MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01T Fall Term 2004 Problem Set 11: Angular Momentum, Rotation and Translation Available on-line November 12; Due: November 23 at 4:00 p.m. Please write your name, subject, lecture section, table, and the name of the recitation instructor on the top right corner of the first page of your homework solutions. Please place your solutions in your lecture section table box. Nov 12 Hour One: Problem Solving Session 16: Angular Momentum Problem Set 10: Due Tues Nov 16 at 4:00 pm. Nov 15 Hour One: Experiment 9: Angular Momentum Reading: Experiment 9 Hour Two: Planetary Motion Reading Class Notes: Planetary Orbits: The Kepler Problem, Energy Diagrams Readings: YF 7.1, 7.5, 12.3-12.5 Nov 17 Hour One: Problem Solving Session 17: Rotation and Translation Galactic Black Hole Reading: YF 10.3 Hour Two: Test Review Nov 18 QUIZ THREE: Energy, Momentum, and Rotational Motion 7:30-9:30 pmNov 19 No Class Problem Set 11: Due Tues Nov 23 at 4:00 pm. Nov 22 Hour One: Kinetic Theory Reading: YF 18.1-18.6 Hour Two: Problem Solving Session 18: Ideal Gas Law Reading: YF 18.1-18.6 1Nov 24 Hour One: Archimedes Principle Reading: YF 14.1-14.3 Hour Two: Archimedes Principle PRS Contest Nov 26 No Class Problem Set 12: Due Fri Dec 3 at 4:00 pm. Problem 1 : Bohr hydrogen atom The Bohr hydrogen atom models the atom as an electron orbiting the proton under the influence of an electric force producing uniform circular motion with radius a0. The mass of the electron is × −m = 9.1 10 31 kg ; the electric charge is e=1.6 ×10 −19 C ; the Planck constant ise× −h = 6.63 10 34 J − s , and the magnitude of the electric force is given by Coulomb’s Law r2 N m C 2 2F = ke r 2 where k = 9.0 ×109 − . The angular momentum is quantized according to 2πLnh = where n =1, 2,... a) Write down the equation that arises from the application of Newton’s 2nd Law to the electron. b) What is the angular momentum of the electron about the center of the atom? c) Using your results from parts a) and b) derive an equation for the radius a0 of the atom as , , ,a function of nehme, and k . , , ,d) What is the energy E for the atom? Express your answer in terms of nehme, and k .ne) A hydrogen atom emits a photon which arise from an energy transition from the n = 3 to n =1 energy level. Calculate the frequency of the light emitted f =−∆ Eh=−(E − E3 ) h .1 f) What is the wavelength of the emitted light from part e)? Problem 2: Experiment 11 Angular Momentum: Analysis Part One: Rotor Moment of Inertia Enter the results from your experiment report into the table below. 2downα 2rad s−⎡ ⎤⋅⎣ ⎦ upα 2rad s−⎡ ⎤⋅⎣ ⎦ a 2ms−⎡ ⎤⋅⎣ ⎦ T [ ]N upτ [ ]Nm⋅ Note: The rotational dynamics for the two stages are given by RT −τf = Iαup −τf = Iαdown Note that αdown< 0. The first two equations above imply that RTIαdown = Iαup .+ The force equation for the first stage is given by mg − T = maup . The linear acceleration and angular acceleration are constrained by a = Rα .up up Combining these last two equations and solving for the tension yields Tmg − mαRup = 3Substituting the tension into the combined torque equation yields (mg − mαupR R + Iα= Iα) down up We can solve this for the moment of inertia 2 2mgR − mαupR mgR − mαupR αI = = up −αdown αup +αdown (Here m = 0.055 kg is the mass of the weight, r = 0.0127 m , αup and α are obtained from downyour measurements). What is your numerical value for IR ? Part Two: Inelastic Collision: Write your measurement results into the table below. 1ω -1⎡ ⎤⋅⎣ ⎦ 2ω -1⎡ ⎤⋅⎣ ⎦ tδ [ ]srad s rad s 2What is your numerical value for IW = 1 mr0 + ri 2 ) ?w (2 1. Use the moments of inertia IR and IW along with ω and ω to calculate the angularifmomentum before and after the collision and compare them. 2. Use the values you found for the friction torque τ and δt to estimate the angular fimpulse of τ during the collision. Compare it to the angular momentum difference that fyou just calculated. 1 3. Calculate the rotational kinetic energies K1 = I w 12 , before, and K2 = 12(IR + IW )w2,22 R after the collision. Part Three: Slow Inelastic Collision: Write your measurement results into the table below. 4c1ω ⎡rad s -1 ⎤⋅⎣ ⎦ 2ω ⎡rad s -1 ⎤⋅⎣ ⎦ δt[ ]s cα ⎡rad s -2 ⎤⋅⎣ ⎦ 1. Use the moments of inertia IR and IW along with ω and ω to calculate the angular 1 2momentum before and after the collision and compare them. 2. Use the value you found for the friction torque τ and δt to estimate the angular impulse fof τ during the collision. Compare it to the difference in angular momentum before and fafter the collision. 3. Use the value you found for the angular acceleration during the collision, α , to estimate cthe total torque τ on the rotor during the collision. f4. The torque τ is made of two parts: the friction torque τf from the bearings and thectorque τ due exerted by the washer you dropped on the rotor. By the 3rd law, the rotor RWexerts an equal and opposite torque on the washer. Since you know τ , subtract it fromfτ to find an estimate for τRW.
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