Dimensions and Dimensional Analysis

Dimensional analysis offers a method for reducing complex physical problems to the simplest (that is, most economical) form prior to obtaining a quantitative answer.

1. Dimensions & Dimensional Analysis Peter Huruma Mammba Department of General Studies DODOMA POLYTECHNIC OF ENERGY AND EARTH RESOURCES MANAGEMENT (MADINI INSTITUTE) –DODOMA p [email protected] m

2. What is dimension? Is a physical property which describes a way any physical quantity is related to fundamental physical quantities E.g. dimension of velocity is 1 in length and -1 in time.

3. What is dimensional analysis? Is the analysis of the relationships between different physical quantities   by identifying their fundamental dimensions such as length , mass, time .

4. Physical quantity I s a physical property of a phenomenon, body or substances that can be quantified by measurement.   C an be expressed as the combination of a  number  and a unit  or combination of units .

5. Types of physical quantity There are two main types of physical quantity which are:- Fundamental (primary/ basic) physical quantities Derived (secondary) physical quantities

6. Fundamental Physical Quantity Fundamental quantities are the quantities which cannot be expressed in terms of any other physical quantity. E.g. Mass , length, Time, Temperature, Intensity of light, Electric Current, etc.

7. Derived physical quantities These are quantities whose definitions are based on other physical quantities ( base quantities ). E.g. Pressure ( ), speed ( ),Young modulus ( ).  

8. Other examples of Derived quantity

9. Unit Unit is the reference used as the standard measurement of a physical quantity. The unit in which the fundamental quantities are measured are called fundamental unit and the units used to measure derived quantities are called derived units .

10. System of unit G.S. system – It is the centimeter gram second system which are units of length, mass and time. P.S. system – it is foot pound second system. Britishers used this system.

11. System of unit… K.S. System – it is metre kilogram second system. European countries like France use this system. S.I. System – It is the international system of unit. It is universally accepted and has seven fundamental units.

12. Law of dimensional analysis (principle of homogeneity) State that “The equation is dimensionally correct if the dimensions on the left hand side of the equation are equal to the dimensions on the right hand side of the equation, if not the equation is not dimensionally correct”.

13. Checking a Result Terms do not match Terms match, this could be a valid formula.

14. Dimensional formulas of physical quantities

15. Four category of physical quantities Dimensional variables Dimensionless variables Dimensional constant Dimensionless constant

16. Dimensional variables  

17. DIMENSIONLESS VARIABLES Those physical quantities which have neither dimensions nor fixed value. E.g.. Specific gravity, strain, angle, etc.

18. DIMENSIONAL CONSTANT Those physical quantities which possess dimensions and have fixed value. E.g. Gravitational contact, planks constant, velocity of light, etc.

19. Dimensionless constant Those quantities which do not possess dimensions but have fixed value. E.g. 1 , 2, 3, 4, 5 , π , e, etc.

20. Uses of dimension equations To check the correctness of the physical relation. To recapitulate a forgotten formula. To derive the relationship between different physical quantities .

21. Uses of dimension equations… To convert one system of unit to another. To find the dimensions of constant in a given relation.

22. Limitations Dimensional analysis only checks the units. Numeric factors have no units and can’t be tested. is also valid. is not valid.

23. Limitations... Dimension analysis cannot be used to derive the exact form of a physical relation if the physical quantity depends upon more than three physical quantities (M,L & T).

24. Limitations... Dimension analysis can not be used to derive the relation involving trigonometrical and exponential function . Dimensional analysis does not indicate whether a physical quantity is scalar or vector.

25. Example 1  

26. Solution ……( i ) But; = …..(ii) ….(iii) = ….(iv)   = ….(v)  

27. Solution … Substitute equation ii, iii, iv & v into equation i ; =k = c = 1……vi 0= b + c; but c = 1 b = -1 ….vii   = But; c = 1 & b = -1 ….viii  

28. Solution… Substitute equation vi, vii & viii into equation i ;  

29. Example 2 A gas bubble from an explosion under water is found to oscillate with a period T , which is proportional to , and where P is pressure, is the density and is the energy of explosion. Find the units of the constant of proportionality.  

30. Solution ….( i ) But; …..(ii) = ….( iii) ….(iv)   = ….(v) Substitute equation ii, iii, vi & v into equation i ; .. vi … a  

31. Solution… … b …c   From quadratic equations x, y & z;  

32. Solution… Type equation here. Substitute the value of x , y and z into equation vi ; no unit (hence shown)  

33. Example 3 If the viscous force F is defined by where is the coefficient of viscosity, A is the cross sectional area and is the velocity gradient. Find the dimensions and units of .  

34. Solution …( i ) But; ... (ii ) …(iii)   = …..( iv) ….( v )  

35. Solution… Substitute equation ii , iii , iv & v into equation i , we get; (Dimension) Kg (Unit)  

36. Example 4 While moving through liquid at speed, v a spherical body experiences a retarding force given by . Where k is constant, is density of liquid and R is radius of the body. Determine the numerical values of x , y and z by the method of dimension.  

37. Solution …..( i ) But; …(ii) ….(iii ) = ….(iv ) = …..(v)   Substitute the dimensions of equation ii , iii , iv & v into equation i ; ; 2  

38. Question 1 The equation below is called Bernoulli's equation which is applied to fluid flow and it is stated that Where P = pressure, h = height, = Density, v = velocity, g = acceleration due to the gravity and k = constant. Show that K = 1  

39. Question 2 The stress between two planes of molecules in a moving liquid is given by . Where v is the velocity difference between the planes, x is their distance apart and is a constant for a liquid. Show that the dimensions of are  

40. Question 3 The velocity of wave of wavelength on the surface of a pool of liquid whose surface tension and density are and respectively is given by . Where g is the acceleration due to the gravity. Show that the equation is dimensionally or not dimensionally correct.