Sunday, September 14, 2014

Errors and approximations in physical quantities

Physics is the science of measurement. We cannot learn science and its applications without measuring physical quantities. Thus measurement plays a vital role in the process of learning physics.

While measuring physical quantities we always take some references and that references are being called as fundamental units. For example to measure the length we always use a meter scale. But every reference that we are taking will have its own limitations. If you consider a meter scale, the smallest reading that we can measure with that is millimeter. It is called the least count of the device. So this device is able to measure any length accurately up to 1 mm only. Hence we are having a compromise in measuring the reading. If you want an accuracy that is less than 1 mm, we shall use a better devise like vernier calipers.

Vernier calipers is a device which can measure the readings accurately up to 1/10 of the millimeter. If you are looking for a better accuracy we shall choose the devices like screw gauges who have an accuracy of one by hundreds of the millimeter.Every devise will have its own least count and we cannot measure accurately less than that using the device. So to what level we compromise in measuring in a reading simply depends on what is a requirement is. We appropriately choose the device as per our requirements.
Thus whatever the devise that we use to measure some physical quantities,the reading is prone to some sort of errors. Here we are going to discuss what are the types of possible errors are there and how can be contained within the possible limitations.

Error is the amount of uncertainty that is present in the measurement made with a measuring instrument. We always tries to either eliminate or minimize the errors. In many cases as it is not possible to eliminate the errors completely we try to minimize them using some mathematical and statistical methods.
Arithmetical mean is the one of that kind of the mathematical method to minimize the possible errors in a measurement. Here we measure the same reading multiple times and we take the mean of them.

Accuracy is the measure of the closeness of the measured value to the true value. It is obvious that the less the error  better the accuracy.
Precision refers to the agreement among the group of the measured values.If the measurement that we are making is consistent it will have a better appreciation and mean to say that all the readings will be very close to each other. Eliminating systematic errors can improve accuracy whereas repeating the experiment number of the Times can improve the precision.

Types of errors:
There are many reasons because of which get we errors. There are those that are always unidirectional are called systematic errors. Least count is a kind of the systematic error that occurs whenever you measure a particular reading with a particular instrument it is always prone to that least count error.

Environmental errors are quite possible as the conditions around experiments in the laboratory will be keep on changing. For example when we are doing an experiment parameters like pressure, temperature, humidity and wind velocity are always having a possibility of changing and because of this also we get some errors in the final readings.
There are some sort of errors called personal errors who comes the only due to the experimenter. This depends on how focused the experimenter and how careful he is while is taking the readings. Every experiment demand some sort of precautions and if they were not taken care, then also we get errors. A simple example of this kind of personal error is parallax error.This error comes because the readings are not taken by keeping the the object parallel to the eye.

To minimize the errors,we can take the arithmetic mean of all the readings. 
Random errors:

The errors which are irregular and whose cause is not known and random in nature in their sign and size are called random errors. There is no way that we can eliminate this kind of errors.The only possibility is to minimize them by using some mathematical methods like statistics.
An error is the difference between the measured value and the actual value.

Absolute error is the arithmetic mean of all their errors that were measured in different readings of an experiment.
Relative error is the ratio of absolute error to the actual value.

To get the percentage error,  we have to multiply the relative error with hundred.

Combination of errors:
A physical Quantity in equation is because of some sort of mathematical operation like addition, subtraction, multiplication and division of some other physical quantities.When this kind of situation is there we have to deal with the combination of their errors in the  resultant.

In addition and  subtraction we always treat the total possible error as the sum of two errors in the given to physical quantities. By taking so we are preparing ourselves to the worst possible scenario so that we can use that readings quite safely. The same is the case of multiplication and division and how the errors are being taken in the process is described as shown in the diagram below.

When physical quantities are the result of rising the physical quantities  to certain powers than also there are possible errors and we can get the value of the error by differentiating the given equation as shown below.

Significant figures:
The digits of a number representing a measurement that are definitely known plus one more digit added at the end which is estimated are called significant digits are significant figures. There are certain rules to follow in determining the number of significant figures.
Rules for identifying significant figures:

  1. All nonzero digits given in a number are significant without any regard to the location of decimal points.
  2. All zeros occurring in between nonzero digits are significant.

  3. If a number is greater than one all the zeros right to the decimal point are significant.
  4.  all the zeros to the right of last nonzero digits after the decimal point are significant.
  5. If a number is not having a decimal point then all zeros to the right of last nonzero digit are not significant. 

  6. Anyway this rule is having an exception that if the reading comes because of an actual measurement who is having a unit then this zeros becomes significant.

Rounding off the number:

In any experimenter in practical life we don’t need so many significant figures which will make your calculation measurements lengthy. In that case we can reduce the number of the significant figures and make the reading more simple and the process is called rounding off.

The preceding digit has to be raised by one if the immediate insignificant is it to be dropped is more than five.
The preceding digit  need not be arise if the significant figure that you want to drop ease less than five.
If that is it that you want to drop is five itself then we have to look at the previous digit. If the previous digit is a even digit we can just leave it as it is otherwise we have to add one to the previous digit.

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