Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mrao.cam.ac.uk/~kjbg1/exams/2008_paper.pdf Дата изменения: Thu Mar 22 14:59:59 2012 Дата индексирования: Tue Oct 2 04:01:41 2012 Кодировка: Поисковые слова: asteroid

NATURAL SCIENCES TRIPOS Part IA Wednesday 16 January 2008 14.00 to 15.30

SELWYN COLLEGE REVISION TEST Attempt all of the questions in Section A and then any 2 questions in SectionB CONSTANTS Except where otherwise stated, constants are denoted by the following symbols and may be taken to have the values below Elementary charge Speed of light in vacuum Planck constant Permeability of vacuum Permittivity of vacuum Unified atomic mass constant Rest mass of proton Rest mass of electron Boltzmann constant Avogadro constant Gravitational constant e c h 0 0 mu mp m
e

1.60 x 10-19 C 3.00 x 108 m s-1 6.63 x 10-34 J s 4 x 10-7 H m-1 8.85 x 10-12 F m-1 1.66 x 10-27 kg 1.67 x 10-27 kg 9.11 x 10
-31

kg

kB NA G

1.38 x 10-23 JK-1 6.02x 1023 mol-1 6.67 x 10-11 N m2 kg

-2


SECTION A ­ answer all questions. A1 In a poorly maintained train, the thin cavity of a double glazed window is partially filled with rain water. As the train decelerates along a horizontal track, a passenger notices that the water surface is at an angle of 15 degrees to the horizontal. What is the deceleration of the train? [5] A2 Assuming that you can jump a maximum of 1 metre into the air on earth, calculate the largest radius of a homogeneous spherical asteroid from which you could jump clear. (Assume a density of 3000 kg m-3 for the asteroid.) [5] The highest energy cosmic rays have energies E=1020 eV. One of these, assumed to be a proton, enters the atmosphere at an altitude of 20km. In the frame of reference of the proton, how long would it take to pass through the atmosphere, assuming it suffers no collisions? [5]

A3


SECTION B ­ Physics 1. State the two forms of Newton's second law, and explain what are meant by the terms force and mass in this context [5]

In a lecture demonstration, you witness Dr Duffett-Smith drop a small glass marble from his outstretched hand. The marble falls vertically from rest onto a hard metal floor and bounces back upwards. Assuming the marble's collision with the floor is elastic, sketch two graphs showing: a) b) The velocity of the marble as a function of time; The acceleration of the marble as a function of time, [5] [5]

for times that cover the period from just before the marble is dropped to the time at which it returns to the height from which it was released. In each case, annotate your sketches, and give quantitative estimates ­ together with your reasoning for these ­ of the velocities, accelerations and times shown.

2.

Write down expressions for the total energy, E, the kinetic energy, T, and the momentum, p , of a relativistic particle of mass m moving with velocity v , defining all the terms in your answers. [4]

Show that the expression for the kinetic energy is consistent with the non-relativistic result you are familiar with from your A-level physics. [2]

Explain what is meant by the term energy-momentum invariant, and why it is so helpful in problems involving relativistic dynamics. [3]

A particle of mass m has a kinetic energy of 2mc2 and strikes another particle of mass m, which is initially at rest in the lab, head-on. It combines with it to form a composite particle of mass M which moves off at speed v. By using the "Energy-momentum invariant" or otherwise, obtain an expression for the mass, M, of the composite particle, and sketch this as a function of . [6]


3.

Write down the expression for the gravitational field, g, at distance r from a point mass of mass m, carefully defining all the terms in your expression. [4]

A uniform thin flat sheet of lead of density , thickness t, and infinite extent in the x and y directions, exists in space, far away from any other gravitating masses. By dividing the sheet into a set of concentric thin annuli, and by summing the contributions to the gravitational field from each of these, show that the gravitational field strength of the lead sheet at a distance h perpendicular to its surface is given by: ^ g = -2 Gt z. [No marks will be given for derivations that utilise Gauss' theorem] [8]

Explain, without any detailed calculations, why this expression for the field strength does not decrease as the inverse square of the perpendicular distance h. [3]