(1)+Measurement+&+Error+Analysis

PAGE EDITORS: Andrew Roberts, Matt Pataro, Frankie Montes

**Natland Note (10/05/09):** Nice job, guys. One thing that still needs to be done is to list from where, under sources, you got your images (e.g. list the websites and which picture came from where. Additionally, if you look on the HOME page, there are a few things missing that are listed under the "Wiki Assignment" list.

**__Error Analysis__**

In general, the result of any measurement of a quantity "x" is stated as:

Measured value for "x" = Xbest ± dx

This statement shows: 1) best estimate 2) range of measurement within which you are reasonably confident the actual measurement lies 3) measurement lies between Xbest + dx and Xbest – dx


 * Uncertainty should only have **one** significant figure
 * Two types of possible errors
 * Systematic error – an error you can correct; usually instrument error. (i.e. did not zero scale so all measurements consistently off)
 * Random error – even after systematic errors are accounted for, there is still error that is experimental uncertainties that can be revealed by repeating the measurements (i.e. slight variations in reaction time)
 * These two types of errors affect two aspects of your results
 * Accuracy- how close to actual measurement you are (low systematic error)
 * Precision- how often your measurements match up (low random error)
 * Propagation of error when Adding or Subtracting**
 * If X = Xbest ± dx and Y = Ybest ± dy then what is q = x+y?
 * q = qbest ± dq
 * qbest = Xbest + Ybest
 * dq = square root: (dx)2 + (dy)2 *dq has **one** sig fig

Suppose you measure the volume of water in 2 beakers as, V1 = 130 ± 6 mL V2 = 65 ± 4 mL You then carefully pour the contents of the first into the second. What is the total volume with the uncertainty dV? Vtotal = Vbest ± dv Vtotal = (130 + 65) ± square root: (6)2 + (4)2 Vtotal = 195 ± 7.2 Vtotal = 195 ± 7 mL ß answer…uncertainty **must** be 1 sig fig
 * Example:**

**Propagation of error when Multiplying and Dividing**
 * If X = Xbest ± dx and Y = Ybest ± dy then what is q = x/y?
 * q=qbest ± dq
 * qbest = Xbest / Ybest
 * ** dq=l qbestl x     square root: (dx/Xbest)2 + (dy/Ybest)2   **

Suppose F = (25 ± 1)N and X = (6.4 ± 0.4)m What is F/X? qbest = Fbest/Xbest = 25N/6.4m = 3.9 N/m dq = (3.9 N/m) **x** square root: (1/25)2 + (0.4/6.4)2 = 0.29 N/M à 0.3 N/M (one sig fig) F/X = (3.9 ± 0.3) N/m
 * Example:**

**Propagation of error in a Power** > > **(Note: you do NOT need to know uncertainty from an average in the table below)** __**Measurement**__
 * If “n” is an exact number and q = xn
 * Then: qbest = Xbestn and dq =    | n |     x (dq/| x |)     x     | q |

There are two types of measurement: scalar measurements and vector measurements.

__Scalar Measurements__
A scalar measurement is one that only takes a value and its corresponding units, such as length, mass, temperature, and time who take on the units of meters, kilograms, degrees Celsius, and seconds, respectively.

__Vector Measurements__
A vector quantity is measured in terms of magnitude and direction. Magnitude is similar to a scalar quantity in that it consists of a value and units. Some examples of vector quantities are velocity, force, acceleration, and displacement.

When describing a vector, it is important to remember the **MAR** rule (**M**agnitude **A**ngle **R**eference). These three things must always be listed when solving a vector problem. A vector is represented by at least one letter with an arrow over it

Each vector is made up of **components** (a projection of a vector onto an axis). A vector has both an x-component (x-comp) and a y-component (y-comp). To find vector components, one uses the pythagorean theorem, sine, and cosine as seen in this picture. When using mathematical operations (i.e. addition/subtraction) with vectors, you will almost always use the components. For example, with velocity problems, the x and y directions of velocity will always be independent of each other, and you will work with the components of the vector.

