HCI 01 Measurement Solutions (Lecture Examples)

Example 1 According to Newton's Law of Gravitation, the force F between two point masses M and m separated by the distance r is given by the formula where G is the universal gravitational constant. Obtain the SI base units for G. 2 GMmF r maFMm FrG  and 2 2-1-3 2-22 s kg mkg m s m of Unit of Unit       Mm marG

Example 2 Bernoulli’s equation, which applies to fluid flow states that P + hρg + ½ ρv2 = k where P is pressure, h is height, ρ is density, g is acceleration due to gravity, v is velocity and k is a constant. Show the left hand side of the equation is homogeneous and state the SI unit for k. 2 12 2 kg m s kg m sm F maUnits of P Units of Units ofAA      3 2 1 2 ( )( )( ) m kg m m s kg m s MUnits of h g Units of h Units of Units of gV          2 3 1 2 1 21 (1) kg m (m s ) kg m s2Units of v           Homogeneous! Analysing each term on the LHS of the equation: Unit for k = kg m-1 s-1

Example 3 Consider the period T of a simple pendulum. The possible factors which may affect it are its length l, its mass m and the acceleration due to gravity g. Use unit analysis to arrive at a plausible relationship between T and these quantities. Suppose the relationship between T, l, m and g is given by : where k is a dimensionless quantity and x, y, and z are constants. x y zT kl m g 22(1)(m) (kg) (m s ) m kg sx y z x z y z      0, 0, 2 1 11, 0, 22 x z y z x y z          Units on the LHS : s Units on the RHS: Comparing the indices : A plausible relationship must be homogeneous. ( / )T k l g

• Assuming the bus travels at 60 km h-1, its KE is • A typical domestic light bulb is between 5 W and 50 W. • 300 K is just only about 27 ᵒC! • Assuming an outer diameter of 30 cm and inner diameter of 20 cm, and a width of 15 cm, Example 4 Which estimate is realistic? A. The kinetic energy of a bus on an expressway is 30000 J. B. The power of a domestic light is 300 W C. The temperature of a hot oven is 300 K. D. The volume of air in a tyre is 0.03 m3.    22 611 ( )( )( ) 1.4 10 J 22 6000010000 3600mv   2 2 3(0.30) (0.20) (0.15) 0.024 m[]V

Example 5 A student makes measurements from which he calculates the speed of sound to be 327.66 m s-1. He estimates that the percentage uncertainty is 3%. Round off the speed to an appropriate number of significant figures. )sf 1(s m 1082.9)66.327(03.0 -1v  -1v = (330 ±10) m s ΔvGiven = 3%v Note that v is expressed to the same place value as v. Hence in this case, the speed is expressed to the tens place (which happens to be 2 sf).

Example 6 The measurements of the dimensions of a particular piece of rectangular cardboard are (18.5 ± 0.5) mm and (12.5 ± 0.5) mm. Determine the area of the cardboard with its associated uncertainty. Area, A = L X B = 18.5 x 12.5 = 231.25 mm2 Fractional uncertainty of the area,  A = (230  20) mm2 = (2.3  0.2) x 102 mm2 (“A” same place value as A) 0.5