Problem Set 11: Angular Momentum, Rotation and Translation

Transcription

1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department o Physis Physis 80T Fall Term 004 Problem Set : Angular Momentum, Rotation and Translation Available on-line November ; Due: November 3 at 4:00 pm Please write your name, subjet, leture setion, table, and the name o the reitation instrutor on the top right orner o the irst page o your homework solutions Please plae your solutions in your leture setion table box Nov Hour One: Problem Solving Session 6: Angular Momentum Problem Set 0: Due Tues Nov 6 at 4:00 pm Nov 5 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, 75, 3-5 Nov 7 Hour One: Problem Solving Session 7: Rotation and Translation Galati Blak Hole Reading: YF 03 Hour Two: Test Review Nov 8 QUIZ THREE: Energy, Momentum, and Rotational Motion 7:30-9:30 pm Nov 9 No Class Problem Set : Due Tues Nov 3 at 4:00 pm Nov Hour One: Kineti Theory Reading: YF 8-86 Hour Two: Problem Solving Session 8: Ideal Gas Law Reading: YF 8-86

2 Nov 4 Hour One: Arhimedes Priniple Reading: YF 4-43 Hour Two: Arhimedes Priniple PRS Contest Nov 6 No Class Problem Set : Due Fri De 3 at 4:00 pm Problem : Bohr hydrogen atom The Bohr hydrogen atom models the atom as an eletron orbiting the proton under the inluene o an eletri ore produing uniorm irular motion with radius a 0 The mass o the eletron is m = kg ; the eletri harge is e = e C ; the Plank onstant is h = J s, and the magnitude o the eletri ore is given by Coulomb s Law r F = ke r where k = N m C The angular momentum is quantized aording to L= nh π where n =,, a) Write down the equation that arises rom the appliation o Newton s nd Law to the eletron b) What is the angular momentum o the eletron about the enter o the atom? ) Using your results rom parts a) and b) derive an equation or the radius a 0 o the atom as a untion o nehm,,, e, and k d) What is the energy E n or the atom? Express your answer in terms o nehm,,, e, and k e) A hydrogen atom emits a photon whih arise rom an energy transition rom the n = 3 to n = energy level Calulate the requeny o the light emitted = Eh= (E E 3 ) h ) What is the wavelength o the emitted light rom part e)? Problem : Experiment Angular Momentum: Analysis Part One: Rotor Moment o Inertia Enter the results rom your experiment report into the table below

3 α down rad s α rad s a m s T [ N ] τ [ N m] Note: The rotational dynamis or the two stages are given by RT τ = Iα Note that α down < 0 The irst two equations above imply that The ore equation or the irst stage is given by τ = Iα down RT + Iα down = Iα mg T = ma The linear aeleration and angular aeleration are onstrained by a = Rα Combining these last two equations and solving or the tension yields T = mg mα R 3

4 Substituting the tension into the ombined torque equation yields We an solve this or the moment o inertia (mg mα R R + Iα ) down = Iα mgr mα R mgr mα R I = = α α down α + α down (Here m = 0055 kg is the mass o the weight, r = 007 m, α and α are obtained rom down your measurements) What is your numerial value or I R? Part Two: Inelasti Collision: Write your measurement results into the table below - ω rad s ω rad s - δ t [ s] What is your numerial value or I W = m r 0 + r ) w ( i? Use the moments o inertia I R and I W along with ω and ω to alulate the angular i momentum beore and ater the ollision and ompare them Use the values you ound or the rition torque τ and δt to estimate the angular impulse o τ during the ollision Compare it to the angular momentum dierene that you just alulated 3 Calulate the rotational kineti energies K = I ater the ollision R w, beore, and K = (I + I R W )w, Part Three: Slow Inelasti Collision: Write your measurement results into the table below 4

5 ω rad s - ω rad s - δt [ s ] α rad s - Use the moments o inertia I R and I W along with ω and ω to alulate the angular momentum beore and ater the ollision and ompare them Use the value you ound or the rition torque τ and δt to estimate the angular impulse o τ during the ollision Compare it to the dierene in angular momentum beore and ater the ollision 3 Use the value you ound or the angular aeleration during the ollision, α, to estimate the total torque τ on the rotor during the ollision 4 The torque τ is made o two parts: the rition torque τ rom the bearings and the torque τ due exerted by the washer you dropped on the rotor By the 3 rd RW law, the rotor exerts an equal and opposite torque on the washer Sine you know τ, subtrat it rom τ to ind an estimate or τ RW 5

6 6