C1h2

Basic quantity is defined as a quantity which cannot be derived from any physical quantities Derived quantity is defined as a quantity which can be expressed in term of base quantity cd candela Luminous Intensity mol mole N Amount of substance A ampere I Electric current K kelvin T/  Temperature s second t Time kg kilogram m Mass m metre l Length Symbol SI Unit Symbol Quantity

d. 29 cm = ? in e. 12 mi h -1 = ? m s -1

Learning Outcome: At the end of this chapter, students should be able to: Use dimensional analysis to check homogeneity and construct equation of physics.

Dimensional Analysis Dimension is defined as a technique or method which the physical quantity can be expressed in terms of combination of basic quantities . It can be written as [physical quantity or its symbol] Table 1.5 shows the dimension of basic quantities. mole N [amount of substance] or [ N ] K  [temperature] or [ T ] A A @ I [electric current] or [ I ] s T [time] or [ t ] m L [length] or [ l ] kg M [mass] or [ m ] Unit Symbol [Basic Quantity]

Dimension can be treated as algebraic quantities through the procedure called dimensional analysis. The uses of dimensional analysis are to determine the unit of the physical quantity . to determine whether a physical equation is dimensionally correct or not by using the principle of homogeneity . to derive/construct a physical equation . Note: Dimension of dimensionless constant is 1 , e.g. [2] = 1, [refractive index] = 1 Dimensions cannot be added or subtracted. The validity of an equation cannot determined by dimensional analysis. The validity of an equation can only be determined by experiment. Dimension on the L.H.S. = Dimension on the R.H.S

Determine a dimension and the S.I. unit for the following quantities: a. Velocity b. Acceleration c. Linear momentum d. Density e. Force Solution : a. The S.I. unit of velocity is m s  1 . Example 1.2 : or

b. Its unit is m s  2 . d. S.I. unit : kg m  3 . c. S.I. unit : kg m s  1 . e. S.I. unit : kg m s  2 .

Determine Whether the following expressions are dimensionally correct or not. a. where s , u , a and t represent the displacement, initial velocity, acceleration and the time of an object respectively. b. where t , u , v and g represent the time, initial velocity, final velocity and the gravitational acceleration respectively. c. where f , l and g represent the frequency of a simple pendulum , length of the simple pendulum and the gravitational acceleration respectively. Example 1.3 :

Solution : a. Dimension on the LHS : Dimension on the RHS : Dimension on the LHS = dimension on the RHS Hence the equation above is homogeneous or dimensionally correct. b. Dimension on the LHS : Dimension on the RHS : Thus Therefore the equation above is not homogeneous or dimensionally incorrect. and and

Solution : c. Dimension on the LHS : Dimension on the RHS : Therefore the equation above is homogeneous or dimensionally correct.

The period, T of a simple pendulum depends on its length l , acceleration due to gravity, g and mass, m . By using dimensional analysis, obtain an equation for period of the simple pendulum. Solution : Suppose that : Then where k , x , y and z are dimensionless constants. Example 1.4 : ………………… (1)

By equating the indices on the left and right sides of the equation, thus By substituting eq. (3) into eq. (2), thus Replace the value of x , y and z in eq. (1), therefore The value of k can be determined experimentally. ………………… (2) ………………… (3)

Determine the unit of  in term of basic unit by using the equation below: where P i and P o are pressures of the air bubble and R is the radius of the bubble. Solution : Example 1.5 :

Since thus Therefore the unit of  is kg s -2

Deduce the unit of  (eta) in term of basic unit for the equation below: where F is the force, A is the area,   v is the change in velocity and  l is the change in distance. ANS. : kg m -1 s -1 A sphere of radius r and density  s falls in a liquid of density  f . It achieved a terminal velocity v T given by the following expression: where k is a constant and g is acceleration due to gravity. Determine the dimension of k . ANS. : M L -1 T -1 Exercise 1.1 :

a. What is meant by homogeneity of a physical equation? b. The escape velocity, v for a tomahawk missile which escape the gravitational attraction of the earth is depend on the radius of the earth, r and the acceleration due to gravity, g . By using dimensional analysis, obtain an expression for escape velocity, v . ANS. : Show that the equation below is dimensionally correct. Where R is the inside radius of the tube, L is its length, P 1 -P 2 is the pressure difference between the ends,   is the coefficient of viscosity ( N s m -2 ) and Q is the volume rate of flow ( m 3 s -1 ). Exercise 1.1 :

Acceleration is related to velocity and time by the following expression Determine the x and y values if the expression is dimensionally consistent. ANS. : x= 1; y=  1 Bernoulli’s equation relating pressure P and velocity v of a fluid moving in a horizontal plane is given as where  is the density of the fluid and k is a constant. Determine the dimension of the constant k and its unit in terms of basic units. ANS. : M L  1 T  2 ; kg m  1 s  2 Exercise 1.1 :

C1h2

More Related Content

Similar to C1h2 (20)

C1h2