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# JEE Main study Notes for Error: Basic Concepts, Important Formulas, and Preparation Tips

The chapter on errors is part of Measurement. We all know that for doing some measurements we need tools but sometimes these measuring instruments result in some uncertainty. This uncertainty is known as Error. Accuracy and precision are two important factors that need to be considered while taking measurements. Both of them indicate how close a measurement is to a known or accepted value.

• Questions asked from this chapter are majorly related to finding dimensions, conversions, and find errors in measurement.  Check JEE Main Physics Syllabus
• The chances of getting a question from this section are quite low. However, we cannot neglect this chapter because if you check some papers from previous year then you can find 1 question from this chapter.

To score positive marks from this chapter of  JEE Main  candidates can go through the important formulas, solved questions and some tips to prepare this chapter.

### Introduction to Error

The measurement is the most important requirement of almost every science experiment and theory. Whether you study the units and dimensions of a body or do theories of electricity and magnetism, accurate measurement is required to understand the basic concepts behind it. Every measurement involves some kind of uncertainty in the experiment.

• For instance, suppose in case you are measuring a building’s height to calculate the velocity of an object when it is thrown vertically upwards at a certain moment.
• In case the kind of instrument you use to measure the building is shattered or defected, then the measurement done by it will be wrong.
• The reason for the wrong measurement is the selection of the wrong instrument.

The uncertainties that happen while experimenting are referred to as errors. Errors mostly occur due to experiments, as every measured value can have some errors in it. Thus eradicating the errors from our results is much important so that it won’t create problems in the future. Based on the nature of the errors, they are divided into two terms, namely Accuracy and Precision.

#### Sample Question

Question: If the percentage of errors in the measurement of mass and speed is 2% and 3% respectively. Then calculate the maximum error in the estimation of the K.E by calculating the mass and speed?

A) 11%

B) 8%

C) 5%

D) 1%

Solution: (B)

## Accuracy and its types

### Accuracy and its types

Accuracy means the closeness of measured value to a standard value.

Example - Suppose you weigh a packet and its weight is 3.1 kg but its known value is 9 kg, then your measurement will not be reliable. The ability of any instrument to judge the accurate value is known as its accuracy. In easier words, the closeness of the measured value to a standard or true value.

Accuracy is calculated by considering small readings as small reading can lower the chances of error of the calculation. The accuracy of the system is classified into three types mentioned below:

Point Accuracy- The accuracy of the instrument at a particular point on its scale is known as the point accuracy. This accuracy doesn’t offer any information about the general accuracy of the instrument.

Accuracy as Percentage of Scale Range- The uniform scale range helps in calculating the accuracy of a measurement. For instance, a thermometer that hasve the scale range up to 500ºC, the thermometer has an accuracy of plus or minus 0.5, which means an increase or decrease by this much value in the instrument is negligible. But in case the reading shows changes of more or less than 0.5, then it might result in a high-value error.

Accuracy as Percentage of True Value- This kind of accuracy of the instruments is explained by considering the measured value regarding their true value. The accuracy of the instruments can be neglected up to plus/minus 0.5 percent from the true value.

#### Sample Question

Question: Let us consider that the random error in the arithmetic mean of 100 observations is x then calculate that what would be the random error in the arithmetic mean of 400 observations?

A) 4x

B) 1/4x

C) 2x

D) 1/2x

Solution: (B)

## Precision of Instruments

### Precision of Instruments

Precision means closeness between two or more measured values to each other.

Example - Suppose you weigh the same packet four times and get close results like 4.1, 4.2, 4.22, and 4.0 then your measurements are accurate. Always remember that Accuracy and Precision are two independent terms; you can be accurate in the measurements but non-precise, or vice-versa. Measurements of units revolve around accuracy and precision, therefore the experiment readings are mentioned in the decimal form.

Repeatability- The variations that arise when the conditions are kept identical and continuous measurements are taken after a short period.

Reproducibility- The variation that arises when the same measurement process among different instruments and operators, and measurements are taken after longer time periods.

