This document discusses dimensional analysis and its applications. It can be used to: 1) Derive equations by ensuring the dimensions on both sides are equal 2) Check if equations are dimensionally correct 3) Find the dimensions/units of derived quantities Examples are provided to illustrate deriving equations based on quantities' dimensions and checking the homogeneity of equations.

3. DIMENSIONS What is “Dimension” ?

7. Derived an Equation (Quantities) Example 1 Velocity = displacement / time [velocity] = [displacement] / [time] = L / T = LT -1 v = s / t

9. T = km a ℓ b g c Write a general equation: T α m a ℓ b g c By using the dimension method, an expression could be derived that relates T, l and g whereby a, b and c are dimensionless constant thus

10. Write out the dimensions of the variables T = M a L b (LT -2 ) c T = M a L b L c T -2c T 1 = M a L b+c T -2c [T] = [m a ][ ℓ b ][g c ] Using indices a = 0 -2c = 1 -> c = -½ b + c = 0 b = -c = ½

11. T = km a ℓ b g c T = km 0 ℓ ½ g - ½ Whereby, the value of k is known by experiment

12. Exercises The viscosity force, F going against the movement of a sphere immersed in a fluid depends on the radius of the sphere, a the speed of the sphere, v and the viscosity of the fluid, η . By using the dimension method, derive an equation that relates F with a, v and η . (given that )

13. To check whether a specific formula or an equation is homogenous Example 1 S = vt [s] = [v] [t] L.H.S [s] = L R.H.S [v] [t] = LT -1 (T) [v] [t] = L Thus, the left hand side = right hand side, rendering the equation as homogenous

14. Example 2 Given that the speed for the wave of a rope is , Check its homogenity by using the dimensional analysis

15. L.H.S [C] = (LT -1 ) 2 [C] = L 2 T -2 R.H.S [F] = MLT -2 , = LT -2 [M] = M Conclusion : The above equation is not homogenous (L.H.S ≠ R.H.S)

17. Find out dimension or units of derived quantities Example Consider the equation , where m is the mass and T is a time, therefore dimension of k can be describe as

18. -> unit: kgs -2 thus, the units of k is in kgs-2

24. Dimensional Analysis (a) (b) (c) (c) This has the correct units!! This must be the answer!! Example continue… Try the third equation