Ps is what type of quantity

One way to analyze the precision of measurements would be to determine the range, or difference between the lowest and the highest measured values. In the case of the printer paper measurements, the lowest value was Thus, the measured values deviated from each other by, at most, 0. These measurements were reasonably precise because they varied by only a fraction of an inch. However, if the measured values had been The measurements in the paper example are both accurate and precise, but in some cases, measurements are accurate but not precise, or they are precise but not accurate.

Let us consider a GPS system that is attempting to locate the position of a restaurant in a city. In Figure 1. This indicates a low precision, high accuracy measuring system. However, in Figure 1. This indicates a high precision, low accuracy measuring system. Finally, in Figure 1. The accuracy and precision of a measuring system determine the uncertainty of its measurements. Uncertainty is a way to describe how much your measured value deviates from the actual value that the object has.

If your measurements are not very accurate or precise, then the uncertainty of your values will be very high. In more general terms, uncertainty can be thought of as a disclaimer for your measured values.

For example, if someone asked you to provide the mileage on your car, you might say that it is 45, miles, plus or minus miles. The plus or minus amount is the uncertainty in your value. That is, you are indicating that the actual mileage of your car might be as low as 44, miles or as high as 45, miles, or anywhere in between. All measurements contain some amount of uncertainty. In our example of measuring the length of the paper, we might say that the length of the paper is 11 inches plus or minus 0.

The factors contributing to uncertainty in a measurement include the following:. In the printer paper example uncertainty could be caused by: the fact that the smallest division on the ruler is 0. It is good practice to carefully consider all possible sources of uncertainty in a measurement and reduce or eliminate them,. One method of expressing uncertainty is as a percent of the measured value. A grocery store sells 5-lb bags of apples. You purchase four bags over the course of a month and weigh the apples each time.

You obtain the following measurements:. We can use the following equation to determine the percent uncertainty of the weight. Consider how this percent uncertainty would change if the bag of apples were half as heavy, but the uncertainty in the weight remained the same. Hint for future calculations: when calculating percent uncertainty, always remember that you must multiply the fraction by percent.

If you do not do this, you will have a decimal quantity, not a percent value. There is an uncertainty in anything calculated from measured quantities.

For example, the area of a floor calculated from measurements of its length and width has an uncertainty because the both the length and width have uncertainties. How big is the uncertainty in something you calculate by multiplication or division? If the measurements in the calculation have small uncertainties a few percent or less , then the method of adding percents can be used.

This method says that the percent uncertainty in a quantity calculated by multiplication or division is the sum of the percent uncertainties in the items used to make the calculation. For example, if a floor has a length of 4. For a quick demonstration of the accuracy, precision, and uncertainty of measurements based upon the units of measurement, try this simulation.

You will have the opportunity to measure the length and weight of a desk, using milli- versus centi- units. Which do you think will provide greater accuracy, precision and uncertainty when measuring the desk and the notepad in the simulation? Consider how the nature of the hypothesis or research question might influence how precise of a measuring tool you need to collect data.

An important factor in the accuracy and precision of measurements is the precision of the measuring tool. In general, a precise measuring tool is one that can measure values in very small increments. For example, consider measuring the thickness of a coin. A standard ruler can measure thickness to the nearest millimeter, while a micrometer can measure the thickness to the nearest 0.

The micrometer is a more precise measuring tool because it can measure extremely small differences in thickness. The more precise the measuring tool, the more precise and accurate the measurements can be.

When we express measured values, we can only list as many digits as we initially measured with our measuring tool such as the rulers shown in Figure 1. For example, if you use a standard ruler to measure the length of a stick, you may measure it with a decimeter ruler as 3.

You could not express this value as 3. It should be noted that the last digit in a measured value has been estimated in some way by the person performing the measurement. For example, the person measuring the length of a stick with a ruler notices that the stick length seems to be somewhere in between 36 mm and 37 mm. He or she must estimate the value of the last digit. The rule is that the last digit written down in a measurement is the first digit with some uncertainty.

For example, the last measured value The number of significant figures in a measurement indicates the precision of the measuring tool. The more precise a measuring tool is, the greater the number of significant figures it can report.

Special consideration is given to zeros when counting significant figures. For example, the zeros in 0. There are two significant figures in 0. However, if the zero occurs between other significant figures, the zeros are significant.

