h1 measurement notes

2011 H1 PHYSICS (8866) Measurements

JJ Physics Dept Page 1 of 23

Measurements SI Units Errors and uncertainties Scalars and vectors Learning Outcomes Candidates should be able to: (a) recall the following base quantities and their units: mass (kg), length (m), time (s),

current (A), temperature (K), amount of substance (mol).

(b) express derived units as products or quotients of the base units.

(c) Show an understanding of and use the conventions for labelling graph axes and table columns as set out in the ASE publication SI units, Signs, Symbols and Abbreviations, except where these have been superseded by Signs, Symbols and Systematics ( The ASE Companion to 5-16 Science, 1995 )

(d) use the following prefixes and their symbols to indicate decimal sub-multiples or multiples of both base and derived units: pico (p), nano (n), micro (), milli (m), centi (c), deci (d), kilo (k), mega (M), giga (G), tera (T).

(e) make reasonable estimates of physical quantities included within the syllabus.

(f) show an understanding of the distinction between systematic errors (including zero errors) and random errors.

(g) show an understanding of the distinction between precision and accuracy.

(h) assess the uncertainty in a derived quantity by simple addition of actual, fractional or percentage uncertainties

(i) distinguish between scalar and vector quantities, and give examples of each.

(j) add and subtract coplanar vectors.

(k) represent a vector as two perpendicular components.



(a) recall the following base quantities and their units: mass (kg), length (m), time (s), current (A), temperature (K), amount of substance (mol).

Physical Quantities & SI units A physical quantity defines some measurable feature of many different items. It

consists of a number and a unit.

Example: Area of the school compound, A = 5 000 m2

Numbers are not physical quantities. Without a unit, numbers cannot be a measure of any physical quantity.

Physical quantities are the building blocks of Physics in terms of which the laws of Physics are expressed.

There are 2 types of physical quantities: (1) base (fundamental) quantities (2) derived quantities

Base Quantities & Units

A base quantity is chosen and arbitrarily defined rather than being derived from a combination of other physical quantities.

Base unit is the standards assigned to each of these basic quantities and to no others. It is not derived from other units, i.e. independent of other units.

The seven base quantities and their SI units are:

Base Quantity / Symbol Unit Symbol SI Unit

Length, l Mass, m Time, t electric current, I temperature, T amount of substance, n *luminous intensity, Iv

m kg s A K

mol cd

metre kilogram

second ampere

kelvin mole candela

* luminous intensity is not in the syllabus

Physical Quantity Numerical magnitude

Unit

Base units are units from which all other units can be defined

All physical quantities consist of a numerical magnitude and a unit.



(b) express derived units as products or quotients of the base units.

A derived quantity is defined based on combination of base quantities and has a derived unit that is the product and/or quotient of these base units.

Example

timentdisplaceme velocity =

Unit of velocity = unit of distance / unit of time

= m / s = m s1

Example Force = mass x acceleration

unit of F = unit of m x unit of a = kg m s-2

Example Potential Energy = mgh

unit of PE = unit of (m x g x h) = kg m s-2 m

so, SI unit of energy, J, can be expressed as kg m2 s-2 Worked Example 1 (J93/I/2, J83/II/28) What are the SI base units of specific heat capacity? Solution Specific heat capacity, c, of a substance is defined as the thermal energy required by a unit mass of the substance (e.g.1 kg) to experience a unit rise in temperature (e.g. by 1 K).

Unit of c

= =2 -2unit of thermal energy J kg m s =

(unit of mass) x (unit of temperature change) kg . K kg . K

2 -2 -1= m s K More on homogeneity of equations

A physical equation is true irrespective of the system of units used for the physical quantities mentioned in the equation. Note that: - Each term of the equation has the same units - Only quantities of the same units can be added, substracted or equated in an

equation - The equation is said to be homogeneous or dimensionally correct if all the terms

in it has the same base units.