Vector Notation
There are two ways of describing a vector that are identical but simply use different letters, î-ĵ-k and x-ŷ-ẑ. ŷ is pronounced "y-hat." Note: Characters for x-hat and k-hat could not be found in the Character Map.

x-hat and i-hat both represent the component directed along the x-axis. y-hat and j-hat both represent the component directed along the y-axis. z-hat and k-hat both represent the component directed along the z-axis.

XYZ coordinate axis



For example, if there is a vector B that measures 7m at 20° North of East, it could be rewritten as Bcos20(x-hat) + Bsin20(y-hat) or Bcos20(i-hat) + Bsin20(j-hat). Bcos20 represents a scalar multiple of the unit vector "x-hat"

Vector Manipulation
If a vector is multiplied by a scalar constant, its components also increase by the same amount. A vector multiplied by -1 results in a vector heading 180º in the opposite direction from the original vector.

Vector Addition and Subtraction
There are two methods to do addition and subtraction. There is the tail-to-head method, as seen in the picture, where one vector is moved so that its tail is at the head of the other vector's head. Then, the length from the tail of Vector A to the head of Vector B is computed to determine the magnitude of the vector C. The other method is the parallelogram method, as seen in the other picture, where two lines are drawn (A and B) to finish the parallelogram and the length of the diagonal (A+B) is computed. Vector addition is commutative, so it does not matter what order you add or subtract the vectors in.

Tail to head Method -- Parallelogram method



Given two vectors, Vector A = 10 m at 50º N of E and Vector B = 8 m at 20º N of E. Compute the magnitude and angle of the resultant vector.
 * Example**:

First, the components of each vector must be calculated. For Vector A, x-comp = 10cos50 = 6.43 m y-comp = 10sin50 = 7.66 m For Vector B, x-comp = 8cos20 = 7.52 m y-comp = 8sin20 = 2.74 m

Second, the components of the resultant vector must be calculated by using the pythagorean theorem with the components for the original values. Vector R's x-comp = sqrt (6.43² + 7.52²) = 9.89 m Vector R's y-comp = sqrt (7.66² + 2.74²) = 8.14 m

Third, the magnitude of the resultant vector is calculated by using the pythagorean theorem. Vector R = sqrt(9.89² + 8.14²) = 12.81 m

Finally, the angle is computed using tangent inverse. Angle of Vector R = tan¯¹(8.14/9.89) = 39.46°

In review, there are 4 steps: 1. Compute the original vectors' components. 2. Compute the resulting vector's components using Pythagorean Theorem. 3. Compute the resulting vector's magnitude using Pythagorean Theorem with Step 2 values. 4. Compute the angle by using tan inverse with the Step 2 values.

For subtraction, addition is performed except the second angle is treated as being multiplied by -1. Instead of being Vector A - Vector B, it is treated as Vector A + (-Vector B).

=__Links__= 1. Vector information with calculators for plugging in components, addition, three vector addition, etc.: []

2. Information on uncertainty and error propagtion, which includes practice problems: []

3. Error propagation calculator for 8 different operations: []

4. A video all about vectors: [] Once you get past the voice, she's relatively smart. Ignore all the information about multiplication in the middle

5. A table with metric conversions: http://www.csgnetwork.com/converttable.html


 * __SOURCES__**
 * 1) Notes
 * 2) Error picture: http://techrepublic.com.com/i/tr/cms/contentPics/error-reporting-5276398a.gif
 * 3) Accuracy vs. Precision : http://www.av8n.com/physics/img48/scatter-2d.png
 * 4) XYZ Axis: http://www.elsaelsa.com/wp-content/uploads/2008/09/xyz-coordinates.png
 * 5) Parallelogram Method: http://mathworld.wolfram.com/images/eps-gif/ParallelogramLaw_1000.gif
 * 6) Head to Tail Method: http://chortle.ccsu.edu/VectorLessons/vch03/headToTailRule1.gif

Error Analysis: Matt Vectors and Pictures and Video and Links: Andrew Pictures and Links: Frank the Tank