![9. Dimensional Analysis
• The physical dimension of the physical quantity 𝑥 (written as 𝑥 ) is the product
of the base quantities constituting it. In Phys101, [𝑥] has the general form
𝑥 = 𝐿𝑙𝑇𝑚𝑀𝑛
where 𝐿 = 𝐋ength, 𝑇 = 𝐓ime & 𝑀 = 𝐌ass . 𝑙, 𝑚 & 𝑛 are rational numbers
(mostly integers).
• Examples:
period = 𝑇
speed =
𝐿
𝑇
Area = 𝐿 × 𝐿 = 𝐿2
𝜋 = 1 (dimensionless!)
18](https://image.slidesharecdn.com/chapter1-240120133554-2818cfb2/85/measurement-units-slideshow-chapter-one-pdf-16-320.jpg)
![9. Dimensional Analysis
𝑃 = 𝑐 𝑔𝑎
𝑙𝑏
.
We have that
𝑐 𝑔𝑎𝑙𝑏 = [𝑔]𝑎[𝑙]𝑏= 𝑇.
Using 𝑔 = 𝐿/𝑇2
and 𝑙 = 𝐿 we get
𝐿𝑎+𝑏
𝑇−2𝑎
= 𝑇,
which gives us
𝑎 + 𝑏 = 0,
−2𝑎 = 1.
22](https://image.slidesharecdn.com/chapter1-240120133554-2818cfb2/85/measurement-units-slideshow-chapter-one-pdf-20-320.jpg)
measurement units slideshow chapter one pdf
- 1. CH 1 MEASUREMENT Abdallah M. Al Zahrani 1
- 2. 2. Measuring Things • We discover physics by learning how to measure physical quantities (length, mass, temperature, etc). • A quantity is measured in its own unit, by comparison with a standard. • The unit is a unique name we assign to measures of that quantity—for example, meter (m) for the quantity length. • There are so many physical quantities! Luckily, not all of them are independent; for example, speed = distance/time. • A few physical quantities were chosen (by an international agreement) to define all other quantities. They are called base quantities. • Base units and base standards are associated to the base quantities. 4
- 3. 3. The International System of Units • In 1971, seven base quantities were chosen as the basis of the International System of Units (SI). • In Phys101, three SI base quantities are used. • The units for all other quantities can be derived from these 3 base units. For example: 1 watt = 1 W = 1kg m2 s3 5
- 4. 3. The International System of Units • Very large and very small quantities can be suitably expressed in the scientific notation: 4530000000 = 4.53 × 109 0.000000013 = 1.3 × 10−8 • Sometimes unit prefixes are used: 4.53 × 109 watts = 4.53 gegawatts = 4.53GW 13 × 10−9𝑠 = 13 nanoseconds = 13 ns 6
- 5. 4. Changing Units • Chain-link conversion: Multiply a measurement by a conversion factor equal to unity so that only the desired units remain. For example: 200 km = 200 km × 1000 m 1 km = 2.00 × 105m 1 h = 1 h × 60 min 1 h × 60 s 1 min = 3600 s 1 m s = 1 m s × 1 km 1000 m × 3600 s 1 h = 18 5 km h 7
- 6. 5. Length • The SI unit of length is meter (m). • A meter is the distance travelled by light in vacuum in a time interval of 1/299 792 458 of a second. The speed of light 𝑐 is 𝑐 = 299 792 458 m/s. 8
- 7. 6. Time • The SI unit of time is second (s). • One second is the time taken by 9 192 631 770 oscillations of the light (of a specified wavelength) emitted by a cesium-133 atom. 9
- 8. 7. Mass • The SI unit of mass is kilogram (kg). • The standard kilogram is the mass of a platinum–iridium cylinder 3.9 cm in height and in diameter. 10
- 9. 7. Mass • A second mass standard: The carbon-12 atom has been assigned a mass of 12 atomic mass units (u), where 1u = 1.660 538 86 ± 10 × 10−27kg • Density: The density 𝜌 of an object of mass 𝑚 and volume 𝑉 is defined as 𝜌 = 𝑚 𝑉 11
- 10. Problems 𝐺𝑅 = 3.7 m 14 d = 0.264 m d = 0.264 m d × 1 𝜇m 10−6 m × 1 d 86400 s = 3.06 μm s
