instruments and measurement

1 Devilim, Physics Grade X, 1st sem, SMA K IBC 2015-2016

QUANTITIES AND UNITS

The goal of physics is to provide an understanding of the physical world by developing theories based on experiments.

In physics, quantities that can be measured are known as physical quantities.

All physical quantities consist of a numerical value and a standard of unit.

Out of so many physical quantities, there are seven fundamental or base

quantities chosen arbitrarily to be the building blocks of all derived physical

quantities. Their corresponding units are called base units.

Beside base units, there are prefixes used to indicate decimal multiples of

submultiples of all units. The standard prefixes are shown in Table 1.4.

A derived quantity is a product or combination of fundamental quantities.

Example: Volume = length x breadth x height (its unit is m3).

Base unit are used to find units of unknown quantities in an equation.

Example:

What is the unit of h (Planck constant) (in terms of base units) in the

equation E = h.f? (E is the energy of a photon of light and f is the frequency).

E = h.f h = E/f

Then, unit of h is 𝑘𝑔.𝑚2/𝑠2

1𝑠⁄

= 𝑘𝑔. 𝑚2/𝑠

Base units can also be used to check the homogeneity of a physical equation.

If the left-hand side of an equation has the same unit as its right-hand side,

then the equation is said to be homogenous or dimensionally consistent.

Example: v = distance / time

the base units in left-hand side is : m/s

the base units in right-hand side is : meter/second = m/s

Because both sides have the same unit, then the equation is homogenous.

Conversion of unit:

Example: 1 inch (in) = 2.54 cm 1 = 2.54 cm/in

1. The width of a table is 21.5 in, and then in cm, the width is:

21.5 𝑖𝑛 = 21.5 𝑖𝑛 ×2.54 𝑐𝑚

𝑖𝑛= 54.6 𝑐𝑚

2. The semiconductor chip has an area of 1.25 square inches. In cm, the area is:

1.25 𝑖𝑛2 = 1.25𝑖𝑛2 ×2.54 𝑐𝑚

𝑖𝑛×

2.54 𝑐𝑚

𝑖𝑛= 8.06 𝑐𝑚2

Exercise:

1. Write down the SI unit of Pressure.

2. The following equation was given by a student during an examination:

𝑣 = 𝑣𝑜 + 2𝑎𝑡2

where v is speed, v0 is the initial speed, a is acceleration, and t is time.

Check and state whether the equation above homogenous or not.

3. Convert the values of the following quantities to the unit given: [3]

a. 0.035 GB = B

b. 50 nm = cm

c. 36 km/h = m/s

d. 5 m/s2 = km/h2

4. If a car is traveling at a speed of 28.0 m/s, is it exceeding the speed limit of 50.0 mi/h? (1 mile = 1609

m)


INSTRUMENTS AND MEASUREMENT

Instruments Used in Experiments on Mechanics

1. Measuring Length

The SI unit for length is meter (m). In length measurement, we choose an instrument that is suitable for the

length to be measured. Table 1 summarises the commonly used instruments and the lengths which they are

suitable for measuring.

Table 1. Instruments used for measuring length

Length to be measured Suitable instrument accuracy of instrument

several meter (m) measuring tape 0.1 cm

several centimeter (cm) to 1 m metre of half-metre rule 0.1 cm

between 1 cm to 10 cm vernier calipers 0.01 cm

less than 2 cm micrometer screw gauge 0.01 mm (or 0.001 cm)

The metre rule

This instrument is commonly used in laboratory to measure the lengths of objects. To use the metre rule, it is

best to measure from 1.0 cm mark, and then subtract 1.0 cm from the reading at the other end. This is because

for most metre rules, the zero mark is at the very end of the rule. For accurate measurement, the eye must

always be placed vertically above the mark being read, to avoid parallax errors which will give rise to

inaccurate measurement (Fig. 1 and 2)

The vernier calipers

This instrument is usually used for accurate measurement up to 0.1 mm or 0.01 cm.

