Chapter 1: INTRODUCTION

• Physics: branch of physical science that deals with energy, matter, space and time.

• Developed from effort to explain the behavior of the physical environment.

• Summary: laws of Physics, Formula, graphs.• Basis of rocket/space travel, modern

electronics, lasers, medical science etc.• Major goal: reasoning critically (as a

physicist), sound conclusions, applying the principles learnt.

• We will use carefully defined words, e.g. velocity, speed, acceleration, work, etc.

• Factor (or ratio) – number by which a quantity is multiplied or divided when changed from one value to another.

• Eg. The volume of a cylinder of radius r and height h is V = r2h. If r is tripled, by what factor will V change?

• Vold = r2h, Vnew = (3r)2h = 9. r2h, Vnew/Vold = 9. V will increase by a factor of 9.

(a)Decreasing the number 120 by 30% gives ----

(b)Increasing the number 120 by 30% gives -----

§ 1.3: The Use of Mathematics

Proportion• If two quantities change by the

same factor, they are directly proportional to each other.

• A B – means if A is doubled, B will also double.

• S r2 – means if S is decreased by factor 1/3, r2 (not r!) will also decrease by the same factor.

Inverse Proportion• If A is inversely proportional to B – means

if A is increased by a certain factor, B will also decrease by the same factor.

• K inversely proportional to r [K 1/r] – means if r is increased by factor 3, K will decrease by the same factor.

• The area of a circle is A = r2. (a)If r is doubled, by what factor will A

change? (b)If A is doubled, by what factor will r

change?

§ 1.4: Scientific notation: • Rewriting a number as a product of a number

between 1 and 10 and a whole number power of ten.

• Helps eliminate using too many zeros.

• Helps to correctly locate the decimal place when reporting a quantity.

• Eg: Radius of earth = 6,380,000 m

= 6.38 x 106 m

Radius of a hydrogen atom

= 0.000 000 000 053 m = 5.3 x 10-11 m.

• In reporting a scientific measurement, it is important to indicate the degree of precision and the accuracy of your measurement.

• This can be done using absolute (or percentage) error, significant figures and order of magnitude, etc.

Precision/Accuracy in Scientific Measurements

(a)Absolute/Percentage error:Eg. Length of a notebook = 27.9 ± 0.2 cm Actual length is somewhere between 27.9

– 0.2 and 27.9+0.2, ie 27.7 and 28.1 cm ± 0.2 is the estimated uncertainty (error). 0.2 is the absolute uncertainty (error). 27.9 is the central value 27.7 and 28.1 are called extreme values.

Percentage Uncertainty

100xValueCentral

ErrorAbsolutePercentage uncertainty =

Eg. Length of a notebook = 27.9 ± 0.2 cm

% Uncertainty = %7.01009.27

2.0x

Fractional Error 9.27

2.0

ueCentralVal

rorAbsoluteEr

• The length of a table was found to be 1.5 m with 8% error. What was the absolute error (uncertainty) of this measurement?

• The mass of a bag was found to be 12.5 0.6 kg. What was the percent error in this measurement?

Examples

Error Propagation in Addition/Subtraction

The absolute error in the sum or difference of

two or more numbers is the SUM of the absolute errors of the numbers.

)()(

)()(

yxyxDifference

yxyxSum

yyandxx

Eg. 8.5 0.2 cm and 6.9 0.3 cm

Sum = 15.4 0.5 cm

Difference = 1.6 0.5 cm

Error Propagation in Multiplication/Division

The fractional error in the product or quotient of two numbers is the SUM of the fractional errors of the numbers.

)(

)(

.Pr

:

2

1

21

y

y

x

xiswhich

Q

QisQerrorFractional

L

LQQuotient

y

y

x

xiswhich

P

PisPinerrorFractional

LLPoduct

y

yyisinand

x

xisxinerrorsFractional

yyyandxxx

Error Propagation in Multiplication/Division

Eg. x = 8.5 0.2 cm and y = 6.9 0.3 cm

Fractional errors:

in x = =5.8

2.0 in y = =9.6

3.0

Find the product, P = x.y and its absolute uncertainty (P).