- 11. Problems
- 12. Problems
- 13. 8. Significant Figures • Every measurement has some uncertainty in it. For a example, a length measurement of 163.4 cm is said to have an absolute uncertainty of 0.1 cm. The uncertainty is sometimes expressed explicitly; 163.4 ± 0.1 cm. • The number of significant figures (also significant digits) in a measurement or a result is the number of figures (digits) that are known with some degree of reliability. For example, 163.4 cm has four significant figures, and 0.041 cm has two significant figures. 15
- 14. 8. Significant Figures Rules for deciding the number of significant figures: • All nonzero digits in a measurement are significant figures. • The leading zeroes are not significant figures. For example, 0.0000325 has three significant figures. • The trailing zeroes not preceded by a decimal point are not necessarily significant figures. For example, 10000 has one to five significant figures. There are five significant figures in 1.0000, however. 16
- 15. 8. Significant Figures • Arithmetics: • Multiplication & Division: The resultant number has as many significant figures as the number with the least number of significant figures. 3.14 2.093 2 = 13.7552 = 13.8. 3.11 0.025 = 124.4 = 120. • Addition & Subtraction: The resultant number has as many digits after the decimal point as the number with the least number of digits after the decimal point. 1.0201 + 8.54 = 9.5601 = 9.56. 14.7 − 15.03 = −0.33 = −0.3. 17
- 16. 9. Dimensional Analysis • The physical dimension of the physical quantity 𝑥 (written as 𝑥 ) is the product of the base quantities constituting it. In Phys101, [𝑥] has the general form 𝑥 = 𝐿𝑙𝑇𝑚𝑀𝑛 where 𝐿 = 𝐋ength, 𝑇 = 𝐓ime & 𝑀 = 𝐌ass . 𝑙, 𝑚 & 𝑛 are rational numbers (mostly integers). • Examples: period = 𝑇 speed = 𝐿 𝑇 Area = 𝐿 × 𝐿 = 𝐿2 𝜋 = 1 (dimensionless!) 18
- 17. 9. Dimensional Analysis • All terms in any correct physical equation must have the same dimension. For example, distance = 1 2 acceleration × time2 distance = 𝐿 acceleration × time2 = acceleration × time 2 = 𝐿 𝑇2 × 𝑇2 = 𝐿 19
- 18. 9. Dimensional Analysis • Example: Using Newton 2nd law Force = mass × acceleration, What is the dimension of force? We have Force = mass × acceleration = 𝑀 × 𝐿 𝑇2 = 𝑀𝐿 𝑇2 . 20
- 19. 9. Dimensional Analysis • Dimensional analysis can be helpful in solving problems and checking solutions. Example: Using that the acceleration due to gravity is 𝑔, what is the period 𝑃 of a pendulum of length 𝑙? We know that 𝑃 = 𝑇. We need a combination of 𝑔 and 𝑙 that has the dimension of 𝑇. We therefore write 𝑃 = 𝑐 𝑔𝑎 𝑙𝑏 , where 𝑐 is a constant. We need to find of 𝑎 and 𝑏. 21
- 20. 9. Dimensional Analysis 𝑃 = 𝑐 𝑔𝑎 𝑙𝑏 . We have that 𝑐 𝑔𝑎𝑙𝑏 = [𝑔]𝑎[𝑙]𝑏= 𝑇. Using 𝑔 = 𝐿/𝑇2 and 𝑙 = 𝐿 we get 𝐿𝑎+𝑏 𝑇−2𝑎 = 𝑇, which gives us 𝑎 + 𝑏 = 0, −2𝑎 = 1. 22
- 21. 9. Dimensional Analysis 𝑎 + 𝑏 = 0, −2𝑎 = 1. Solving for 𝑎 and 𝑏 we get 𝑎 = − 1 2 and 𝑏 = 1 2 . The period of a pendulum is therefore 𝑃 = 𝑐 𝑙 𝑔 . 23
- 22. 9. Dimensional Analysis: Problems 24 𝑦 = 𝐿 The dimensions of 𝐴𝑡 and 𝐵𝑡2 must be Length too: 𝐴𝑡 = 𝐴 𝑇 = 𝐿 ⟹ 𝐴 = 𝐿 𝑇 . 𝐵𝑡2 = 𝐵 𝑇2 = 𝐿 ⟹ 𝐵 = 𝐿 𝑇2 .
- 23. 9. Dimensional Analysis: Problems 25
- 24. 9. Dimensional Analysis: Problems 26
- 25. 9. Dimensional Analysis: Problems 27