1.3-1.4 Scalar n Vector

8/3/2019 1.3-1.4 Scalar n Vector

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Chapter 1

1.3 Scalar and Vector Quantities


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Understanding Scalar and Vector

Quantities

1. A scalar quantity is a quantity which

has only magnitude or size.

Mass = 58 kg


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Quantities

2. A vector quantity has both

magnitude/size and direction.

Velocity = 900 km/h

down south.


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Quantities

3 When we say that the temperature of a room is 28C,

or a bottle contains 500 cm3of milk, we are dealing with

scalar quantities. On the other hand, a force of120 Nacting downwards is a vector quantity.

120N


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Quantities

4. Time, temperature, mass, volume, distance,density and power are examples ofscalar

quantities. These quantities can be added usingsimple mathematical rules.

40W45 cm3


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Quantities

5. Force, velocity, displacement, acceleration andmomentum are vector quantities.

Force

Displacement,

AC


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Quantities

5. Force, velocity, displacement, acceleration andmomentum are vector quantities. To find a

resultant vector, all vector quantities are eitheradded or subtracted taking into account themagnitude and direction of the individual vector.


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Chapter 1

1.4 Measurements


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Understanding Measurements Nature of Measurement

1 Measurementsare trials to determine the true value

of a particular physical quantity.


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Understanding Measurements 2 The difference between the true valueof a

quantity and the value obtained in measurement

is the error.Actual mass = 60 kg

Weighing machine = 59 kg

Error = 60 - 59 = 1kg


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Understanding Measurements Nature of Measurement

3 No measurement can be absolutely accurate;

there will be some sort of error in a measurement.

Thickness of book

1.5 cm

1.52 cm

1.518 cm


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Errors in Measurement

1. There are two main types of errors.

(a) Systematic errors

(b) Random errors


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Systematic Errors

1 Systematic errors are cumulative errors

that can be compensated for, if the errorsare known.


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Systematic Errors

2 Systematic errors in measurement result from

(a) an incorrect position of the zero point, or known aszero error, and


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Systematic Errors

2 Systematic errors in measurement result from

(a) an incorrect position of the zero point, or known aszero error, and

(b) an incorrect calibrationof the measuring instrument.


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3 Systematic errors always occur(with

the same value) when we continue to use

the instrument in the same way.


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4 A zero errorarises when the measuring

instrument does not start from exactly zero.


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5 Zero errors are consistently presentinevery reading of a measurement so that the

results obtained may be precise but lack inaccuracy.


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6. Systematic errors cannot be eliminated by repeating

the measurements and averaging out the results. It only

can be eliminated or corrected if the measuring

instruments are calibrated or adjusted frequently.


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Random Errors

1 Random errors occurs due to mistakes made when

making measurement either through incorrect positioningof the eye or the instrument. It will produce a different

errorevery time you repeat the experiment. They may

vary from observation to observation.

You measure the mass of a ring three times using thesame balance and get slightly different values: 17.46 g,

17.42 g, 17.44 g


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Random Errors

2. Random errors can be minimised by repeating the

measurements several times and taking the average ormean value of the readings.

You measure the mass of a ring three times using the

same balance and get slightly different values: 17.46 g,

17.42 g, 17.44 g

Average/mean = g44.173

44.1742.1746.17


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Random Errors

3. A parallax erroris an error caused by incorrect

positioning of the eye when reading a measurement.

Error = + 0.2ml

Error = - 0.1ml

Error = + 0.1ml


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Random Errors

4. If he repeats his reading several times, and takes the

average of the results, he will end up with an answer thatis closer to the true value; butrepeating measurements

does nothing at all for the first observer.


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Random Errors

5 (a) To avoid parallax errors, the position of the eye

must be in line with the reading to be taken, as in positionC.


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5 (b) To overcome parallax

errors in instruments with a

scale and pointer, e.g. an

ammeter, often have a mirror

behind the pointer. The correct

reading is obtained by making

sure that that the eye is exactly

in front of the pointer, so thatthe reflection of the pointer in

the mirror is behind it (refer

Figure 1.3).


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5 (b)

EyeEye

Consistenc Acc rac and


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Consistency, Accuracy and

Sensitivity

Consistency/Precision

1 The consistency of a measuring instrument is its

ability to register the same reading when a measurement

is repeated.

Consistency Accuracy and


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Sensitivity


2 A set of readings from identical instruments will havea small relative deviation or no deviation from the meanvalue.

High consistency => Small deviation from the mean value

Big deviation: 54kg, 56kg, 57kg

Small deviation: 54kg, 54kg, 55kgPrecise



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Sensitivity


3 A deviation is the difference between a measured

value and its mean value or the average value.

Average reading of diameter = 3.24 cm

One of the reading = 3.26 cm

Deviation = 3.263.24 = 0.02 cm



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Sensitivity


4 Relative deviation is defined by the

formula below.

