WJEC Physics for AS Level Student Book 2nd Edition

11 1.1 Basic physics Here is another example, this time using an unfamiliar quantity: Example The drag force, F D , on a sphere moving through a fluid is given by Stokes’ formula, F D = 6 ʌȘDY , where a is the radius of the sphere, v the velocity and Ș [eta] is the coefficient of viscosity of the fluid. Find the unit of Ș in terms of the base SI units. Answer Rearranging the equation Ș = F D 6 ʌDY . Both 6 and ʌ have no units, so [ Ș ] = [ F D ] [ a ][ v ] [ F D ] = kg m s – 2 , [ a ] = m and [ v ] = m s – 1 \  [ Ș ] = kg m s – 2 m 2 s – 1 = kg m – 1 s – 1 . (c) Using SI multipliers and standard form Many problems arise in which the quantities are either much larger or much smaller than the basic units. The data are thus given either in standard form or using SI multipliers. This example has data in mixed forms. Example Calculate the energy transmitted by a 44 kV power cable in one day if it carries a current of 2.5 × 10 2 A . [Use P = IV and E = Pt ] Answer From the two equations, E = IVt . \ Converting to the base units: E = 2.5 × 10 2 A × 44 × 10 3 V × 86 400 s = 9.5 × 10 11 J (2 s.f.) 1.1.2 Checking equations for homogeneity Consider the equation: v 2 = u 2 + 2 ax , where u and v are the initial and final velocities, a the acceleration and x the displacement of a uniformly accelerating object. We’re going to take this equation apart and look at the units of its various bits. 1. The u 2 term: Now [ u ] = m s – 1 , so [ u 2 ] = ( m s – 1 ) 2 = m 2 s – 2 . 2. The 2 ax term: [2 ax ] = [ a ] × [ x ] = m s – 2 × m = m 2 s – 2 Let’s just stop here for a moment: the u 2 term and the 2 ax term have the same units! Why is this important? Because it means that they can be added together. See Rule 1 in the margin. This means that the unit of the right-hand side of the equation is m 2 s – 2 . 3. The v 2 term: [ v ] = m s – 1 , so [ v 2 ] = ( m s – 1 ) 2 = m 2 s – 2 Notice that the left-hand side has the same unit as the right-hand side. Why is this important? Two things can only be equal if they have the same units; 53 V can never be equal to 53 A – similarly 1 day and 1 cm could never be the same! We say that this equation is homogeneous – only terms with the same units are added or subtracted and the units of the two sides are the same. If the ‘equation’ isn’t homogeneous it cannot be right – you must have remembered it incorrectly. Study point Homogeneity – Rule 1 Two quantities a and b can only be added together if they have the same units – and then the answer has the same units. The same goes for subtraction. Homogeneity – Rule 2 An equation is homogeneous only if the units of the two sides are the same. Maths tip See Section 4.2.1 (c) and (d) for SI multipliers and standard form. Knowledge check Show that the equation x = ut + 1 2 at 2 is homogeneous. (Remember that 1 2 has no units.) 1.1.4 Study point Warning Just because an equation is homogeneous doesn’t mean that it is right, e.g. v 2 = u 2 + 3 as is homogeneous and incorrect! Top tip In the example, we could have written 44 kV as 4.4 = 10 4 V . However, to avoid mistakes it is easier to write it as 44 = 10 3 and let the calculator handle it!