The Chudnovsky Brothers

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It is often interesting in mathematics lessons to talk about curious characters who have done original things in mathematics. David and Gregory Chudnovsky are two brilliant brothers who will not be found in the index of most mathematics textbooks. In 1991 the brothers built a supercomputer in their apartment in Manhattan. They used mail order parts delivered in boxes, the building superintendent being unaware of what they were doing. They called their computer m-zero and claimed that it was just as powerful as a Cray supercomputer, the Cray costing $30 million and theirs $70,000. Why did they do this? Why fill their apartment with electrical leads and circuitry and raise its temperature to intolerable levels? To calculate pi. The brothers had a passion for mathematics and used their computer to calculate pi to two billion decimal places. Is it necessary to find this value to such high accuracy for everyday calculations? No. When we perform calculations our overall accuracy is determined by the least accurate number that we input. Most scientific constants are only known to at most 10 decimal places. The brothers were explorers determined to venture into new territory. It is only by pushing the limits of knowledge that new, and unexpected, discoveries, are made.

Physics in the Lab I

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Doing experiments in a laboratory allows us to understand how the laws of physics work. We develop a knowledge of the quantity that we are measuring and how it depends on other factors. Experimental work involves making measurements and looking for patterns in the measurements. There are two concepts in experimental work that students have difficulty with. These are uncertainty (error) analysis and plotting data. As the NSW HSC syllabus no longer includes uncertainties in practical work many students arrive at first year university labs not fully prepared for the requirements of experimental work in both calculator skills and uncertainty analysis. The international examining boards, Cambridge International Examinations and the International Baccalaureate, still include questions with uncertainties in their examination papers. To help students learn uncertainty analysis a tutorial problem set is provided below.

  1. A student drops a ball from the same height and measures the time of fall. Their measurements are 1.75s, 1.85s, 1.60s, 1.70 s and 1.71 s. Determine the average time of fall.
  2. A student measures the dimensions of a desk top. The average value of the length was found to be 2.524 ± 0.004 m and the average value of the width was found to be 0.622 ± 0.004 m. Determine the perimeter and area of the desk top.
  3. A student releases a ball from rest and measures the time it takes to fall to the ground. The average time was found to be 1.32 ± 0.08 s. Given that the height of release is 8.61 ± 0.05 m, determine the acceleration due to gravity.
  4. A student measures the mass of a block as 117.56 ± 1.24 g. The volume of the block was measured as 22.67 ± 0.36 cm3. Determine the density of the block.
  5. In a laboratory experiment a student measures the time of 10 oscillations of a simple pendulum of length 3.25 ± 0.03 m. Their time was 37.21 ± 0.86 s. Use this data to determine the acceleration due to gravity.
  6. A dynamics trolley is moving along a smooth laboratory bench at a speed of 0.26 ± 0.03 m/s. The trolley accelerates at 0.84 ± 0.04 m/s2 for 6.53 ± 0.08 s. Determine the distance travelled by the trolley.
  7. Determine the volume of a right circular cylinder of radius 3.215 ± 0.025 m and height 7.512 ± 0.025 m.
  8. Determine the volume of a sphere of radius 3.219 ± 0.038 m.
  9. The density of plutonium is 19.8 ± 0.4 gcm-3. Determine the radius in centimetres of a sphere of plutonium of mass 15.0 ± 0.5 kg.

  10. When the radius r of the bob of a simple pendulum of length L is included in the calculation the period T of the small oscillations of the pendulum is given by the equation

    T = 2𝜋√(L/g + 2r2/(5gL))

    If L = 2.00 ± 0.02 m, g = 9.81 ± 0.03 ms-2 and r = 10.0 ± 1.0 cm, determine the value of T.

  11. When the angle of oscillation of a simple pendulum is not small, the approximate period of the oscillations is given by the equation

    T = 2𝜋√(L/g) (1 + 𝜽2/16)

    where L is the length of the string, g is the acceleration due to gravity and 𝜽 is the angle of release of the string from the vertical measured in radians. Taking L = 6.57 ± 0.05 m, g = 9.81 ± 0.04 and 𝜽 = 22 ± 3 ∘, determine the value of T.

  12. Determine the area of a triangle of sides 1.236 ± 0.015 m, 3.256 ± 0.023 m and 2.887 ± 0.023 m.