The vernier calipers must be checked for any zero error. Before

placing the object to be measured between the jaws of the

vernier calipers, close the jaws fully to check whether the zero

mark on the main scale coincides with the zero mark on the

sliding vernier scale. If the two zero marks coincide as in

Figure 6, then there is no zero error.

Fig. 1 Fig. 2


However, if the zero mark on the sliding vernier scale is slightly to the right (see Figure 7) or to the left (see

Figure 8), then there is a zero error for each case.

In Figure 7, the third mark on the vernier scale coincides with a mark on the main scale. When using this

instrument, 0.03 cm should be subtracted from every reading to obtain the correct measurement. In Figure 8,

although the seventh mark on the vernier scale coincides with a mark on the main scale, the distance between

the two zero markings is not 0.07 cm, but 0.03 cm. Therefore, when using this instrument, 0.03 cm should be

added to every reading to obtain the correct measurement.

The micrometer screw gauge

The micrometer screw gauge has accuracy of 0.01 mm (or 0.001 cm). Figure 10 shows the main features of

this instrument and its use.

Figure 10. using micrometer

The micrometer must be checked for any zero error. Before placing

the object to be measured between the anvil and spindle, turn the

thimble until anvil and spindle meet. If the zero on the thimble scale

lies directly opposite the datum line of the main scale (Figure 11),

we say that there is no zero error. If the zero on the thimble scale

does not lie directly opposite the datum line (as in Figure 12 and

13), then we say that he instrument has zero error.

For Figure 12, the zero error is + 0.03 mm, so all measurements

should be reduced by 0.03 mm.

For Figure 13, the zero error is – 0.03 mm, so all measurements

should be increased by 0.03 mm.

In general, for a micrometer with zero error, the equation to obtain

the corrected value is given by:

Corrected value = reading shown – zero error


Exercise

A. State the reading shown in the diagrams of vernier calipers below:

B. State the reading shown in the diagrams of the micrometer below:

5. The diagrams below show the initial zero reading of a micrometer and a reading showing the diameter of a

ball bearing.

What is the diameter of the ball bearing?

6. The diagrams below show the initial zero reading of a micrometer and a reading showing the diameter of a

small coin.

What is the diameter of the coin?


2. Measuring Time The SI unit of time is second (s). For most experiments, the stopwatch is adequate.

The stopwatch

There are two types of stopwatch, namely analog and digital stopwatch. To measure time interval, the

stopwatch needs to be started and stopped by hand. This introduced an error called human reaction time

which can be quiet a large fraction of a second. For most people, the reaction time is about 0.3 s.

In experiments involving regular oscillations, you will need to use stopwatch to time a large number of

oscillations. In timing oscillations, take note of the following points:

a. ignore the first few oscillations. Time only steady oscillations.

b. When oscillations become ‘abnormal’ (e.g. elliptical oscillations in pendulum), the timing must be rejected.

c. Repeat timings and using the average time. This will reduce errors due to human reaction and due to

counting oscillations (either slightly more or less than the required number).

3. Measuring Mass The SI unit of mass is kilogram (kg).

The sliding mass balance (Ohau’s / Four-beam balance) and the electronic balance

For Ohau’s balance, the unknown mass is placed onto the pan and its mass is obtained by sliding the movable

masses on the beams until the beams are balanced. It is basically a beam balance. For electronic balance, the

unknown mass is placed on top of the pan and its mass is read directly from a screen.

4. Measuring Volume

Regular solids

To measure regular solids, we can measure the dimensions of object using metre-rule, calipers, or micrometer,

and apply appropriate formula to compute the volume.

Irregular solids

Instrument used to measure volume of irregular solids is measuring cylinder (if necessary, with displacement

can).


ERRORS, UNCERTAINTIES, SIGNIFICANT FIGURES

No measurement can ever be perfect; there will always be a degree

of uncertainty. Therefore, in measurements and experiments, we

have to consider precision and accuracy.

If you make several measurements and they are all very similar

(each measurement is clustered around the average), then the level

of precision is high.

If the result of measurement is close to the true value of the

object (value from data or literature), then the result of measurement

is called accurate.