Derived quantity

Base quantities

Derived unit



- Consider the following equation

21s = ut + at2

Unit for s = Unit for ut = Unit for at2 = Note that all the three terms have the same unit. Therefore, we can conclude that the equation is homogeneous. An equation which is not homogeneous must be physically wrong. On the other hand, if the units for the various terms in an equation are the same, it does not imply that the equation is physically correct. Cases where an equation can be homogeneous and yet incorrect are: a. Incorrect Coefficient (s)

lT = 2g

is dimensionally correct and physically correct, but lT = 3g

is

dimensionally correct but physically wrong. b. Missing terms

The relationship between s, u, t and a may just be written as 21s = at2

which is

incomplete and wrong even though the equation is homogeneous c. Extra Terms

An equation may be wrongly written with an extra term which has the same units

as the other terms for e.g. +l lT = 2g g

The correctness of a physical equation is determined experimentally. To test for the homogeneity of physical equations, the units of the terms on the right hand (RHS) of the equation must be equal to the units of the terms on the LHS.

Worked Example 2 (N82/II/29) At Temperatures close to 0 K, the specific heat capacity of a particular solid is given by c = aT3, where T is the thermodynamic temperature and a is a constant. Find the units of a in terms of SI based units. Solution c = aT3 unit of c = unit of (aT3) m2 s-2

K-1 = unit of a unit of K3

unit of a = m2 s-2 K-4



Worked Example 3 The experimental measurement of the specific heat capacity c of a solid as a function of temperature T is to be fitted to the expression c = T + T3. What are the possible units of and expressed in SI base units? Solution

c = T + T3

Base unit of c = base unit of T = base unit of T3

m2 s-2 K-1 = (unit of ) (unit of T ) unit of = m2 s-2 K-2

Also base unit of c = base unit of T3

m2 s-2 K-1 = (unit of ) (unit of T3 ) unit of = m2 s-2 K-4 Worked Example 4 Which of the following units are not equivalent to that of force? A. Pa m2 B. W m1 s C. J m s1 D. kg m s2 Solution

Working Base units A) Pa m2 kg m-1 s-2 x m2 kg m s-2 B) W m-1 s kg m2 s-3 x m-1 s kg m s-2 C) J m s-1 kg m2 s-2 x m s-1 kg m3 s-3 D) kg m s-2

Answer is C.

For the equation above to be dimensionally correct or homogeneous, the base units of the terms on the L.H.S. of the equation must be the same

as the base units of the terms on the R.H.S.



Worked Example 5 The critical density of matter 0 in the universe may be written as

What are the SI base units of H ?

Solution

3G8H 0=

unit of H = ( kg m-3 )( N m2 kg-2 ) = ( kg m-3 )( kg m s-2 m2 kg-2 )

= s-2 Worked Example 6 The energy E of a damped oscillator of mass m varies with time t is given by What are the units of k and b?

Solution

e-bt/2m is a number and hence has no unit

Power has no unit and hence 2btm

= 1 =(unit of b).s 1kg

unit of b = kg s-1

unit of E = (unit of A2 ) ( unit of k )

( unit of k ) = J m-2 = ( kg m2 s-2 ) m-2 = kg s-2

G8H

3

0 =H = Hubble constant G = gravitational constant (unit: N m2 kg2)

G8H

3

0 =

2mbt

2ekAE

= A = amplitude (unit: m) k, b = constants



(c)

Signs, Symbols and Systematics ( The ASE Companion to 5-16 Science, 1995 )

Show an understanding of and use the conventions for labelling graph axes and table columns as set out in the ASE publication SI units, Signs, Symbols and Abbreviations, except where these have been superseded by

as the leading column in a table, e.g. pre-selected physical quantity such as length.

readings recorded from experiment or calculated values from previous columns. e.g.

Time taken for 20 oscillat ns

A table of ordered pairs is used to represent a relationship. In a table, it is conventional to have the independent variable to be recorded

The second and subsequent columns should be the dependent variable(s), e.g.

ioPendulum length

L

Period, T = ( t1 + t2 ) / 40

/ cm

t1 / s

t2 / s

T / s 20.0 10.3 10.5 0.520 40.0 15.7 15.8 0.788 60.0 20.9 21.2 1.05

Independent

ces (d.p.) according to the type of the measuring instrument i.e. the etre rule.

count of the human reaction time of ~ .2 s in starting and stopping the stopwatch.

ll have the ame number of significant figures (s.f.) as its raw score ( raw data ).

e horizontal axis ( x-axis ) and the dependent variable on the vertical axis ( y-axis ).

