Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mrao.cam.ac.uk/~kjbg1/exams/2007_paper.pdf Дата изменения: Thu Mar 22 14:59:59 2012 Дата индексирования: Tue Oct 2 04:01:24 2012 Кодировка:

NATURAL SCIENCES TRIPOS Part IA Wednesday 17 January 2007 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 1. A uniform solid cylinder is released at the top of a slope down which it rolls without losing contact or slipping. How fast is it travelling after losing 10m in height? [5]

2. Calculate the radius of a geostationary orbit (an orbit in which a satellite is always above the same point on Earth). The mass of the Earth is 6 x 1024 kg. [5]

3. Two equal masses, m, are connected by a spring of force constant k. An impulse is applied to one mass along the direction of the spring. Describe the subsequent motion. [5]



SECTION B ­ Physics 4. Explain what is meant by the term linear momentum [2]

In the lab frame, S, a mass, m travelling with a velocity v, undergoes a (non-head-on) elastic collision with an equal stationary mass. In the zero momentum frame S', after the collision the first mass now has velocity u. Calculate the following: (i) (ii) (iii) (iv) (v) a, the velocity of the zero momentum frame with respect to the lab frame. b and c the velocities of the two masses in S' before the collision. d, the velocity of the second mass in S' after the collision (in terms of u). |u| (in terms of v). e and f, the velocities of the two masses in S after the collision (in terms of v and u) Show that the two masses move off perpendicular to each other in S. Show that kinetic energy is conserved in S. [5] [4] [4]


5.

Give the linear and rotational equations of motion resulting from a force, F ,acting on a rigid body at a displacement, r ,from its centre of mass, defining any terms you use. [4]

A yoyo of mass m consists of a uniform density disc of radius d and a light axel of radius b around which string is wound. The moment of inertia of the yoyo is I = md 2 / 2 . At time t=0, the yoyo is released from rest and falls under gravity.

By considering the distance moved by the yoyo's centre of mass when the yoyo has undergone a rotation of one radian, deduce a relationship between the yoyo's linear acceleration, a, and its angular acceleration . Show that the time, t, taken for the yoyo to fall through a distance h is given by: 2h( b 2 + d 2 / 2 gb 2 [1]

t2 =

)

[6]

Show that at time t the yoyo's gain in kinetic energy is equal to its loss of gravitational potential energy. [4]


6.

A spaceship passes the Earth travelling towards a space-station at a speed of 0.6c. As it passes the Earth, clocks on both the ship and Earth read 12:00. In the Earth frame, the space station is a distance of 1 light second from the Earth. A pulse of light of frequency f is emitted from the space station towards the ship at exactly the time the ship passes the Earth in the Earth frame. How far is the ship from the Earth when the light pulse hits the ship? The light pulse is then perfectly reflected and travels back to the station. What is the time on the ship clock when the pulse arrives at the station? What is the frequency of the pulse received by the station? [6] [4] [5]