Uncertainty in a Reading

Generally, a reading can be estimated to half of the smallest division on a

measuring scale. In case of ruler, the estimation would be up to 0.5 mm or

0.05 cm. Hence, value of the reading from Figure 19 is 64.5 mm or 6.45 cm.

All experimental data has element of uncertainty. The maximum range within

which the reading is likely to lie is known as the maximum uncertainty. In

the above case, it is between 64.0 mm to 65.0 mm or 6.40 cm to 6.50 cm.

Uncertainty in a Measurement

A measurement is the process by which a physical quantity is compared to a standard unit. Often, this

involves determination from two readings. Example:

The uncertainty associated with this length is obtained as follows:

Smallest division on the metre rule = 1 mm or 0.1 cm

The uncertainty for 1 reading = ± 0.5 mm or ± 0.05 cm

Since two readings are involved, uncertainty in the measurement = ± (0.5 + 0.5) mm = ± 1 mm

The measured length = (64 ± 1) mm or (6.4 ± 0.1) cm

The measured value has the same number of decimal places as the maximum uncertainty while the latter

is generally expressed to one significant figure only.

The maximum uncertainty is an indicator of the scale sensitivity or the accuracy of the measuring

instrument used.

The maximum uncertainty determines the number of significant figures a measurement should have. If the

maximum uncertainty is 1 mm, then the value of measurement of length should be written up to 1 mm (e.g.

100 mm or 10.0 cm). To write the measurement as 10.00 cm is unwarranted, because it is beyond the

accuracy of the instrument.

The number of significant figures in a measurement is the number of meaningful digits whose values are

known with certainty.

Experimental Errors

Random Errors

o Random errors produce unpredictable deviations from the actual value. It cannot be eliminated, but

can be reduced by finding the average of all the readings obtained. If a set of readings has small

random errors, it is precise.

Systematic Errors o Systematic error causes a set of readings to be distributed either above or below the actual value. It is

predictable. When a set of readings has small systematic error, it is accurate.

Look at Table 1

Figure 19


Table 1. Examples of random and systematic error

Example 1 2

Random Error ………….. …………..

Systematic Error ………….. …………..

Diagram

Remarks ………….. …………..

Example 3 4

Random Error ………….. …………..

Systematic Error ………….. …………..

Diagram

Remarks ………….. …………..

Treatment of Errors

Remember! If a measurement has value of x and maximum uncertainty of Δx, then write: (x ± Δx).

Example :

(10.0 ± 0.1) cm correct

(10.00 ± 0.1) cm not correct

(100 ± 1) mm correct

(100 ± 0.1) cm not correct

In this case, 0.1 cm or 1 mm is called absolute uncertainty of a measurement.

2 ways to compare uncertainty with the reading:

Maximum fractional uncertainty : (∆𝑥

𝑥)

percentage uncertainty = (∆𝑥

𝑥) . 100%

If we want to add or subtract two physical quantities, their maximum uncertainties must be added

together.

Example:

Length A : (55 ± 1) mm

Length B : (4.2 ± 0.1) cm >>> convert it to mm, to become: (42 ± 1) mm

Length A – length B = (55 ± 1) – (42 ± 1)

= (13 ± 2) mm

Its percentage uncertainty : (2/13) x 100% = 15%

3A – 2B = 3(55 ± 1) – 2(42 ± 1)

= (165 ± 3) – (84 ± 2)

= (81 ± 5) mm

Its percentage uncertainty : (5/81) x 100% = 6.2%

If two quantities A and B are multiplied together, the maximum fractional uncertainty of the product is the

sum of two fractional uncertainties of A and B.

If quantity A is divided by quantity B, the maximum fractional uncertainty of the product is the sum of two

fractional uncertainties of A and B.


Example 1:

The dimensions of a piece of A4 paper are as follows:

Length A = (297 ± 1) mm

Width B = (209 ± 1) mm

Calculate:

a. the fractional uncertainty of A.

b. the percentage uncertainty of A.

c. the area of paper with its uncertainty.

d. The ratio of A/B with its uncertainty.

