Maths for A Level Physics - updated edition

The variables in this relationship are A and t . Comparing the two equations, if we plot ln  A against t , we should get a straight line graph of gradient − λ and an intercept of ln  A 0 on the ln  A -axis (see Figure 4.5). This type of graph is called a semi-log or a log-linear graph. An equivalent, and very useful, way of writing the decay equation is A = A 0 2 − n , where n is the number of half-lives: see Example I. Example I: Show that the radioactive decay equation can also be written as A = A 0 2 − n . n is the number of half-lives. ∴ t = nt 1 2 ∴ From equation (1) in 4.4.7, t = n  ln 2 λ Substitute into equation (2) in 4.4.7: ln  A = ln  A 0 − n  ln 2 ∴ ln  A = ln ( A 0 2 − n ) using laws 1 and 3 of logs (see Section 4.4.3) ∴ A = A 0 2 − n  QED ln A t ln A 0 Ϋ λ Fig 4.5 Test Yourself 4.1 Questions 1–10 relate to indices only. ➊ Express as a number or fraction, (a) 125 1 3  , (b) 125 2 3  , (c) 125 − 1 3  , (d) 125 − 2 3  , (e) 125 4 3 ➋ Express as a number or fraction, (a) 16 1 2  , (b) 16 − 1 2  , (c) 16 3 4  , (d) 16 7 4  , (e) 16 − 3 4 ➌ Express in the form a P , (a) 4 √ a ,, (b) 1 4 √ a , (c) a 3 √ a  , (d) 5 √ a 2  , (e) ( 1 4 √ a 3 ) 2 ➍ Evaluate, (a) ( 900 4 ) 1 2 , (b) √ 900 √ 4 , (c) ( 16 625 ) 1 2 , (d) 4 √ 625 4 √ 16 , without using a calculator. ➎ The intensity, I , of radiation at distance r from a star is given by I = Kr p . I falls by a factor of 4 when r doubles. What is p ? ➏ The square of the period, T , of revolution of a planet around the Sun is proportional to the cube of its mean orbital radius, a . This can be expressed in the form T = ka p . State the value of p . ➐ The volume, V of a sphere can be expressed in terms of the sphere’s surface area by V = KA p . Determine the constant, k , and the index p . [Hint: Start with the usual formulae for volume and area of a sphere, and eliminate the radius r .] ➑ The resistance, R , of a wire is given by R = ρ l A  , where ρ is the resistivity, l the length and A the cross-sectional area. A volume, V , of metal is to be made into a wire but the diameter, d , of the wire can be any value. Write the relationship between R and d in the form R = kd n , determining the values of k and n . ➒ The luminosity, L , of a star, that is its total emitted power depends upon its temperature and surface area according to Stefan’s law: L = σ AT 4 , where σ is a constant. The luminosity of the Sun, L = 4 × 10 26  W. Calculate the luminosity of, (a) a red giant star with a temperature 2 3 times and a radius 100 times those of the Sun and (b) a white dwarf star with a temperature 3 and a radius 1 100 times those of the Sun. ➓ The current, I , through a non-ohmic resistor varies with pd, V , according to I = kV 3 , in which k = 8.0 × 10 −3  A V −3 . Express the resistance, R , of the device [ deϐined by R = V I ] in terms of the current. Hint: Start with R = cI n and determine the values of c and n. ⓫ Given that log 2 = 0.3010 use the laws of logarithms to determine: (a) log 4, (b) log 0.04, (c) log 8, (d) log 200, (e) log 2.5. [Hint: log b b n = n ] Mathematics for Physics 44