Examples

The area of a circle is A = r2.

(a)If r is doubled, by what factor will A change?

(b)If A is doubled, by what factor will r change?

(b) Significant Figures:

Number of reliably known digits in a measurement. Includes one “doubtful” or estimated digit written as last digit.

Eg. 2586

[6 is the last digit. It is the doubtful digit].

Eg. 25.68

[8 is the last digit. It is the doubtful digit].

• All nonzero digits are significant.• Zeros in between significant figures are significant.

[2,508]• Ending zeros written to the right of the decimal point

are significant. [0.047100]• Zeros written immediately on either sides of decimal

point for identifying place value are not significant. [0.0258, 0.258]

• Zeros written as final digits are ambiguous.[25800] To remove ambiguity, rewrite using scientific notation.

• Eg. (i) 58.63 – 4 sf, (ii) 0.0623 – 3 sf, (iii) 5.690 x 105 – 4 sf. (iv) 25800 – 2.58x 104 = 3 sf, 2.580x 104 = 4 sf, 2.5800x 104 = 5 sf.

Significant Figures contd:

Significant Figures in Addition/Subtraction

The sum/difference can not be more precise

than the least precise quantities involved.

ie, the sum/difference can have only as many

decimal places as the quantity with the least

number of decimal places.

Eg: 1) 50.2861 m + 1832.5 m + 0.893 m =

2) 77.8 kg – 39.45 kg =

“keep the least number of decimal places”

Significant Figures in Multiplication/Division

The product/quotient can have only as many sf as the number with the least amount of sf.

Eg: 1) What is the product of 50.2861 m

and 1832.5 m?

2) What is 568 m divided by 2.5 s?

“keep the least number of significant figures”

– (roughly what power of ten?) To determine the order of magnitude of a number:

• Write the number purely as a power of ten.• Numbers < 5 are rounded to 100

• Numbers 5 are rounded to101

• Eg. 754 =7.54 x 102 ~101 x 102 = 103. The order of magnitude of 754 is 3.

• 403,179 = 4.03179 x 105 ~100 x 105 = 105 = 5 O/M

• 0.00587 ~ orders of magnitude = - 2 (how?).

(c) Order of Magnitude

What is the difference between accuracy and precision?

Precision: • Reproducibility or uniformity of a result.• Indication of quality of method by which a set of results is obtained.• A more precise instrument is the one which gives very nearly the same result each time it is used.• A precise data may be inaccurate!!

Accuracy:• How close the result is to the accepted value.• Indication of quality of the result.• A more accurate instrument is the one which gives a measurement closer to the accepted value.

Precise/Accurate

Precise/Not Accurate Not Precise/Not Accurate

Not Precise/Accurate

§ 1.5: Units We will use the SI system of units

which is an international system of units adapted in 1960 by the General Council of Weights and Measures.

• In SI system:Length is measured in meters (m).Mass is measured in kilograms (kg).Time is measured in seconds (s).

• Other fundamental quantities and their units in the SI system includes Temperature (in Kelvin, K),

Electric current (in Amperes, A)

Amount of substance (in mole, mol) and Luminosity (in Candela, cd).

• The SI system is part of the metric system which is based on the power of ten.

Converting Between Units

Eg. Convert 65 miles/hour to SI units.

1 mile = 1.609 km = 1609 m.

1 hour = 3,600 seconds

sms

m

hour

miles/1.29

3600x1

1609x65

1

65

§ 1.6: Dimensional Analysis

Dimensions – Units of basic (Fundamental) quantities:

Mass [M], Length [L], Time [T]

We can only add, subtract or equate quantities with the same dimensions.

Eg. 1 Check if the expression v = d2/t is correct, where v = speed (in m/s), d is the distance (in m) and t is time (in s).

Quantity Dimension

V

d2 [L]2

T [T]

v = d2/t

][

][

T

L

][

][

][

][ 2

T

L

T

L

Hence eqn is not correct

Eg. 2: If the equation was now correctly written as v = kd2/t, what must be the units of k?

][

1

][

][

][

][ 2

Lk

T

Lk

T

L

The units of k must be m-1

§ 1.7-1.9: Reading Assignment