Answer:

a. Fractional uncertainty of A is : 1

297 = 0.00337

b. Percentage uncertainty of length A is : 0.00337 x 100% = 0.337%

c. Area = length x width = A x B

= 297 x 209 = 62 073 mm2

Fractional uncertainty of area is : 𝛥𝐴

𝐴 =

1

297+

1

209 ∆𝐴 = (

1

297+

1

209) (62 073) = 506 𝑚𝑚2

The area should be expressed as (62 100 ± 500) mm2 or (6.21 ± 0.05) x 104 mm2.

d. Ratio of A/B : 𝐴

𝐵 =

297

209 = 1.42105

Fractional uncertainty of area is : 𝛥𝐴

𝐴 =

1

297+

1

209 ∆𝐴 = (

1

297+

1

209) (1.42105) = 0.011584

The ratio should be expressed as (1.42 ± 0.01)

Example 2:

The density of a rectangular block is

𝑑𝑒𝑛𝑠𝑖𝑡𝑦 = 𝑚𝑎𝑠𝑠

𝑙𝑒𝑛𝑔𝑡ℎ 𝑥 𝑤𝑖𝑑𝑡ℎ 𝑥 ℎ𝑒𝑖𝑔ℎ𝑡

From measurements, it is known that:

Mass = (240.0 ± 0.1) g

Length = (6.00 ± 0.01) cm

Width = (5.00 ± 0.01) cm

Height = (2.00 ± 0.01) cm

a. Calculate the density of the block.

b. Find the uncertainty of the density of the rectangular block and indicate how the value of density

should be written.

Answer:

a. Density = mass / volume

= 240.0 / (6.00 x 5.00 x 2.00)

= 4.00 g/cm3

b. Uncertainty: ∆𝑑

𝑑=

𝛥𝑚

𝑚+

𝛥𝑙

𝑙+

𝛥𝑤

𝑤+

𝛥ℎ

ℎ

∆𝑑

4.00=

0.1

240.0+

0.01

6.00+

0.01

5.00+

0.01

2.00

∆𝑑 = 0.036 ≈ 0.04 𝑔/𝑐𝑚3 Density of the block = (4.00 ± 0.04) g/cm3

Significant Figures

Significant figures are reliable known digits in a number. This is related to uncertainty in reading. Example:

From measurement of the dimensions of a paper:

length = (297 ± 1) mm maximum value = 298 mm ; minimum value = 296 mm

width = (209 ± 1) mm maximum value = 210 mm ; minimum value = 208 mm

Area of paper: maximum value = 298 x 210 = 62 580 mm2 ; minimum value = 296 x 208 = 61 568 mm2

The area is (297 x 209) mm2 = 62 073 mm2

the uncertainty is 500 mm2

Claiming to know anything precisely until several mm2 doesn’t make sense, because we can’t be certain of the

units place. Therefore, the 80 in 62 580 mm2 is not significant, the 73 in 62 073 mm2 is also not significant,

because it is uncertain. So, we drop the numbers and take only the first three numbers and let the rest numbers

be zero, that is 62 100 mm2. The value of the area is (62 100 ± 500) mm2. This value is within the maximum

value and minimum value of the area from the calculation above.


Exercise 2:

1. From the figure beside, determine, which one is:

a. precise and accurate

b. precise but inaccurate

c. not precise but accurate

d. not precise and inaccurate

2. A rectangular plate has a length of (21.3 ± 0.2) cm and a width

of (9.8 ± 0.1) cm. Calculate the perimeter and area of the plate,

including its uncertainty.

3. The speed of light is now defined to be 2.99792458 x 108 m/s. Express the speed of light to (a) three

significant figures, (b) five significant figures, and (c) seven significant figures.

