NTURL SCIENCES TRIPOS Part I Saturday 9 June 2007 1.30 pm to 4.30 pm PHYSICS nswer the whole of Section and four questions from Sections B, C and D, with at least one question from each of these Sections. Numbers in the right hand margins indicate the approximate distribution of marks available. Note that Section carries approximately one third of the total marks. nswers must be tied into four bundles, one for each Section of the paper, and each bearing a clearly labelled cover sheet. STTIONERY REQUIREMENTS Script paper Linear graph paper Rough workpad Blue coversheets (4) Tags SPECIL REQUIREMENTS Mathematical formulae handbook pproved calculator allowed You may not start to read the questions printed on the subsequent pages of this question paper until instructed that you may do so by the Invigilator.
2 Values of constants speed of light in a vacuum c 3.00 10 8 m s 1 permeability of a vacuum µ 0 4π 10 7 H m 1 permittivity of a vacuum ε 0 8.85 10 12 F m 1 elementary charge e 1.60 10 19 C Planck constant h 6.63 10 34 J s h/2π h 1.05 10 34 J s vogadro constant N 6.02 10 23 mol 1 unified atomic mass constant m u 1.66 10 27 kg mass of electron m e 9.11 10 31 kg mass of proton m p 1.67 10 27 kg mass of neutron m n 1.67 10 27 kg Bohr magneton µ B 9.27 10 24 J T 1 molar gas constant R 8.31 J K 1 mol 1 Boltzmann constant k B 1.38 10 23 J K 1 Stefan Boltzmann constant σ 5.67 10 8 W m 2 K 4 gravitational constant G 6.67 10 11 N m 2 kg 2 Other data acceleration of free fall g 9.81 m s 2
3 SECTION nswers should be concise, and relevant formulae may be assumed without proof. 1 ball is thrown horizontally with a speed 30 m s 1 from a height of 70 m. How far has the ball travelled horizontally when it first hits the ground? 5] Neglect air resistance. ] 2 The Schwarzschild radius R s of a black hole depends only on its mass M, the speed of light c and the gravitational constant G. Given that R s M α c β G γ use dimensional analysis to find α, β and γ. 5] 3 Six identical wires of resistance R form the edges of a tetrahedron and are connected at its vertices. Show that the resistance between any two vertices is R/2. 5] 4 10 mh inductor and a 10 Ω resistor are connected together in a loop. How long does it take for a current I 0 flowing at t = 0 to fall to half of its initial value? 5] 5 What is the spacing of fringes seen on a screen 20 cm beyond a pair of narrow slits having separation 50 nm illuminated by a beam of electrons accelerated through 2 kev? 5] You may assume that 2 kev electrons are non-relativistic. ] 6 One litre of air at a temperature of 300 K and a pressure of 1 atmosphere is mixed with another litre of air at the same pressure but with a temperature of 600 K. If the mixture is confined to a volume of 2 litres, what is the pressure after cooling to 300 K? 5] ssume that air behaves as a perfect gas. ] (TURN OVER
4 SECTION B B7 Outline how the classical concepts of momentum and energy have to be modified according to the theory of special relativity. 4] n object with energy E and momentum of magnitude p collides with a mass m at rest and coalesces with it to form a single body without loss of energy. Show that the speed of the composite body is p m + E/c 2 3] In a spherically symmetric explosion in free space, all of the matter in an object having mass M is converted to a mixture of photons and particles. The photons account for 25% of the original energy, and the rest energy of the particles another 25%. Show that the Lorentz factor of the particles is γ = 3. Hence find the time difference between the arrival times of the photons and the particles at a stationary spacecraft located at a distance r from the explosion. 4] fraction f of the original particles collide with the spacecraft. What is their total momentum and total energy? If the spacecraft has mass m, and the photons pass through it but the particles are absorbed, what is the final speed of the spacecraft? 4] B8 Give an expression for the gravitational potential energy of two point masses. 2] mass m is at a distance 2r from the centre of a planet of radius r and mass M (M m), and is moving at speed u 0 radially away from the planet. Find the minimum value of u 0 that allows it to escape from the planet. 3] The mass now moves in orbit from a point at distance 2r from the centre of the planet, with speed u > u 0 at angle θ to the inward radial direction. What is its angular momentum about the centre of the planet? 2] Sketch the form of the orbits for θ = 0, θ = π/2, and an intermediate angle θ such that the mass just grazes the surface of the planet (i.e. has a distance of closest approach equal to r). 2] For this grazing orbit show that, at the moment of closest approach, the speed is u 2 + GM r Hence show that the orbit just grazes the surface of the planet if sinθ = 1 2 1 + GM ru 2 4] n impact with the planet is narrowly avoided in such a grazing orbit and the mass continues beyond the planet. Find the distance b of the normal between the line of its final path and the centre of the planet. 2]