#### Sample Question

Question: The significant figures in 0.310 x 103 are

A) 2

B) 3

C) 4

D) 6

Solution: (B)

## Difference between Accuracy & Precision

### What is the difference between Accuracy and Precision?

AccuracyPrecision
Accuracy means the level of agreement between the actual measurement and the absolute measurement.Precision means any variations that lie in the values of several measurements of the same factor.
Accuracy means how closely the results agree with the standard valueIt represents how closely results agree with one another
Single factor or measurementDifferent measurements or factors are required
For any measurement to be accurate, it must precise.Results can be precise without actually accurate. Alternatively, the results can be precise and accurate.

### Unit, Dimensions, and Measurement Notes

• To measure or express a physical quantity, a standardized measurement is considered so that different measurements of one physical quantity can be related to each other. The standard unit is being referred to as the unit of the specific physical quantity.
• While measuring any physical quantity, we need two parts including Numerical value (n) × Unit (u)
• Numerical value tells us the total number of times the physical quantity is measured in regard to its standard unit. The second part gives the name of the unit.
• Base Units- The units for the fundamental or base quantities are known as fundamental or base units.
• Derived Units- The units of all other physical quantities is mentioned as the combinations of the base units. The units obtained for the derived quantities are known as derived units.

### What are the kinds of Error?

The errors are divided into two types including:

• Systematic Errors
• Random Errors

Systematic Errors- The types of errors that arise only in one direction are referred to as Systematic Errors. Their direction might be positive or negative but will not be the same at the same time. Systematic errors are also referred to as Repetitive Error as they might occur due to some error in the machines and due to the selection of the wrong experiment apparatus. These errors take place in case the device used to take measurements is wrong.

Random Errors: The errors which happen due to lack of accuracy in an instrument are termed as instrumental errors. These kinds of errors happen due to beneath reasons:

• In case the instruments are designed properly, it will not result accurate results
• The calibration of the instrument is incorrect
• If the scale is worn off at edges or broken from somewhere
• If an instrument is giving a wrong reading instead of the actual one

The error in calculating the value of a physical quantity is divided as

Absolute error - The magnitude of variations between the measured and the true value of the quantity is called the absolute error of the measurement.

Percentage error - When the relative error is described in percentage, it is known as the percentage error.

Relative error- The ratio of absolute error to the true value in measuring a physical quantity is referred as relative error.

#### Examples

• In case the marking of a thermometer is mentioned wrongly, it results in Instrumental error. For instance, if the marking on the thermometer is 108°C instead of 100°C, then it is an instrumental error
• If the pressure of the atmosphere is 1 bar and the instrument shows 1.5 bars, then it results in an instrumental error
• In a Vernier caliper, if the 0 of the main scale don’t coincide with that of Vernier scale then also it is an instrumental error as the design of Vernier caliper is not proper

## Imperfection in Technique

### Imperfection in Technique

In case an experiment is not done as per the guidelines or physical conditions around are not favourable, then it will result in imperfection in technique errors. These kinds of errors arise due to:

• In case the instrument is not used properly
• If the instructions are not followed as per the rules of the experiment
• In case the environment is not favourable with external physical conditions
• If the technique is not accurate

#### Some examples of errors

Example 1

In case we keep the thermometer under the armpit rather than under the tongue, the temperature will always come out to be lower than that of body, as the way of calculating the temperature is wrong.

Personal Errors: These errors happen due to improper setting of apparatus, of sometimes due to lack of observation skills as well. Personal errors depend on the user or student who is experimenting and don’t know much about the instrument settings.

Example 2

Also, in some cases, while measuring the height of an object, if the student has not correctly placed his head, it might result to parallax and incorrect readings.