Relative deviation = x 100%valueAverage

deviationAverage



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Sensitivity

Consistency/Precision Example 1

The diameter of an object was measured 5 times using vernier caliper.The results are 3.14 cm, 3.15 cm, 3.12 cm, 3.09 cm and 3.05 cm.

Calculate the relative deviation.

Average diameter =

= 3.11 cm

3.14 3.15 3.12 3.09 3.05

5



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Sensitivity

Example 1:Diameter/cm Deviation/cm

3.14 (3.143.11) cm = 0.03 cm

3.15 (3.153.11) cm = 0.04 cm

3.12 (3.123.11) cm = 0.01 cm

3.09 (3.09

3.11) cm = |

0.02 cm| =0.02 cm

3.05 (3.05 3.11) cm = | 0.06 cm| =

0.06 cm



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Sensitivity

Example 1:

Mean deviation = = 0.03 cm

Relative deviation = x 100%

= x 100%

= 0.96%

0.03 0.04 0.01 0.02 0.06

5

0.03

3.11

valueAveragedeviationAverage



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Sensitivity


5 The consistency of a measuring instrument can beimproved by:

(a) eliminating parallax errors during measurement.



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Sensitivity



(b) exercising greater care and effort when taking

readings.



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Sensitivity



(c) using an instrument which is not defective.



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Sensitivity

Accuracy

1 Accuracy is the degree to which a

measurement represents the actual value.

Gravity = 9.81 ms-2

Experimental valueA = 9.76 ms-2

B = 9.62 ms-2 9.819.769.62



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Sensitivity

Accuracy

2 An accurate instrument is able to give

readings close to or almost equal to the actualvalue of a quantity.

9.819.769.62

Closer



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Sensitivity

Accuracy

3 An instrument with 100% accuracy does not

exist.



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Sensitivity

Accuracy

4 The error is the difference between the

measured valueand the actual or true value

9.81A.9.769.62

Error A = 0.05



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Sensitivity

Accuracy

5 The level of accuracy is related to the relative

error which is defined as the ratio of the error tothe actual value.

Relative error = x 100%valueactual

valueerror

9.81A.9.76B.9.62

Error A = 0.05

Error B = 0.19

R. Error A = %100x81.9

05.0

R. Error B = %100x81.9

19.0

=0.5%

=1.9%



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Sensitivity

6 A measured value with a very small error has a highaccuracy. If the relative error is of a small value, the levelof accuracy is high and vice versa.

Relative error Accuracy R. Error A = %100x

81.9

05.0

R. Error B = %100x81.919.0

=0.5%

=1.9%

Accuracy

high



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Sensitivity

Accuracy

7 How to improve the accuracy of a measurement?

(a) Repeated readings are taken and the average value iscalculated.



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Sensitivity

Accuracy

7 How to improve the accuracy of ameasurement?

(b) Avoid parallax errors,



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Sensitivity

Accuracy

7 How to improve the accuracy of ameasurement?

(c) Avoid zero errors.



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Sensitivity

Accuracy

7 How to improve the accuracy of a measurement?

(d) Use measuring instruments with a higher accuracy.For example, a vernier caliper is more accurate than a

ruler.



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Sensitivity

Sensitivity

1 The sensitivity of a measuring instrument is its ability

to detect quickly a small change in the value of a

measurement.



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Sensitivity

Sensitivity

2 A measuring instrument that has a scale withsmaller divisionsis more sensitive.



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Sensitivity

Sensitivity

3 As an example, the length of a piece of wire is

measured with rulers A and B which have scales

graduated in intervals of0.1 cm and 0.5 cm respectively,as shown in Figure 1.5. Which of the rulers is more

sensitive?



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Sensitivity

Sensitivity

3 Results:

Ruler A: Length = 4.8 cm

Ruler B: Length = 4.5 cm

Ruler A is more sensitive as it can measure to an accuracy

of 0.1 cm compared to 0.5 cm for ruler B



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Sensitivity

4 In addition to the size of the divisions on the scale of

the instrument, the design of the instrument has an effect

on the sensitivity of the instrument. For example, a

thermometer has a higher sensitivity if it can detect small

temperature variations. A thermometer with a narrowcapillary and a thin-walled bulb has a higher sensitivity.



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Sensitivity

Comparisons between Consistency, Accuracy,

and Sensitivity

1 The drawings in Figure 1.5, which show the

distribution of gunshots fired at a target board, serve toillustrate the meaning of consistency and accuracy.



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Sensitivity

2 A consistent measuring instrument is not necessarily

accurate. For example, a measurement with a metre rule is

consistent but not accurate due to end errors. In this

respect, this type of instrument gives readings which,

however, do not represent the true value of the measuredquantity.



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Sensitivity

3 A sensitive measuring instrument too, may not

be accurate or consistent. This is due to external

variations which cause variations in the readings.

Documents

1.3-1.4 Scalar n Vector