Maths for A Level Physics - updated edition

⓬ Given that log 2  3 = 1.585, use the laws of logarithms to ϐind: (a) log 2  9, (b) log 2 1 3 , (c) log 2  6, (d) log 2  1.5, (e) log 3  4 ⓭ Calculate: (a) log 3  9, (b) log 3 1 3 , (c) log 3 √ 3, (d) log 3 √ 27, (e) log 3  1 4 √ 243 ⓮ Calculate: (a) log 4  2,(b) log 4  32, (c) log 4  1 64 , (d) log 4 √ 2, (e) log 4  20 [given log 2 = 0.3010] ⓯ Express as a multiple of log 2: (a) log 4 + log 8, (b)log 4 − log 8, (c) log 2 + log  1 2 , (d) log 2 + log  1 4 ⓰ . Express in terms of ln 2: (a) ln 4 + ln  e , (b) ln 8 e , (c) ln 32 − ln  e , (d) ln 16 e , (e) ln  √ 2 e 2 ⓱ Use a calculator to solve the following equations for x : (a) e 2 x = 6, (b) e 2 x = 1 6 , (c) 0.1 = 10 e − x , (d) 20 = 5 × 10 3 × 2 − x , (e) ln  √ x = 3 ⓲ (a) Show that a x = e x  ln  a . [Note this is how we deϐine a x , for a > 0 and irrational x. ] (b) Use the above result to calculate 2 π , without using the x y button on your calculator. ⓳ Solve the following: (a) log 2 x = 4, (b) log 2 x 2 = 6, (c) log 4 3 √ x = 6, (d) log 4 x 2 = −1, (e) log 6 1 x = −2 ⓴ The sound intensity level, in ‘dB SIL’ is deϐined by L 1 = 10 log sound intensity 10 –11  Wm –2  dB SIL [10 −12  Wm −2 is roughly the threshold of human hearing.] (a) Calculate the sound intensity level in dB SIL of a sound of intensity 1Wm −2 (dangerous to hearing). (b) Show that an increase of 3 dB SIL represents an increase in sound intensity by a factor of 2. 21 The activity, A , in becquerel, of a radioactive nuclide varies with time according to A = A 0 e − λ t , where A 0 = 6MBq and λ = 3.5 × 10 −7  s −1 . Calculate: (a) the half-life of the decay, (b) the activity after 1 year and (c) the time taken for the activity to drop to 100 kBq. 22 The frequency, f , of oscillation of a loaded cantilever depends upon its projection length, l , according to the equation f = kl n . In an experiment to investigate this relationship a student timed 20 oscillations for two different lengths and obtained the following results: For l = 35 cm, time = 12.9 s. For l = 45 cm, time = 18.9 s. (a) Use the results to calculate values of f . (b) Show that n is approximately −1.5 and k is approximately 0.3 [with l in m]. (c) If the student had obtained a series of values for l and f what graph should she plot to test the relationship? State how values k and n would be obtained from the graph. 23 A gamma source is shielded by a 0.50 cm thick sheet of lead. A G-M tube outside the shielding registers radiation. A medical physicist adds additional lead layers and obtains the following results, which are corrected for background: Total thickness / cm 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Count rate / min −1 415 350 250 195 165 115 100 77 (a) Assuming that the count rate, C , is related to the thickness x by an equation of the form C = C 0 e − ( x L ) , where L is called the characteristic length of the relationship, plot a suitable graph and use it to ϐind values for C 0 and L . (b) If the safe level of radiation outside the box emanating from the source within is deemed to be 25min −1 , calculate the thickness of lead shielding required. 4 Indices and Logarithms 45