For example, both zeros in Therefore, the The zeros in may or may not be significant, depending on the style of writing numbers. They could mean the number is known to the last zero, or the zeros could be placeholders. So could have two, three, or four significant figures. To avoid this ambiguity, write in scientific notation as 1. Therefore, we know that 1 and 3 are the only significant digits in this number. In summary, zeros are significant except when they serve only as placeholders.

When combining measurements with different degrees of accuracy and precision, the number of significant digits in the final answer can be no greater than the number of significant digits in the least precise measured value. There are two different rules, one for multiplication and division and another rule for addition and subtraction, as discussed below. For multiplication and division: The answer should have the same number of significant figures as the starting value with the fewest significant figures.

Then, using a calculator that keeps eight significant figures, you would get. But because the radius has only two significant figures, the area calculated is meaningful only to two significant figures or. For addition and subtraction : The answer should have the same number places e. Suppose that you buy 7. Then you drop off 6. Finally, you go home and add How many kilograms of potatoes do you now have, and how many significant figures are appropriate in the answer?

The mass is found by simple addition and subtraction:. The least precise measurement is This measurement is expressed to the 0. Thus, the answer should be rounded to the tenths place, giving The same is true for non-decimal numbers. For example,. We cannot report the decimal places in the answer because 2 has no decimal places that would be significant. Therefore, we can only report to the ones place. It is a good idea to keep extra significant figures while calculating, and to round off to the correct number of significant figures only in the final answers.

The reason is that small errors from rounding while calculating can sometimes produce significant errors in the final answer. Keeping all significant during the calculation gives Rounding to two significant figures in the middle of the calculation changes it to 5, — 5. You would similarly avoid rounding in the middle of the calculation in counting and in doing accounting, where many small numbers need to be added and subtracted accurately to give possibly much larger final numbers.

In this textbook, most numbers are assumed to have three significant figures. Furthermore, consistent numbers of significant figures are used in all worked examples. You will note that an answer given to three digits is based on input good to at least three digits. If the input has fewer significant figures, the answer will also have fewer significant figures. Care is also taken that the number of significant figures is reasonable for the situation posed.

In some topics, such as optics, more than three significant figures will be used. The U. Most of us do not have any concept of how much even one trillion actually is. If you made bill stacks, like that shown in Figure 1.

One of your friends says 3 in. What do you think? Since this is an easy-to-approximate quantity, let us start there. We can find the volume of a stack of bills, find out how many stacks make up one trillion dollars, and then set this volume equal to the area of the football field multiplied by the unknown height. Calculate the number of stacks. The number of stacks you will have is.

Calculate the area of a football field in square inches. Because we are working in inches, we need to convert square yards to square inches. Note that we are using only one significant figure in these calculations. The height of the money will be about in.

Converting this value to feet gives. The final approximate value is much higher than the early estimate of 3 in. How did the approximation measure up to your first guess?

What can this exercise tell you in terms of rough guesstimates versus carefully calculated approximations? What can this exercise suggest about the value of rough guesstimates versus carefully calculated approximations?

Most results in science are presented in scientific journal articles using graphs. Graphs present data in a way that is easy to visualize for humans in general, especially someone unfamiliar with what is being studied. They are also useful for presenting large amounts of data or data with complicated trends in an easily-readable way. One commonly-used graph in physics and other sciences is the line graph , probably because it is the best graph for showing how one quantity changes in response to the other.

Our two variables , or things that change along the graph, are time in minutes, and distance from the station, in kilometers. Remember that measured data may not have perfect accuracy. Next, you must determine the best scale to use for numbering each axis.

Because the time values on the x -axis are taken every 10 minutes, we could easily number the x -axis from 0 to 70 minutes with a tick mark every 10 minutes. Likewise, the y -axis scale should start low enough and continue high enough to include all of the distance from station values.

A scale from 0 km to km should suffice, perhaps with a tick mark every 10 km. A space or half-high dot is used to signify the multiplication of units. Variables and quantity symbols are in italic type. For more details, see Typefaces for symbols in scientific manuscripts. She exclaimed, " That dog weighs 10 kg!

He exclaimed, " That dog weighs 10 kg! Superscripts and subscripts are in italic type if they represent variables, quantities, or running numbers. The combinations of letters "ppm," "ppb," and "ppt," and the terms part per million, part per billion, and part per trillion, and the like, are not used to express the values of quantities. Unit symbols or names are not modified by the addition of subscripts or other information.

