Uncertainties in calculated results Determine the uncertainties in results. Uncertainties in graphs
Friedman, Gabe, Federal Litigation Reporter has reference to this Academic Journal, PHwiki organized this Journal Uncertainties in calculated results Uncertainties. Absolute uncertainty is the value of r in addition to om uncertainty reported in a measurement (the larger of the reading error or the st in addition to ard deviation of the measurements). This value carries the same units as the measurement. Fractional uncertainty equals the absolute error divided by the mean value of the measurement (it has no units). Percentage uncertainty is the fractional uncertainty multiplied by 100 to make it a percentage (it has no units). Determine the uncertainties in results. A simple approximate method rather than root mean squared calculations is sufficient to determine maximum uncertainties For functions such as addition in addition to subtraction absolute uncertainties may be added When adding or subtracting measurements, the uncertainty in the sum is the sum of the absolute uncertainty in each measurement taken An example: suppose the length of a rectangle is measured as 26 ± 3 cm in addition to the width is 10 ± 2 cm. The perimeter of the rectangle is calculated as L + L + W + W = (26 + 26 + 10 + 10) ± (3 + 3 + 2 + 2) = 72 ± 10 cm
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Determine the uncertainties in results. Cont A simple approximate method rather than root mean squared calculations is sufficient to determine maximum uncertainties For multiplication, division in addition to powers percentage uncertainties may be added When multiplying in addition to dividing, add the fractional or percentage uncertainties of the measurements. The absolute uncertainty is then the fraction or percentage of the most probable answer. An example: find the area of the same rectangle. Area = L x W. The most probable answer is 26 x 10 = 260 cm2. To find the absolute uncertainty in this answer, we must work with fractional uncertainties. The fractional uncertainty in the length is 3/26. The fractional uncertainty in the width is 2/10. The fractional uncertainty in the area is the sum of these two = 3/26 + 2/10 = 41/130. The absolute uncertainty in the area is found by multiply this fraction times the most probable answer = (41/130) x 260 cm2 = 82 cm2. Finally, the area is 260 ± 82 cm2. Determine the uncertainties in results. Cont A simple approximate method rather than root mean squared calculations is sufficient to determine maximum uncertainties For other functions ( as long as example, trigonometric functions) the mean, highest in addition to lowest possible answers may be calculated to obtain the uncertainty range. For functions of the as long as m xn, the fractional uncertainty in the result is equal to n x fractional uncertainty in x. An example: find the volume of a cube with a side of length 5 ± .25 cm. Volume = L3. So, the most probable value is 53 = 125 cm3. The fractional uncertainty in the length is .25/5, so the fractional uncertainty in the volume is 3 x .25/5 = .75/5. Then, the absolute uncertainty in the volume is (.75/5) x 125 = 18.75 cm3. Finally, the volume is reported as 125 ± 19 cm3. If one uncertainty is much larger than others, the approximate uncertainty in the calculated result may be taken as due to that quantity alone. Uncertainties in graphs Based on the ideas above about drawing trend lines it is possible to draw several different trend lines through the data. To find the range of values as long as the slope & intercept you should draw the trend line with the largest value as long as the slope that still crosses all of the error bars in addition to the trend line with the smallest value as long as the slope. Then simply measure the slope in addition to intercept as long as each of these new lines. These represent the range of values as long as the slope in addition to intercept. Take the average of these two values as the absolute uncertainty as long as the value.
Trans as long as m equations into generic straight line as long as m (y = mx + b) in addition to plot the corresponding graph To use the slope, intercept, or area under a graph to determine the value of physical quantities, it is easiest to work with a straight-line graph. Most graphs can be trans as long as med from a curved trend to a linear trend by calculating new variables. If the relationship between two variables is known in theory, then the known relationship can be used to trans as long as m a graph into a straight-line graph. An example: Suppose that the speed of a falling ball is measured at various distances as the ball falls from a height of 10 m toward the ground on the moon. From energy conservation, we can predict that the speed of the ball is related to the height above the ground as v2 = 2gh. If we were to plot v vs. h we would get a sideways facing parabola. How can we make this a linear equation If we calculate a new quantity (call it w) so that w = v2, then our relationship becomes w = 2gh. If we plot w vs. h, we will get a straight line. (see the graphs below). This straight line can be used to find g on the moon (the slope of the graph = 2g). Analyze a straight-line graph to determine the equation relating the variables Once the straight-line graph has been achieved, using the intercept in addition to slope the equation of the trend line can be written (y = mx + b). Then substitute into the linear equation as long as the quantity that was used as the y in addition to /or x variable to produce the straight-line graph. In our example, the equation of the trend line is y = 3.2x. This means that our relationship is v2 = 3.2h. And, the slope of the trend line (3.2) should be twice the acceleration of gravity on the moon, so 2g = 3.2, or g on the moon = 1.2 ms-2.
Trans as long as m equations involving power laws in addition to exponentials -into the generic straight line as long as m y = mx + b plot the corresponding -log-log -semi-log graphs from the data Analyze log-log in addition to semi-log graphs to determine the equation relating two variables The relationship between two variables may not be readily seen in addition to if the relationship between the variables is not linear, then it might be difficult to find the correct relationship. One method to find the relationship is guess in addition to check. If y vs. x is not a straight line, then maybe y vs. x2 or y vs. x3 might work. Luckily, mathematics provides us a more reliable method. Analyze log-log in addition to semi-log graphs to determine the equation relating two variables If the relationship between two quantities, x in addition to y, is of the as long as m y = kxn, where k is a constant in addition to n represents some unknown exponent, then we can use logs (either base 10 logs or ln may be used) to trans as long as m this equation into a linear relationship as follows. take the log of both sides of the equation: log y = log (kxn) use the properties of logs to simplify: log y = log k + n log x substitute log y = Y in addition to log x = X: Y = log k + n X
Graph of Y vs. X, we would get a straight line. AND, the slope of that line is equal to n, the exponent in our original relationship. The vertical intercept (b) = log k. The value of the vertical intercept, b, can be used to find the constant k: b = log k, so 10b = 10log k = k. Simply substitute the values of n in addition to k determined from the trans as long as med log-log graph into the original relationship to get the equation relating the two variables, y in addition to x. y = kenx Again, this will produce a curved graph in addition to we can use logs (this time it has to be ln) to trans as long as m the graph into a straight line. Let’s see what logs will do as long as us this time. take the ln of both sides of the equation: ln y = ln (kenx) used the properties of ln to simplify: ln y = ln k + (nx)ln e = ln k + nx substitue ln y = Y Y = ln k + nx. if we were to make a graph of Y vs x, we would get a straight line. AND, the slope of that line is equal to n (the constant in the exponent of our relationship). The vertical intercept (b) = ln k. The value of the vertical intercept, b, can be used to find the constant k: b = ln k, so eb = eln k = k. This type of graph is called a semi-log graph because we only had to use the ln function to trans as long as m one of the variables. Semi-log graphs will trans as long as m exponential relationships into straight-line graphs. Simply substitute the values of n in addition to k determined from the trans as long as med semi-log graph into the original relationship to get the equation relating the two variables, y in addition to x.
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Friedman, Gabe Federal Litigation Reporter
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