tercept of a graph, y = 0 is substituted into the graph or equation

tercept of a graph, x = 0 is substituted into the graph or equation to solve for y.

variable

Notice the pendulum length readings are recorded with the same number of decimal plam The time taken by a stopwatch however was only recorded to 1 d.p. throughout instead of the usual 2 to 3 d.p. as indicated in the time instrument. This is because the time readings recorded must take into ac0 The period of one oscillation is a calculated value ( processed data ) taken from records of the time taken for 20 oscillations, hence, the values recorded wis A graph is an infinite set of order pairs plotted on a Cartesian plane, representing a relationship between the elements of the ordered pair. It is conventional to plot the independent variable on th

To find the x-in

to solve for x. To find the y-in

Dependent variable



The selected scales of the axes should be easy to read and large enough to enable the curve(s) (Curve refers to both curves and straight lines) to occupy at least half of the graph grid. The selected scale should usually be in multiples of 1, 2, 5 or 10 unit(s). A false origin is sometimes used in order to accommodate the curve into the graph.

When determining the gradient of the curve at a particular point or straight line, the chosen reference triangle should fall within the recorded data points and be at least half the size of the curve. The chosen points should not coincide with any of the recorded data points from the experiments. Label these chosen points for easy checking.

(d) use the following prefixes and their symbols to indicate decimal sub-multiples or

multiples of both base and derived units: pico (p), nano (n), micro (), milli (m), centi (c), deci (d), kilo (k), mega (M), giga (G), tera (T).

Prefixes are used to simplify the writing of very large or very small orders of

magnitude of physical quantities.

Fraction / multiple Prefix Symbol 10-12 pico p 10-9 nano n 10-6 micro 10-3 milli m 10-2 centi c 10-1 deci d 103 kilo k 106 mega M 109 giga G 1012 tera T

L / cm

Reference triangle

Independent variable

Dep

ende

nt v

aria

ble

False origin

T / s

10.0

15.0

20.0

25.0

30.0

0.0 20.0 40.0 60.0

x

x

xx

x x



Examples

1500 m = 1.5 x 103 m = 1.5 km 0.00077 V = 0.77 x 10-3 V = 0.77 mV 100 x 10-9 m3 = 100 x (10-3)3 m3 = 100 mm3

(e) make reasonable estimates of physical quantities included within the syllabus.

The following are examples of the estimated values of some physical quantities:

Diameter of an atom ~ 10-10 m Diameter of a nucleus ~ 10-15 m Air pressure ~ 100 kPa Wavelength of visible light ~ 600 nm Resistance of a domestic lamp ~ 1000 Mass of a car ~ 2000 kg Maximum speed of a car on an expressway ~ 90 km h-1

(f) show an understanding of the distinction between systematic errors (including

zero errors) and random errors.

Measuring any physical quantity requires a measuring instrument. The reading will

always have an uncertainty. This arises because a) Experimenter is not skilled enough b) Limitations of instruments c) Environmental fluctuations

As a result, measurements can become unreliable if we do not use good measurement techniques. Some of the common practices to minimise the errors made in measurements are as follows: a) Taking average of many readings b) Avoid parallax errors c) Take readings promptly

Analogue & Digital displays

Often, when we measure a quantity with an instrument, we can make an estimate of the uncertainty by using the following rule:

Uncertainty of a Reading = Half the smallest scale division



For example, take a look at the following analogue scales: Reading = 5.7 Reading = 2.36

Even when instruments with digital displays are used, there are still uncertainties present in the measurements. So for example, a digital ammeter may show the current to be 358 mA. This does not mean that the current is 358 mA exactly. For digital displays, the uncertainty is usually provided by the manufacturer.

Errors & Uncertainties

Errors or uncertainties fall generally into 2 categories: a) Random errors b) Systematic errors

Random Errors

The readings are equally likely to be higher or lower than the MEAN value. Random errors are of varying sign (higher or lower than average) and magnitude and cannot be eliminated. Averaging repeated readings is the best way to minimize random errors.

Examples: Measuring the diameter of a wire due to its non-uniformity

x x x

xx

xx

x

xx

5

7

6

2.2

2.4

2.3

Uncertainty = 0.1 Uncertainty = 0.01

Random error is one that occurs without a fixed pattern resulting in a scatter of readings about a mean value.



