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 0.5= + = + = 0.067018.5 12.5 ΔA ΔL ΔB A L B ΔA 2= (0.0670)(231.25) = 20 mm (to 1 s.f.) When we write 20 mm2, it is unclear if this is a 1 or 2 s.f. value. Writing the answer in standard form removes this ambiguity, even though both are acceptable.

Example 7 The radius of a circle is r = (3.0 ± 0.2) cm. Find the circumference with its associated uncertainty. Circumference, C = 2r = 2 (3.0) = 18.8 cm Absolute uncertainty of the circumference, C = 2r = 2 (0.2) = 1.26 cm = 1 cm (1 s.f.) C = (19  1) cm (“C “same place value as C)

Example 8 Given a sphere of radius R = (18.5 ± 0.5) mm, find the volume of the sphere with its associated uncertainty. 3 3 4 344 33Volume V = π R = π (18.5 )= 2.652 ×10 mm ΔV ΔR 0.5= 3 = 0+ 3( )= 0.0811V R 18.5 344 mm 102.0)10652.2(0811.0 V V 43= (2.7 ± 0.2)×10 mm In general, if a measured number is so large or small that it calls for scientific notation, then it is simpler and clearer to express the number and its uncertainty in the same form, i.e. both are expressed to x 104 mm3.

Example 9 In an experiment, the external diameter d1 and internal diameter d2 of a hollow tube are found to be (64 ± 2) mm and (47 ± 1) mm respectively. Calculate the thickness of the tube and the associated uncertainty. What is the corresponding percentage error? d1 d2 Thickness t = = 8.5 mm Uncertainty in t, Δt = = 2 mm (1 s.f.) Thus t = (9 ± 2) mm Percentage uncertainty in t = = 22% Δt 2×100% = ×100%t9 12d - d 64 - 47=22 12 11Δd + Δd22

Example 10 The period of oscillation of a simple pendulum is A student conducts an experiment to find the acceleration of free fall, g. He measures the length of the pendulum, l = 0.23 ± 0.01 m and the time for 20 oscillations, t = 19.24 ± 0.01 s. Find g and its associated uncertainty. g lT 2 Make g the subject of the equation* (this step is a must!) 1- 2 2 2 2 s m 81.9)2024.19( )23.0(44   T lg 0445.024.19 01.0223.0 01.02  T T l l g g T = t/20, so ΔT / T =Δt / t 4.081.90445.0 g -2s m )4.08.9( g