are three things to watch:
- It is not true that cosine always goes with the x-axis and sine with the y-axis. It depends on which angle you use. You can always avoid mistakes by going back to the definition equations above.
- Using this method of drawing the picture, you will need to assign + and – signs for directions of the components explicitly.
- If you are given the components of a vector and asked for the direction, you will need to “un-do” one of the last three equations. You can do this, for example, by using the “inv tan” or “tan-1” function on your calculator.
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5. Understand the Results
A good double check on your math is that no component is greater than the length of the vector, and that the shorter component on your triangle has a value less than the longer one. It is also a good idea to go back to the picture to make sure that you correctly assigned signs to each component.
Note that when you replace a vector with its components you have the same physical effect as for the original vector. For example, if you move either along a displacement vector or along both of its components, you will end up in the same place; if you exert a force at an angle or replace it with one force for each of its components, you will get the same acceleration.
How to Solve Definition Problems
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1. Identify the Problem
The reason that this website has a separate section for definition problems, rather than putting them as easier problems in other sections (e.g. putting a definition of kinetic energy problem in the energy section) is because the single most important thing you should do as you begin a problem is to identify the best approach to use. Everything else that you do follows from there. Dynamics (force,) conservation of energy, kinematics (motion,) etc. problems all share a common underlying understanding and a common underlying problem-solving approach with each other.
Definition problems all have in common an essentially plug-and-chug approach. Whenever you are given one quantity and asked for a very closely related second quantity, you will approach the problem in the same way. Almost always, the two quantities are, as the category suggests, directly related through the definition of one of the quantities (e.g. density is defined as mass/volume, so mass and density are very closely linked.) A specific category in which you see this close relationship is rate questions (velocity is the rate at which displacement changes so velocity and displacement are very closely linked.)
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2. Draw a Picture
You seldom need a picture for definition equations—you tend to be told two quantities and asked for a third that is related through a single equation. Exceptions are cases in which you can’t visualize what the equation means or where it came from, or in which you are asked to look at multiple points and need to keep the given information straight. In those cases, the picture tends to be a well-labeled picture of the actual situation, rather than an abstract picture such as a free body diagram.
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3. Select the Relation
There are many, many relations that are used in definition problems. However, by the process of identifying the problem as being plug-and-chug, you will already have the relationship between your closely related variables in mind. If not, try looking in the index for the first page of your text book where the unfamiliar variable is mentioned.
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4. Solve the Problem
In most cases, you will just need to put the given information into your equation and algebraically solve for the unknown. There are several notable exceptions:
- Average velocity: you may need to use v information in the definition of velocity equation for the legs of a trip to find x and t for each leg, and then put those values into the definition of velocity equation one more time for the trip as a whole.
- Comparing multiple points: In some cases, you are asked to compare values at two different points. It is very common in those cases not to be given information about all of the variables. However, if you divide the definition equation at one point by the definition equation at another point, any unknown quantities that don’t change in value between the two will divide out.
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5. Understand the Results
For most definition problems, your main goal will be to become comfortable with the close relationship between the two variables—a goal that happens just by identifying the definition nature of, and relation for, the problem. However, you should stop at the end to think about the units and the size of your answer which will also add to your understanding of the variable that the problem explores.
Help! I can’t find an example that looks like the problem I need to work!
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Are you certain your problem is a definition problem?
If you are asked how one variable changes as the result of the change in another it could, indeed, be a definition problem, but only if the two variables are very closely related through the definition of one of them. (E.g., how does potential energy change with position?) If, however, the two variables are not closely related, you might need to look for an example problem that looks at the interaction affecting that change (such as force or energy.)
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Yes, my problem is definitely a definition problem.
In that case, think more broadly about what makes a useful example. Remember, you were given your assignment to practice the problem solving approach, not because the answers to your problems are particularly interesting. An example in which you merely substitute your numbers for those in the problem will give you practice entering numbers on your calculator but will teach you nothing about physics, and when you take your exam every problem on it will feel new and different to you. So think about your example as support for helping you to practice the problem solving approach. If your problem is a straightforward definition, any definition problem will be an effective example even if the variable of interest is entirely different. (E.g. you solve a density-mass problem in exactly the same way you solve a pressure-force or a kinetic energy-velocity problem.)
That said, different situations require you to do different side problems along the way. If you are asked for average velocity, you will definitely want to use another average velocity problem as an example. If you have a great deal of unknown information, it would be more useful to use a ratio problem as an example than to use a problem of similar topic.