Systematic errors

The readings are consistently higher or lower than the ACTUAL value. Systematic error is consistent in both magnitude and sign and results in readings taken being faulty in one direction.

Systematic errors cannot be reduced or eliminated by taking the average of

repeated readings. It could be reduced by techniques such as making a mathematical correction or correcting the faulty equipment.

Examples: zero error of a measuring instrument, a clock running too fast or too slow.

(h) assess the uncertainty in a derived quantity by simple addition of actual, fractional or percentage uncertainties

If we denote the uncertainty or error of P as P, then we write the measured quantity as

P + P

Fractional error of P = PP

Percentage error of P = %100PP

x x

x x

xx

x x

x

xx

xx

x x

x x

x

Systematic error is one that occurs with a fixed pattern resulting in a consistent over-estimation or under-estimation of the actual value.

Absolute error

Mean Value



Worked Example 7

The length of a piece of paper is measured as 297 1 mm. Its width is measured as 209 1 mm. (a) What is the fractional uncertainty in its length? (b) What is the percentage uncertainty in its length? Solution Note : 297 + 1 mm

(a) Fractional uncertainty in its length = 1

297 = 0.00337

(b) Percentage uncertainty in its length = 1 x 100%

297 = 0.337 %

Note:

Rule of thumb: answers should always be rounded off to 3 significant figures (3.S.F.) except for absolute errors, which are always rounded off to 1.S.F. MARKS will be deducted for expressing the answers to an incorrect number of significant figures!

The average value is always rounded off to the number of decimal figures of the absolute error when expressed in scientific notation.

(g) show an understanding of the distinction between precision and accuracy.

Measurements are often described as precise or accurate. In layman terms, these 2 terms are used interchangeably to mean the same thing. In Physics, it is possible to have precise but inaccurate measurements and accurate measurements that are not precise.

Suppose we do some experiments to find g, the acceleration of free fall. The expected result is 9.81 m s-2. The results obtained are shown below:

Absolute error

Mean Value



8.63, 8.78, 8.82, 8.59, 8.74, 8.88 9.76, 9.79, 9.83, 9.85, 9.88, 9.90

( precise, not accurate ) ( accurate & precise )

9.64, 9.81, 9.95, 10.02, 9.77, 9.68 7.65, 8.92, 10.00, 9.12, 8.41, 9.45

( accurate but not precise ) (neither precise nor accurate)

To summarise, Precision: A set of measurements is precise if a) the measurements have a small spread or scatter b) there are small random errors in the measurements Accuracy: A set of measurements is accurate if a) the measurements are close to the actual value b) there are small systematic errors in the measurements

Determining Uncertainty in Derived Quantities

No. of readings, n

Value of reading, x Expected 9.81



Addition & Subtraction

sured with uncertainties A and B respectively. C = A + B, find the uncertainty in C.

If D = A B, find the uncertainty in D.

Multiplication and Division in E.

If F =

Suppose A and B are mea If

If E = AB, find the uncertainty

AB

, find the uncertainty in F.

orked Example 8

W

cm and Y = (5.0 0.2) cm, find D and the associated uncertainties if ) D = X + Y

D = X + Y = 6.0 cm

D = X + Y = 0.2 + 0.2 = 0.4 cm Therefore, D = (6.0 +

Given X = (1.0 0.2) (a

0.4) cm

) D = Y X

(b

D= A + B

Generalising:

If A = Bn, then BBn

AA =

If A = Bm Cn , then CCn

BBm

AA +=

If A = Bm / Cn , then CCn

BBm

AA +=

BB

AA

EE +=

BB

AA

FF +=

C = A + B



D = Y X = 4.0 cm

D = Y + X = 0.2 + 0.2 = 0.4 cm Therefore, D = (4.0 +

0.4) cm

) D = 2Y 3X D = 2Y 3X = 7.0 cm D = 2(Y) + 3(X) = 2(0.2) + 3(0.2) = 1.0 cm

(c 1 cm (to 1 sig. fig)