5 B9 n impulse of magnitude P is applied to a rigid body of mass m and moment of inertia I about an axis through its centre of mass, along a line at perpendicular distance b from its centre of mass. What is the effect of the impulse on the linear and angular velocities? 4] ball of mass m and radius a resting on a horizontal surface is set rolling without slipping at speed u by a single horizontal impulse. How large is the impulse, and where is it applied? What is the final kinetic energy? 4] The ball rolls down the surface of a rough fixed sphere of radius r, starting from rest at the top. What is its speed when the line between the two centres makes an angle θ with the upward vertical? 3] Find the normal component of the reaction force on the ball at this time, and show that the ball leaves the surface of the sphere when cosθ = 10/17. 4] ] The moment of inertia of a sphere with mass m and radius a about an axis through its centre is 5 2ma2. SECTION C C10 Show that the velocity v of transverse waves on a string of density ρ per unit length under tension T is T v = ρ 5] For a continuous travelling wave of the form y = asin(ωt kx) along a string stretched along the x-axis, show that the mean kinetic energy per unit length is K.E. = 1 4 ρω2 a 2 3] string is stretched along the x-axis and is at rest. pulse of sinusoidal shape is then generated on the string by moving one end of it (at x = 0) through the displacement y = asinωt for 0 t 2π/ω Show that the y-component of the force exerted by the string is akt cosωt and hence show that the work done in generating the pulse is W = πvρωa 2 5] Hence show that the mean potential energy per unit length for a wave of the form y = asin(ωt kx) is P.E. = 1 4 ρω2 a 2 2] (TURN OVER
6 C11 system consists of a particle of mass m connected to a fixed point by an ideal spring, which has a spring constant k. Show that the mass will undergo simple harmonic motion with angular frequency k ω 1 = m 3] Now consider a system consisting of two particles of mass m 1 and m 2 which are connected by an ideal spring of spring constant k. Show that the angular frequency of oscillation of the system is ( 1 ω 2 = k + 1 ) m 1 m 2 5] Write this angular frequency in terms of the reduced mass µ = m 1m 2 m 1 + m 2 2] and show this is consistent with the result for ω 1. carbon monoxide molecule consists of a 12 C atom and a 16 O atom. The plot below shows how the potential energy of this molecule varies with the square of the extension of the molecular bond length. The symbol indicates positive extensions and the symbol + indicates negative extensions. Identify the range of extensions over which the molecule will undergo simple harmonic motion, and estimate the angular frequency of the oscillations. 5]
7 C12 State mpère s law. 2] Use mpère s law to show that: (a) the magnetic flux density B wire a distance r from a long straight wire carrying a current I is B wire = µ 0I 2πr 3] (b) the magnetic flux density B in inside a very long solenoid of length l consisting of N turns and carrying a current I is The self-inductance L of a circuit is defined by B in = µ 0NI l 3] Φ = LI where Φ is the magnetic flux linked by the circuit when it carries a current I. Show that the work done in producing a current I in the circuit is 1 2 LI2 4] Hence find the energy per unit volume in a magnetic flux density of B. 3] SECTION D D13 Show that the pressure P of an ideal gas is P = 1 3 mn v 2 where n is the molecular density, m the molecular mass and v 2 the mean square velocity. 5] Explain why the flux J, defined as the number of molecules hitting unit area of surface per unit time, is proportional to n v, where v is the mean molecular velocity. 3] The relative concentration of isotopes of uranium 235 U and 238 U can be altered by allowing the gas UF 6, uranium hexafluoride, to pass from one container to a second container at low pressure, through a barrier with small holes in its surface. The relative concentration in the second container can be taken to be equal to the ratio of the fluxes J( 235 UF 6 ) and J( 238 UF 6 ) on the surface of the barrier. Using the expression for P above, or otherwise, calculate the ratio of the values of v 2 for 235 UF 6 and 238 UF 6. If the relative concentration of 235 UF 6 to 238 UF 6 in the first container is r, what is the relative concentration in the second container? 4] How many times would the above process need to be repeated in order to double the concentration of 235 U from its natural abundance of 0.7%? 3] The ratio of v 2 ] / v 2 is the same for both 235 UF 6 and 238 UF 6. The mass of a fluorine (F) atom is 19 m u. (TURN OVER
8 D14 D15 Write notes on two of the following: (a) applications of the Boltzmann factor; 7 1 2 ] (b) wave particle duality; 7 1 2 ] (c) the equipartition of energy and the heat capacities of gases. 7 1 2 ] The quantum-mechanical wavefunction ψ(x) satisfies the Schrödinger equation: h2 2m d 2 ψ +V ψ = Eψ dx2 Discuss briefly the significance of this equation and the interpretation of the wavefunction ψ(x). 4] particle is confined to a three-dimensional box having dimensions 0 x a, 0 y b and 0 z c. This box can be modelled as an infinitely deep three-dimensional square-well potential, with V = 0 in the interior of the box, and V = elsewhere. Show that the wave function for this potential well must have the general form: ψ(x,y,z) = sin(k x x)sin(k y y)sin(k z z) and find the permitted values of (k x,k y,k z ). 3] Hence derive an expression for the energy of the system, and show that the lowest energy state has energy E 0 = π2 h 2 2m ( 1 a 2 + 1 b 2 + 1 ) c 2 By considering the variation of E 0 with respect to the dimension a, show that a particle in this state exerts a force on the sides x = 0 and x = a given by 3] F = π2 h 2 ma 3 3] Consider the classical motion of a particle confined in this box, travelling in the ±x-direction with momentum ±p x given by p x = hk x for the ground state. Show that the time average of the force on each wall due to the momentum exchange on collision with the walls is given by the same expression as the quantum-mechanical result above. 2] END OF PPER