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NATURAL SCIENCES TRIPOS Part IA Wednesday 18 January 2006 09.30 to 12.30

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


1) The rotor arm of a torsional oscillator in an antique clock consists of two solid spheres, each of mass M, attached to the ends of a rod of mass m and length a. Its moment of inertia is given by I= a2 (6 M + m ) 12

If M and m are approximately 50g and 10g respectively, what is the fractional error in the calculated value of I when a, M and m are each measured with a random error of 5 percent? 2) Estimate the mean distances from the Sun of Venus and Mars, as fractions of the Earth's orbit, given that their orbital periods are 0.62 and 1.88 years respectively. 3) 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?


SECTION B ­ Physics 4. Explain what are meant by the terms linear momentum and kinetic energy. Explain what this is and why its use helps simplify such calculations. [3] [4]

It is often easier to make calculations involving collisions in the zero momentum frame. A mass m travelling at speed u through the laboratory makes a head on collision with a mass M travelling at speed v in the opposite direction. The two masses stick together after the collision. Show that the change in the kinetic energy is the same when calculated in the laboratory frame as it is when calculated in the zero-momentum frame. [8]

5. Two space craft are travelling in opposite directions towards each other, one at speed u and the other at speed v with respect to the Earth. The captain of one space craft measures the speed of the other to be w. Show that w is given by w= u+v uv . 1+ 2 c [8]

The captain of one of the space craft measures the time interval, t, between the moments when the front and back of the other space craft pass him. Given that the length of the other space craft is L0 as measured in its own frame, find an expression for w in terms of t and L0, and show that, as t tends to zero, w tends to the speed of light, c. 6. Explain what is meant by the term gyroscope, and give two examples. [7] [4]

A gyroscope with moment of inertia I is spinning quickly about its axis with angular velocity

and is observed to be precessing slowly about another axis at angular velocity . Show that
a torque of moment G must be acting about a third axis which is perpendicular to both the other axes, and that G = I в . [6] The North pole of the Earth is precessing with a period of about 25,800 years because of a torque from the combined action of the gravitational pulls of the Moon and the Sun. Estimate the magnitude of the average moment of this torque, given that the Earth can be regarded as a sphere with radius 6,400 km, and average density 5,500 kg m-3. [5]