Therefore, D = (7 + 1) cm

D =

(d) 4

X-Y

D = 4

X-Y = 1.0 cm

D = (Y + X) = 0.1 cm

Therefore,

= (0.2 + 0.2)

D = (1.0 + 0.1) cm (e) D = XY

YY

XX

DD +=

0.2 0.2= + 1.0 5.0

= 0.24

Average D = X Y = (1.0) (5.0) = 5.0 cm2

DDDD =

= 0.24 x 5.0 = 1.2 cm2 (to 1 sig. fig)

Therefore, D = (5 +

21 cm

1) cm

) D = 4XY

2

(f

YY

XX

DD +=

= 0.52.0

0.12.0 + = 0.24

= 4(1.0)(5.0) = 20 cm2 D

D = 0.24 x 20 = 4.8 cm2 25 cm (to 1 sig. fig)

= (20 5) cm2

D



(g) D = X2Y

YY

XX

DD += 2

0.2 0.2= 2( ) + 1.0 5.0

= 0.44

Average D = X2 = (1.0)2 3

Y (5.0) = 5.0 cm

DDDD =

= 0.44 x 5.0 = 2.2 cm3 (to 1 sig. fig)

Therefore, D = (5 +

32 cm 2) cm3

ple 9Worked Exam experiment to determine g the equation used is where

ind the value of g and its uncertaint

olution

In a simple pendulumT = (2.16 0.01) s and l = (1.150 0.005) m. F y.

S

Note: T was the original subject of the equation but since we are interested in g, before

we form the error equation, we have to make g the subject of the equation first.

glT 2=

TT

ll

gg += 2

22

Tl4g =

16.201.02

150.1005.0 +=

gg

216.2)150.1(24g = 0136.0

gg =

g = 9.73 m s-2 g = 0.132 = 0.1 m s-2 (to 1 sig. fig)

g = (9.7 0.1) m s-2

glT = 2



Worked Example 10 (N89/II/2) The length of a piece of paper is measured as 297 1 mm. Its width is measured as 209 1 mm. What is the area of one side of the piece of paper? State your answer with its uncertainty. Solution

Area of the paper = 297 x 209 = 6.21 x 104 mm2

A lA l b

b = + 209

1297

1 += 410x 6.21A

A = 506 = 0.0506 x 104 mm2 (to 1 sig. fig) 4 20.05 x 10 mm

Therefore, A = (6.21 + 0.05 ) x 104 mm2

Worked Example 11 In an experiment to measure the Youngs modulus of a wire, E, the following measurements were made:

Length of wire, l 3.025 0.005 m Diameter of wire, d 0.84 0.01 mm Mass, M 5.000 0.002 kg Extension, e 1.27 0.02 mm Acceleration of free fall, g 9.81 0.01 m s-2

If E is calculated as 24

edMglE = , calculate E with its associated uncertainty.

Solution

24

edMglE =

dd

ee

ll

gg

MM

EE ++++= 2

0426298.084.001.02

27.102.0

025.3005.0

81.901.0

000.5002.0

EE =

++++=

E = 0.0426298 x 2.11 x 1011

11 2 -3 -3 2 2.11 x 10 Pa

4 4(5.000)(9.81)(3.025)(1.27 x 10 )(0.84 x 10 )

MglEed == =



E = 8.78 x 109 Pa

E = 0.09 1011 Pa (to 1 sig. fig) E = (2.11 0.09) x 1011 Pa

Note (REPEATED):

Rule of thumb: non-exact answers should always be rounded off to 3 significant figures (3.S.F.) except for absolute errors, which are always rounded off to 1.S.F. MARKS will be deducted for expressing the answers to an incorrect number of significant figures!

The average value is always rounded off to the number of decimal figures of the absolute error when expressed in scientific notation.

(i) distinguish between scalar and vector quantities, and give examples of each.

Scalars and vectors are both physical quantities (they have both a number and a unit).

Examples: mass and charge

Examples: displacement and force Examples

Scalars Vectors Distance displacement speed velocity time acceleration frequency force density momentum (encountered in the topic of

Dynamics)

Note:

A vector can be placed anywhere in a diagram as long as it keeps its same length and direction.

Two vectors with the same length but different directions are different. Directions for vectors must be given clearly without ambiguity.

