Measurements

Errors in Measurements

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Errors in Measurements

Since no measurement can be absolutely accurate, there will always be some sort of error in a measurement. An error is the difference between the measured value and the actual value.

  1. If an error cannot be eliminated or reduced, it must be considered as part of the measurement.
  2. There are two main types of errors in measurements, namely systematic errors and random errors.

Systematic Errors

  1. Systematic errors are errors in measurements due to the condition of the measuring instrument or the state of the environment in which the measurements are taken.
  2. A systematic error is usually unavoidable and is always the same. This means that the reading of a measurement is always bigger than the actual value or it is always smaller than the actual value.
  3. A cause of systematic error is errors in calibration of instruments. The meter rule in Figure may look normal at first glance but a careful study shows that it is wrongly calibrated as its first 1 cm is actually only 0.9 cm. Hence, the readings of the measurements using this metre rule are always bigger than their actual value.
    Errors in Measurements 1
  4. Another cause of systematic error is zero error. Zero error is due to non-zero reading when the actual reading should be zero. This error is usually due to the fact that the pointer of the instrument does not return to zero when it is not being used. Figure shows a meter rule with a zero error. The end of the ruler has worn out and gives a reading of 0.1 cm instead of 0 cm. Hence, the readings of the measurements using this meter rule are always bigger than their actual value.Errors in Measurements 2

Random Errors

  1. Random error occurs due to mistakes made by the observer using incorrect positioning of the eye or the instrument when making a measurement. It may also occur when there is a sudden change of environmental factors like temperature, air circulation or lighting.
  2. When taking a reading, you should look at the scale of the instrument from a point directly in front of it. Otherwise, error will occur in your measurement.
  3. Parallax error is a random error due to the inaccurate positioning of the eye when taking the reading of a measurement. Figure shows how parallax error occurs when taking a reading from a ruler.
    Errors in Measurements 3
  4. An electrical meter with a mirror just beneath the needle helps to reduce parallax error. Parallax error is eliminated when the needles image is hidden behind the needle itself.Errors in Measurements 4

Example 1. Figure shows three different positions of the eye of an observer when measuring the length of a physics book.
Errors in Measurements 5What should be the correct reading in cm? Explain your answer.
Solution:
The correct reading should be 23.5 cm. This is because the correct position of the eye should be directly perpendicular to the edge of the book where the measurement is being made. The other two positions will result in parallax

Techniques to Reduce Errors in Measurements

  1. It is important to reduce errors in measurements as much as possible in order to increase their accuracy.
  2. When a measurement is made, the choice of the measuring instrument is an important factor in reducing errors. For example, Figure shows a water bottle. It is more appropriate to measure the diameter of the cap using a pair of vernier callipers than a metre rule. However, it is more appropriate to use a meter rule to measure the height of the bottle than a pair of vernier callipers.
    Errors in Measurements 6
  3. When no appropriate measuring instruments are available, proper techniques in measurements are desirable to ensure better accuracy. Figure shows how a stack of 10 identical books can be measured using a meter rule to determine the thickness of one of the books. If the height of the stack of books is h cm, then the thickness of a h book is h/10 cm.Errors in Measurements 7
  4. Another way to reduce error is to repeat the measurement a number of times and find the mean (average). This will serve to reduce experimental errors caused by poor alignment of object and instrument, misreading and non-uniform objects.
    (a) Figure shows an athlete clearing a high jump bar. To determine the height of the bar, measurements are taken at different places and the average value is calculated to ensure greater accuracy.Errors in Measurements 8(b) Figure shows a bone. To determine the diameter of the bone, measurements are taken at various parts of the bone using a pair of vernier callipers and the average value is calculated to reduce error due to non-uniformity of the shape of the bone.Errors in Measurements 9(c) Figure 1.34 shows a boy using a stopwatch to determine the period, T, of oscillation of a pendulum. As the value of T is very small and is difficult to determine accurately using a stopwatch, it is easier to measure the time, f, for 10 oscillations and determine the
    period, T, using the formula T = t/10.Errors in Measurements 105. It is important to check a measuring instrument for zero error before using it. For some instruments, like the voltmeter in Figure, zero error can be taken care of by adjusting the pointer to zero before making any measurements.Errors in Measurements 116. When adjustments cannot be done to take care of zero error, the value of this error must be noted and all the subsequent readings have to be corrected by deducting this value.

To Increase the Accuracy of Measurements
When we reduce errors, we will increase the accuracy of measurements. Hence, the following are the ways to increase the accuracy of measurements:

  1. Use appropriate measuring instruments.
  2. Use appropriate techniques in taking measurements.
  3. Make repeated measurements and take the average value.
  4. Avoid parallax error.
  5. Take into account zero errors.