### How to reduce systematic errors?

Systematic errors can be reduced greatly by:

• If we improve the experimental techniques and experiment based on the guidelines and precautions of the experiment
• Use correct and accurate instruments and get the old instruments maintained.
• Focus more while performing an experiment so that the chances of silly mistakes in taking the readings of the measurement can be avoided.
• Try to remove personal mistakes and keep the instruments safe post reading the experiment

#### What are random errors?

Random Errors are not fixed on general perimeters and differ from measurements to measurements. Since these errors differ from measurement to measurement, they are being described as random errors as they are random in nature. Random errors are the variations in statistical readings due to limitations of precisions in the instrument. Random errors happen due to:

• Sudden and unexpected shifts in experimental conditions of the environment
• Some errors which even the student is not aware of

#### Some example of random errors includes:

• A spring balance will give different readings, in case the temperature of the environment is not the same
• If the same experiment is repeated, there are higher chances to get different observations

We can only lower the cases of random errors but can’t remove them completely as they are unpredictable and not fixed in nature unlike systematic errors.

#### Least Count Error

The smallest value that can be measured in an instrument is referred to as the Least Count of the Instrument. Least count means the main part of a measurement and occurs in both random as well as systematic Errors. These errors are based on the resolution of the instrument and can be calculated in case we know the observations and the least count of instruments.

## Previous year Solved Sample Questions

### Previous year Solved Sample Questions

Question 1- The volume of a liquid is 26 mL. A college student calculates the volume and finds it to be 26.3 mL, 26.2 mL, 25.8 mL, and 26.3 mL in different rounds of measurements. Which of the beneath statements is true for his measurements?

• They are not precise and accurate.
• They have poor accuracy.
• They have good precision.
• They have poor precision.

Question 2- The volume of a liquid is 20.5 mL. Which of the beneath sets of measurement has good accuracy?

• 18.6 mL, 17.4 mL, 19.6 mL, 17.3 mL
• 19.3 mL, 19.4 mL, 18.48 mL, 18.9 mL
• 18.7 mL, 19.4 mL, 19.7 mL, 18.0 mL
• 20.2 mL, 20.4 mL, 20.5 mL, 20.0 mL

Answer: The set 20.2 mL, 20.4 mL, 20.5 mL, 20.0 mL has good accuracy.

Question 3- A student uses a simple pendulum of 1m length to calculate G, the acceleration due to gravity. He preferred a stopwatch with minimum count of 1 sec for the same and recorded 40 seconds for 20 oscillation. For this observation, which of the beneath observations will be true?

• Error, the time will be 0.05 seconds
• Error, the time will be 0.05 seconds
• The percentage error in the determination of g is 5%
• The percentage error in the determination of g is 2.5%

Answer: Error, the time will be 0.05 seconds

## Important formulas for Measurements

### Some important formulas for Physics And Measurements

1. Absolute Error for nth reading = = true value - measured value
2. Relative error =
3. Percentage Error =
4. Absolute error in sum (x=a+b) =
5. Absolute error in difference (x=a-b) =
6. Relative error in product (x=a.b) =
7. Relative error in division (x = a/b) =

### Best Books for JEE Main Physics

We suggest  JEE Main 2020  aspirants not to spend too much time reading the books; rather prefer reading some important books that can help in understanding more about the exact exam pattern, and help them in preparing a systematic strategy.

Some of the best and most recommended books for Physics include:

BooksAuthors
Concepts of Physics (volumes 1 and 2)H.C Verma
Guide to AIEEE Problems in General PhysicsG. K Publishers I.E Irodov
D.C Pandey (Complete Set)Arihant.
Problems in PhysicsS. S Krotov

## Useful tips for Measurement

### Some useful tips for Physics and Measurement

• This chapter includes more practicing and memorizing of formulas and units. One can always remember the dimensions, in case they have practiced enough questions.
• Questions asked from this chapter are majorly related to finding dimensions, conversions and find errors in measurement.
• Solve the easiest questions first and practice questions of a different style. First clear the concepts, before rushing to solve practice problems.
• Solve previous year questions of various exams from this chapter.
• Make a plan to prepare for this chapter and adhere to the timetable you prepared.

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