A scalar quantity consists of a magnitude only.

A vector quantity consists of a magnitude and a direction.



3 different ways to give directions clearly are: i) Compass points, e.g. due east, 75o north of west, 20o east of south ii) Bearings, e.g. bearing of 090o, 345o, 160o

iii) X-Y plane, e.g. positive x-axis, 75o above the negative x-axis, 70o below the positive x-axis

(j) add and subtract coplanar vectors.

Dealing with Vectors

When vectors are added however, the result is NOT just the sum of the numbers (magnitudes). The directions of the vectors must be considered, especially when they point in different directions.

Vectors Addition by Scaled Drawing

Examples

75o

A

B

R

i) Due East ii) Bearing of 090o

i) 75o north of west ii) Bearing of 345o iii) 75o above the -ve x-axis

i) 40o south of east or ii) Bearing of 130o iii) 40o below the +ve x-axis 40o

A

B

R



Vectors Addition by Triangle Law

Worked Example 12 Find R.

Solution

180 84 96 = = 96 40

120sin sin

R =

R2 = 1002 + 1202 2(100)(120) cos 40 R = 77.6 m

R is 77.6 m at an angle of 84.0o north of east (other possible answers?)

Vectors Subtraction Example : P - Q = P + (- Q)

(k) represent a vector as two perpendicular components.

When 2 perpendicular vectors are added, they give a resultant as shown:

P+(-Q)

P

- Q

P

Q

V + H = R

H

V R

100 m

120 m R

40o 84o



Finding the components of a vector is the reverse process of vector addition. Instead of combining 2 vectors into one, a vector can be split into 2 components, the horizontal component and the vertical component.

Worked Example 13 Find the horizontal and vertical components of the forces A, B & C. Solution

Vector Horizontal / N Vertical / N A - 400 cos 25o 400 sin 25o

B 500 cos 35o 500 sin 35o

C - 200 cos 20o - 200 sin 20o

Total - 141 387

Note:

Check that your calculator is in the correct mode i.e. radian or degree if you cannot get the final answer for the computation of trigonometric functions.

Worked Example 14 (N87/I/3, J94/I/1) Which of the following pairs contains one vector and one scalar quantities? A. displacement; acceleration B. force; kinetic energy C. power; speed D. work; potential energy Ans : B

Rx = H = R cos Ry = V = R sin

tan = VH

R

H

V

35o25o

20o

B (500 N)

C (200 N)

A (400 N)



Worked Example 15 (N90/I/1) Which list contains only scalar quantities? A. mass, acceleration, temperature, kinetic energy B. mass, volume, kinetic energy, temperature C. acceleration, temperature, volume, electric charge D. moment, velocity, density, force Ans : B

Worked Example 16 (N90/I/1) (a) Two vectors A and B are at right angles to each other. Draw a vector diagram to

show how the sum of the vectors could be found. (b) A car changes its velocity from 30 m s1 due East to 25 m s1 due South.

(i) Draw a vector diagram to show the initial and final velocities and the change in velocity.

(ii) Calculate the change in speed. (iii) Calculate the change in velocity.

Solution

bii) Change in speed = 25 30 = -5 m s-1

biii) Change in velocity = ( 252 + 302 )0.5

= 39.1 m s-1

angle = tan-1 (25 / 30) = 39.8

The change in velocity is 39.1 m s-1 at an angle of 39.8 south of west.

(other possible answers?)

fvJJK

ivJK

bi)

fvJJK

ivJJJK

( )f iv v+ JJK JK

AJK

BJK

A B+JK JK

a)



Note (IMPORTANT):

Change in a physical quantity = Final physical quantity Initial physical quantity: For scalars, one can simply take the difference in the magnitude; For vectors, one need to consider the directions involved!

In the previous example, speed is a scalar whereas velocity is a vector!

Link to mathematical requirements found in O Level 4016 Mathematics (2010) Syllabus:

Acknowledgements - Updated by Chong K.W. , 2011

Analogue & Digital displaysRandom Errors Systematic errors

Worked Example 7Precision: Accuracy:

h1 measurement notes

Documents

physical quantities

following base quantities

combination of base

base fundamental quantities

vector quantities

types of physical quantities

basic quantities

derived quantity