4. State the number of significant digits in the following measurements:

(a) 12,345 cm;

(b) 0.123 g;

(c) 0.5 mL;

(d) 102.0 s

5. Do the following calculation and express your answer using the appropriate number of significant figures.

6. A student weighed 5 empty test tubes for a chemistry experiment. Their masses are recorded in the table

below:

a. Calculate the total mass of all five test tubes. Express your answer using the appropriate number of

significant figures.

b. Calculate the average mass of all five test tubes. Express your answer using the appropriate number

of significant figures.


PRACTICAL SKILLS BUILD-UP

Basic Experimental Skill 1: Taking Readings and Recording Readings

1. Every measurement consists of two parts: number and unit. It is important that a measurement or

calculated quantity must be accompanied by a correct unit, where appropriate.

2. When an instrument is used to take reading, it should be used to its full precision. However, all physical

measurements have uncertainty. The degree of uncertainty can be indicated by recording the readings

using appropriate number of significant figures.

The table below shows the uncertainty of some common measuring instruments.

instrument smallest

division

uncertainty examples of how reading is recorded

ruler / metre rule 0.1 cm 0.1 cm 12.0 cm, 12.1 cm

vernier calipers 0.01 cm 0.01 cm 3.00 cm, 2.13 cm

micrometer screw 0.01 mm 0.01 mm 2.00 mm, 2.10 mm, 2.11 mm

stopwatch (analogue) 0.1 s 0.1 s 36.0 s, 36.1 s

measuring cylinder (100 mL) 1 mL 0.5 mL 18.0 mL, 18.5 mL

measuring cylinder (250 mL) 2 mL 1 mL 70 mL, 71 mL

spring balance (0-1 N) 0.01 N 0.01 N 0.36 N, 0.40 N

spring balance (0-10 N) 0.1 N 0.1 N 3.6 N, 4.0 N

thermometer (-10oC – 110oC) 1oC 0.5oC 23.0oC,23.5oC

3. As far as possible, results should be tabulated. In the heading of each column of the table should include

the name or symbol of measured/calculated quantity together with its unit.

Example: to record the height h measured in centimeters, the heading of the column is written as h/cm.

4. It is important that all observations actually made are recorded directly. Do not manipulate/calculate the

observations before they are recorded. For example:

i. For the measurement of the diameter of a ball bearing using micrometer where readings are taken in

three different directions and the average calculated, all the three readings should be recorded.

ii. For the measurement of periodic time of simple pendulum, the time for 20 oscillations is recorded

before the time for one oscillation is calculated.

iii. For the measurement of the extension of a spring, both initial and final pointer reading are recorded

before the extension is calculated.

Basic Experimental Skill 2: Significant Figures for Calculated Quantities

1. For addition and subtraction:

the number of decimal places given for the calculated quantities should be the same as the least number of

decimal places in the raw data used.

Example:

a. 28.7 + 2.75 = 31.5 (calculator value = 31.45)

b. 3.168 + 2.9 = 6.1 (calculator value = 6.068)

c. 1.43 + 5.0 + 0.368 = 6.8 (calculator value = 6.798)

d. 173 – 0.479 = 173 (calculator value = 172.521)

e. 0.04529 + 0.0028 = 0.0481 (calculator value = 0.04809)

2. For multiplication and division:

the number of significant figures given for the calculated quantities should be the same as the least

number of significant figures in the raw data used.

Example:

a. 2.4 X 8.23 = 20 (calculator value = 19.752)

(2 sf) (3 sf) (2 sf)

b. 0.0872 X 7.552 = 0.659 (calculator value = 0.6585344)

(3 sf) (4 sf) (3 sf)

c. 20.0 X 5.28 = 106 (calculator value = 105.6)

(3 sf) (3 sf) (3 sf)

d. 0.93 : 0.07837 = 12 (calculator value = 11.86678576)

(2 sf) (4 sf) (2 sf)


Basic Experimental Skill 3: Drawing Graphs

Graphs are plotted to show the relationship between two quantities. The quantity you can control / change is

usually plotted on the x-axis. You should vary this quantity in regular steps in the experiment. The quantity

that is dependent on the quantity you control is plotted on the y-axis.

