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Lecture 1: Scientific Method

Outline: 1- Units: Unit systems, unit conversions 2- Dimensions: Dimensional analysis 3- Uncertainty

UnitsYou need to express your results in comparison to certain internationally accepted standards. These standards are called units.

- Units for length, mass and time are defined as "fundamental units".

Scientific Method - Scientific method accepts only theoretical calculations and experimental measurements. - what method did you use - what principles did you employ - what is your strategy

Time → Second, day, minute, lightyear

Length → meter, centimeter, inch, foot Mass → kg , gram, ounce, “hokka", pound

International Standards of units (SI)Length → meter mass → kg time → second

Force → Newton (N) Energy → Joule (J)

CGS :Centimeter-Gram-SecondLength → centimeter Mass → gram Time → second

force- dyn Energy → erg

FPS : Foot- Pound-SecondLength → FootMass → PoundTime → Second

- If an equation is correct, then the both sides of this equation should be written in the same unit.- you can add or subtract parameters if and only if they have the same unit.

1 meter + 3 centimeter1 meter + 1 kg → definitely wrong

when units are not consistent, then youmay need to convert them to each other

1 mile = 1.609km, 1 km= 1000 m1inch = 2.54 cm, 1cm = 0.01 m

ExampleThe speed limit in a highway in Canada is 100 km/h. Express this speed limit in unit of mile/h

Solution

Example: If a car moves with a speed of 30 mile/h, express it in unit of meter/second

Example: A painter is panting a wall of 8ft width and 12ft height. Calculate the painted area in unit of meter-square.

nano n kilo kmicro M Mega M

mini m Giga G

Teru T

Solution

There are many different options in choice of unit, when you express the same parameter. Physics does not depend on the choice of unit system, because the unit system does not determine nature of the parameter.

→ Proton radius→ electron mass

DimensionsWe define three fundamental dimensions.-Mass (M), Length (L), Time (T).Why are these three are fundamental? Because we experience the mechanical world in terms of mass, length and time.

TLMlLT

2

ExampleWhen you walk, you see a wall on your way. Your brain starts calculating. First it is taking your mass. Your distance from the wall and time. Your brain needs to see if you can speed up enough to jump over the wall.

Dimensional Analyses

- Dimensions reveal nature of parameters- You can check your equations and your results by comparing the dimensions of both sides, if your results are correct or need some corrections.

Example: Determine how the period of a simple pendulum depends on its mass, its length and the gravitational acceleration.

Solution T → period L → length M → mass g → gravitational acceleration

⇒The period of a simple pendulum does not depend on its mass

Example: The gravitational force between two objects is given as

G → Newton's constantm1, m2 → masses of the objectsr → distance between the objectsFg → Gravitational force

Find the dimension of Newton's constant.

Solution:

W W y R G CARRY8

G M tip 2

R L

y's Mj me 43 13numberofoscittations Cw T t

time

W C GdRB r

with'T74413 mi'ty L M

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IEEE o Hiaasen1

IN CGdRBI co Roy e

N C G

Example: It is known that stars undergo some mode of oscillations. How does the frequency of these oscillations depend on mass density (ρ) , radius (R) and Newton's constant (G).

Solutions:

dimensionless constant

in dearlyE E7ITaE ces mE mis

R z LG ML

3

w TL

orTt MT 2

hL MEP xto 0

T't Mdt8 LB 38 yza B 38 0

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Example: These stars can also be considered as a large body of liquid. Now determine the frequency of its oscillations in terms of surface energy density (ε), radius(R) and mass density (ρ).

Solution:

e D

Uncertainty

Scientific Method

Theoretical Calculations

Experimental Measurements

- Since the tools used in our experimental measurements have limited sensitivity, there is always some uncertainty in our measurements. for instance if you have a ruler, which canmeasure at least 1 cm, then you cannot measure with this ruler something , whose length is less than 1 cm.

- Since every measurements involve some uncertainty, it should be reflected in your results. In other words, your results should be expressed in a way that reveals the uncertainty in your measurements.

In science, we cannot calculate everything throughtheoretical calculations, thus we need to feed ourcalculations with inputs which are experimentallymeasured.

Example:If you have a stick of 25.2cm, then your ruler reads only 25 cm, and 0.2 cm (2 mm) is accounted for the uncertainty in your measurement.

If you measure a length of 16.5 meter with 20% uncertainty, then the total uncertainty in your measurement is

When you are multiplying, dividing, adding or subtracting parameters with different significant figures, the result should have the least number of the significant figures.

meter

meter

Example:

If your experimental measurements involve some uncertainties, then these uncertainties are transferred to your theoretical calculations when you use experimentally measured

Significant Figures

In your results:- All non-zero digits are significant,- Zeros are significant when they are between other non-zero digits,- Zeros are also significant, when they come after another significant figure.

3 significant figures5 significant figures

- Significant figures reveal the accuracy in your measurements and/or calculations.

Accuracy = number of significant figures

vr r

Example:Rectangular plate: 4.5 cm by 7.3 cmCalculate the area.

Solution:a=4.5 cm 2 Significant figures

b = 7.3cm 2 Significant figures

Area = (4.5 cm) (7. 3cm)

However, this gives the area with 4 significant figures, while the problem gives the parameters with 2 significant figures. Thus the area should be rounded such that the final result will have 2 significant figures.

Area

Even though 4 Significant figures seem more accurate than 2 significant figures, the problem does not provide a tool whosesensitivity is capable to yield results with 4 significant figures. If we could measure the area with the same tool, the measurement would involve 2 Significant figures.

Example: 135 + 6.213 = 141

3 Significant figures 4 Significant figures

3 Significant figures