Coeffcient of Kinetic Friction - Physics - Past Paper, Exams of Physics

Download Coeffcient of Kinetic Friction - Physics - Past Paper and more Exams Physics in PDF only on Docsity! 2011 CAP Prize Examination Compiled by The Department of Physics and Astronomy, University of Calgary Tuesday, February 8, 2011 Duration : 3 hours. Instructions : 1. You are permitted to use calculators for the exam. 2. There are 10 questions on 12 pages. Page 12 is a list of constants. 3. Each question will be marked by a different examiner. The answer to each question should be written on a separate page. If more than one page is required for any question, then those pages should be stapled together separately from other questions. 4. The number of the question, the name of the candidate, and the name of the candidate’s university and department should be clearly indicated on the first page of each answer. 5. Each question has equal value. You are not expected to attempt all of the questions ! Relax, and attempt the questions on material that you are most familiar with or those questions that just look most interesting to you. 6. The completed examination papers should be sent by the Department Chairpersons to : Dr. William J.F. Wilson Department of Physics and Astronomy University of Calgary 2500 University Drive NW Calgary, AB T2N 1N4 1 2011 CAP Prize Examination Department of Physics and Astronomy, University of Calgary page 2 of 12 Question 1 : Mechanics Here are two mechanics problems. Try both. (a) A bead of mass m slides down a wire helix of radius r and pitch angle θ at a constant speed, v. The wire has a circular cross-section, and the long axis of the helix is vertical. Ignore air resistance. What is the coefficient of kinetic friction, µk, between the bead and the wire ? Express your answer in terms of symbols given in this problem and any required constants. (b) A loop ribbon of circumference c, width w and mass m, is spun with an angular velocity ω about its axis of symmetry. What is the tension in the ribbon ? 2011 CAP Prize Examination Department of Physics and Astronomy, University of Calgary page 5 of 12 Question 4 : Oscillations and Waves A wave demonstrator consists of an array of closely-spaced, horizontal rods, all parallel to each other a distance ∆x apart (centre-to-centre), and connected together by a torsional spring that runs along the centre of array, perpendicular to the rods (see figure below). The torsional spring constant is κ, and each rod is of length l and of mass m. The first rod is twisted some small amount ξo, creating a torsional wave which propagates without loss along the wave demonstrator. View from above (a) Derive the wave equation for this situation. (b) What is the wave speed, cw ? (c) If the displacement of the wave obeys the equation ξ = ξo sin(kx−ωt), how much power is transmitted with this wave as a function of m, l, ω, cw, ∆x and ξo ? 2011 CAP Prize Examination Department of Physics and Astronomy, University of Calgary page 6 of 12 Question 5 : Thermodynamics and statistical mechanics If liquid quartz is cooled slowly, it crystallizes at a temperature Tm, and releases latent heat. If it cools more rapidly, the liquid is supercooled and becomes glassy. (a) The liquid and solid phases of quartz are almost incompressible, so there is no work input and changes in internal energy satisfy dE = TdS + µdN . Use the extensivity condition to obtain the expression for µ in terms of E, T , S and N . (b) The heat capacity of crystalline quartz is approximately Cx = αT 3, while that of glassy quartz is roughly CG = βT , where α and β are constants. Assuming that the third law of thermodynamics applies to both crystalline and glass phases, calculate the entropies of the two phases at temperatures T ≤ Tm. (c) At zero temperature the local bonding structure is similar in glass and crystalline quartz, so that they have approximately the same internal energy, E0. Calculate the internal energies of both phases at temeratures T ≤ Tm. (d) Use the condition of thermal equilibrium between two phases to compute the equilibrium melting temperature, Tm, in terms of α and β. (e) Compute the latent heat, L, in terms of α and β. (f) Is the result in the previous part correct ? If not, which of the steps leading to it is most likely to be incorrect ? 2011 CAP Prize Examination Department of Physics and Astronomy, University of Calgary page 7 of 12 Question 6 : Electromagnetism The figure below shows a non-conducting cone with uniform surface charge density σ = 10.6 C/m2. The axis of the cone is along the z-axis, and the point of the cone is at the origin, O. The half-opening angle of the cone is α = 30◦ (see the figure below), and the height of the cone along the z-axis is z0 = 20 cm. The cone is spinning around the z-axis with angular frequency ω = 60.0 Hz in the direction indicated in the figure. (a) Show that at height z (0 ≤ z ≤ 20 cm), an infinitesimal part of the surface of the cone between z and z + dz can be considered a circular current loop with radius r = z tanα carrying a current with magnitude i = ωrσdz cosα (b) Calculate the magnitude and the direction of the magnetic field at the origin, O. Hint : Use the relation (tan2 α + 1) = 1/(cos2 α). (c) At time t = 0 an increasing uniform magnetic field B = 3.00 × 10−4t ẑ T is switched on. Calculate the rate of change of the rotational energy of the spinning cone. Also explain whether ω increases or decreases. Hint : You do not need to know the actual rotational energy of the spinning cone. 2011 CAP Prize Examination Department of Physics and Astronomy, University of Calgary page 10 of 12 Question 9 : Quantum mechanics : Schrödinger’s cat (a) Why do we not usually use quantum physics in our daily lives ? For example, consider a moving cat. Estimate how well we know its position and momentum in practice. Compare the product of your estimated uncertainties in position and momentum for the cat with the Heisenberg uncertainty relation. (b) Now consider a radioactive nucleus that can emit an alpha particle in a random direction. Suppose that the kinetic energy of the alpha particle is 1 MeV. Estimate its momentum uncertainty due to the fact that the direction of emission is uncertain. (c) Suppose that if the alpha particle is emitted in a certain direction, it will trigger a mechanism that kills the cat. If it is emitted in other directions, it will not. We thus don’t know if the cat is still moving or not. Explain how the original small momentum uncertainty for the alpha particle now leads to a large momentum uncertainty for the cat. (d) Can you think of other examples where small quantum uncertainties are amplified to the macroscopic level ? 2011 CAP Prize Examination Department of Physics and Astronomy, University of Calgary page 11 of 12 Question 10 : Physics Applications Consider the production of a new particle Φ created from the head-on relativistic collision of two identical particles p1 = p2 = p by the interaction : p+ p→ Φ It is known that the rest mass of the Φ particle is eight times that of the rest mass of the p particle. (a) What is the minimum relativistic kinetic energy of each p-particle required to produce a Φ if the collision occurs between the two particles moving with the same speed but in opposite directions ? (b) If the collision occurs such that one of the p’s is initially at rest, what is the minimum kinetic energy of the incident particle required to produce a Φ particle from the collision ? (c) What is the final velocity of the Φ particle in both cases ? 2011 CAP Prize Examination Department of Physics and Astronomy, University of Calgary page 12 of 12 Constants used in this examination : c = 3.00× 108 ms−1 e = 1.60× 10−19 C ~ = 1.05× 10−34 J s µ0 = 4π × 10−7 WbA−1m−1 1 u = 1.66× 10−27 kg