Plotting Graph

When plotting a graph, the following should be noted:

a) name the graph

b) label the axes, with quantities and appropriate units (name of quantity/unit). Example: time/second or t/s

c) Choose suitable scale in order to make full use of the graph paper.

d) Mark the experimental points sharp and clear. Small cross (+ or x) and encircled dots are recommended.

Large dots ( ) are penalized.

e) Draw a best straight line or a best smooth curve through the plotted points. Make sure that the number of

points on each side of the line is roughly the same. If the graph is a curve, do not simply join one point to

another point, but draw a smooth curve through the points.

The following are examples of desirable and undesirable graphs:


Obtaining Information from a Graph

To determine the gradient of a straight line graph, draw a large triangle (with broken lines) so that the gradient

determined is more accurate. Data values should be read from the line. The calculation of gradient should be

to two or three significant figures.

Uses of Straight Line Graph

a) The general equation of a straight line graph is y = mx +c where m = a/b is the gradient and c is the y-

intercept.

b) When c = 0, then the equation of the line become: y = mx. In this case, we say that y is directly

proportional to x.

c) The graph shown below is still a straight line graph, but with negative gradient. This graph does not obey

the law: y is inversely proportional to x.


d) The law: y is inversely proportional to x, is shown by hyperbole graph, like the following.

e) Because it is difficult to judge whether a curve is hyperbola, we can plot another graph of y against 1/x to

test it. If y is inversely proportional to x, then the graph will be a straight line passing through origin.

Basic Experimental Skill 4: Experimental Precautions to improve Accuracy

General Precautions:

a) Check and correct zero errors in measuring instruments.

b) Avoid parallax error in reading scale by looking at right angles to it.

c) To minimize error, take few readings at different angles and different places of the item then determine

the average.

d) If the condition is possible, repeat the procedure at least once.

e) The range of readings selected to independent variables should be as big as possible.

Mechanics:

For experiments involving oscillation:

i) For simple pendulum experiment, the oscillations should always be in one plane only.

ii) To determine the period of an oscillation, a more accurate reading is obtained by measuring the time for at

least 20 oscillations and then divide by 20.

iii) Each timing should be taken at least twice to avoid miscounting.

iv) To reduce error due to reaction time, start counting before zero count, i.e. 3,2,1,0,1,2,3,… and start the

stopwatch at the count of zero.

v) The amplitude of oscillation should be small.

vi) Avoid wind or anything that will disturb the oscillations.

Heat:

i) Take readings only when steady condition is achieved.

ii) To measure the temperature of heated liquid, stir the liquid continuously to ensure the temperature is

uniform throughout the liquid.

iii) Place thermometer vertically in the middle of container so that it does not touch the sides or bottom of the

heated beaker.

iv) Avoid droughts or wind.

Exercise:

𝑦 ∝1

𝑥

then : 𝑦 =𝑘

𝑥

k = constant

1. Plot graph of V against h based on the values given

in the table above. You should expect a straight line.

V/cm3 h/cm

6.5 1.8

12.0 3.7

15.0 4.8

23.5 6.7

30.5 8.6

38.0 10.5

46.5 13.0

2. Plot graph of V against Ɵ based on the values given

in the table above. You should expect a curve.

V/cm3 Ɵ/oC

7.0 7.0

7.2 14.0

7.6 20.0

8.2 37.0

8.8 45.0

9.6 55.0

10.5 64.0

12.2 70.0


3. What would you plot on the x-axis and the y-axis if you wanted to obtain a straight-line graph from the

equation:

1

𝑢+

1

𝑣=

1

𝑓

If values are known for the variables u and v, and the graph is to be used to find the constant f?

4. The readings in Table below were obtained from an experiment to determine the acceleration of a vehicle.

The velocity v of the vehicle at times t was recorded.

t / s v / ms-1

0 3.7

0.5 5.7

1.0 7.5

1.5 9.1

2.0 11.1

2.5 13.0

3.0 14.8

3.5 16.8

Plot a graph and use it to find:

a. The acceleration

b. The velocity when t = 0.8 s

c. The velocity when t = 4.0 s

instruments and measurement

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