Example 1. Figure (a) shows the position of the pointer of a voltmeter when it is not taking any measurements. Figure (b) shows the same voltmeter when taking a measurement.Errors in Measurements 12(a) What is the value of the zero error of the voltmeter?
(b) What is the value of the voltage measured?
Solution:
(a) Zero error = 0.1 V
(b) Value measured = 3.6 – 0.1 = 3.5 V

Example 2. A worn out ruler is used to measure the length of a school badge as shown in Figure.Errors in Measurements 13(a) What is the value of the zero error of the ruler?
(b) What is the length of the badge?
Solution:
(a) The value of zero error due to the worn out end of the ruler is 0.2 cm.
(b) Therefore, the measured length of the badge is = 3.6 cm – 0.2 cm
= 3.4 cm.

Understanding Measurements

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Understanding Measurements

 

  1. Physics is based on measurements. Experiments which are the key to the development of physical theories involve measurements of various physical quantities.
  2. Various measuring instruments are used for different types of measurements and the reading obtained by each instrument must be presented in suitable units.
  3. Choosing an appropriate instrument to measure a physical quantity is important to ensure that the measurement is accurate and reproducible.
  4. Consistency, accuracy and sensitivity are three important aspects of a measurement.

Consistency and Accuracy

  1. No measurement is exact. Every measurement is an estimation of the actual value. For example, even the most accurate clock used in the Olympics does not give the exact time of a 100 m runner. It only gives an accurate estimate of the actual time.
    Understanding Measurements 1
  2. Consistency and accuracy are two important properties of measurements. Consider a target as in Figure.
    (a) The bullseye in the centre of the target represents the actual or true value of the quantity to be measured.
    (b) Individual measurements of the quantity under identical conditions are represented schematically by individual shots taken by the marksman on the target.

    Understanding Measurements 2
    Fig. Consistency

    (c) When the shots are closely clustered together, it shows that repeated measurements yield almost the same reading. We say that the measurements are consistent. Consistency of an instrument is the ability of the instrument to measure a quantity with little or no deviation among measurements.
    Language support:
    Deviation means the difference between the measured value and its mean value.

    Understanding Measurements 3
    Fig. Accuracy

    (d) The nearer a shot is to the bullseye, the nearer is the measurement to the actual value. We say that the measurement is accurate. Accuracy of a measurement is how close the measurement made to the actual value.

Example 1. Figure shows four sets of measurements on boards A, B, C and D each of which represent a rifle target.Understanding Measurements 4
(a) Comment on the consistency and accuracy of the measurements of boards A, B and C.
(b) Suggest one possible reason for the set of measurements shown on board D.
Solution:
(a) Board A – Consistent and accurate
Board B – Consistent but not accurate
Board C – Not consistent and not accurate
(b) Three measurements are accurate and consistent. The other measurement may be due to a wrong reading.

Example 2. Instrument X and instrument Y are used to measure the thickness of a dictionary. Table shows two sets of readings taken by the respective instruments.Understanding Measurements 5Which of the two instruments is more consistent? Explain your answer.
Solution:
Instrument X is more consistent. This is because the measurements taken by instrument X are closer to each other (or there is little deviation between the measurements).

Sensitivity of Measuring Instruments

  1. Sensitivity of a measuring instrument is its ability to detect quickly a small change in the value of quantity to be measured.
  2. The smaller the minimum scale division of a measuring instrument, the more sensitive is the instrument.
  3. Table shows the sensitivity of some common instruments for measuring length.
    Understanding Measurements 6

Example 3. A hawker has two measuring scales, A and B. A has a measuring range of 0 – 400 g and gives the smallest reading of 10 g or 0.01 kg. B has a measuring range of 0 – 4 kg and gives the smallest reading of 0.1 kg. A lady wants to buy a whole chicken of about 1.5 kg and red chillies of about 200 g.

Understanding Measurements 7State which scale is suitable for each of the items and explain your answer.
Solution:
Scale B is more suitable for measuring a whole chicken because the mass of the chicken is out of the range of measurement of scale A.
Scale A is more suitable for measuring the red chillies because it is more sensitive and can detect small changes in the mass.

Example 4. A, B, C and D show the scales of four different meters respectively. Which meter is the most sensitive?Understanding Measurements 8Answer: A
Meter A has the smallest minimum scale division out of the four meters.

Understanding Measurements 9Voltmeter A is more sensitive than voltmeter B because:

  1. Volmeter A has a smaller minimum scale division compared to voltmeter B.
  2. Voltmeter A can detect a smaller change in the voltage than voltmeter B.
  3. For the same variation of voltage, voltmeter A gives a bigger deflection than voltmeter B.