Information is not mixed with unit symbols or names. It is clear to which unit symbol a numerical value belongs and which mathematical operation applies to the value of a quantity. Unit symbols and unit names are not mixed and mathematical operations are not applied to unit names. Values of quantities are expressed in acceptable units using Arabic numerals and symbols for units.

There is a space between the numerical value and unit symbol, even when the value is used in an adjectival sense, except in the case of superscript units for plane angle.

The digits of numerical values having more than four digits on either side of the decimal marker are separated into groups of three using a thin, fixed space counting from both the left and right of the decimal marker. SI units are part of the metric system. The metric system is convenient for scientific and engineering calculations because the units are categorized by factors of Table 2 gives metric prefixes and symbols used to denote various factors of Metric systems have the advantage that conversions of units involve only powers of There are centimeters in a meter, meters in a kilometer, and so on.

In non-metric systems, such as the system of U. Another advantage of the metric system is that the same unit can be used over extremely large ranges of values simply by using an appropriate metric prefix. For example, distances in meters are suitable in construction, while distances in kilometers are appropriate for air travel, and the tiny measure of nanometers are convenient in optical design.

With the metric system there is no need to invent new units for particular applications. The term order of magnitude refers to the scale of a value expressed in the metric system. Each power of 10 in the metric system represents a different order of magnitude. For example, 10 1 , 10 2 , 10 3 , and so forth are all different orders of magnitude. All quantities that can be expressed as a product of a specific power of 10 are said to be of the same order of magnitude. Thus, the numbers and are of the same order of magnitude: 10 2.

Order of magnitude can be thought of as a ballpark estimate for the scale of a value. The diameter of an atom is on the order of 10 -9 m while the diameter of the Sun is on the order of 10 9 m. The fundamental units described in this chapter are those that produce the greatest accuracy and precision in measurement.

There is a sense among physicists that, because there is an underlying microscopic substructure to matter, it would be most satisfying to base our standards of measurement on microscopic objects and fundamental physical phenomena such as the speed of light. A microscopic standard has been accomplished for the standard of time, which is based on the oscillations of the cesium atom.

The standard for length was once based on the wavelength of light a small-scale length emitted by a certain type of atom, but it has been supplanted by the more precise measurement of the speed of light.

If it becomes possible to measure the mass of atoms or a particular arrangement of atoms such as a silicon sphere to greater precision than the kilogram standard, it may become possible to base mass measurements on the small scale. There are also possibilities that electrical phenomena on the small scale may someday allow us to base a unit of charge on the charge of electrons and protons, but at present current and charge are related to large-scale currents and forces between wires.

The vastness of the universe and the breadth over which physics applies are illustrated by the wide range of examples of known lengths, masses, and times in Table 1. Examination of this table will give you some feeling for the range of possible topics and numerical values.

See Figure 5 and Figure 6. Figure 5. Tiny phytoplankton swims among crystals of ice in the Antarctic Sea.

They range from a few micrometers to as much as 2 millimeters in length. Gordon T. Figure 6. Galaxies collide 2. The tremendous range of observable phenomena in nature challenges the imagination. Mahdavi et al. Hoekstra et al. It is often necessary to convert from one type of unit to another. For example, if you are reading a European cookbook, some quantities may be expressed in units of liters and you need to convert them to cups.

Or, perhaps you are reading walking directions from one location to another and you are interested in how many miles you will be walking. In this case, you will need to convert units of feet to miles. Let us consider a simple example of how to convert units. Let us say that we want to convert 80 meters m to kilometers km. The first thing to do is to list the units that you have and the units that you want to convert to.

In this case, we have units in meters and we want to convert to kilometers. Next, we need to determine a conversion factor relating meters to kilometers. A conversion factor is a ratio expressing how many of one unit are equal to another unit. For example, there are 12 inches in 1 foot, centimeters in 1 meter, 60 seconds in 1 minute, and so on.

In this case, we know that there are 1, meters in 1 kilometer. Now we can set up our unit conversion. We will write the units that we have and then multiply them by the conversion factor so that the units cancel out, as shown:.

Note that the unwanted m unit cancels, leaving only the desired km unit. You can use this method to convert between any types of unit. Suppose that you drive the Note: Average speed is distance traveled divided by time of travel. First we calculate the average speed using the given units.

Then we can get the average speed into the desired units by picking the correct conversion factor and multiplying by it. The correct conversion factor is the one that cancels the unwanted unit and leaves the desired unit in its place. Average speed is distance traveled divided by time of travel. Take this definition as a given for now—average speed and other motion concepts will be covered in a later module.