Classical Mechanics I
Classical Mechanics I PHY 321
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This 6 page Class Notes was uploaded by Quinn Larkin on Saturday September 19, 2015. The Class Notes belongs to PHY 321 at Michigan State University taught by Pawel Danielewicz in Fall. Since its upload, it has received 69 views. For similar materials see /class/207608/phy-321-michigan-state-university in Physics 2 at Michigan State University.
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Date Created: 09/19/15
PHY321 Homework Set 5 1 5 pts Determine the differential cross section 09 E dadQ and total cross section at for elastic scattering of a point particle from a strong repulsive potential sphere of radius R 0 7 lt R UM U0 7 gt R where U0 gt 00 For a particle scattered from a strong potential the law of reflection is valid see the gure Hint From geometry work out the relation between the scat tering angle 9 and the impact parameter b NY 2 10 pts Start with the Rutherford scattering cross section d0 1 kql 12 2 1 dQ 1 6 lt T gt sin4 92 where k 147T60 and T is kinetic energy within the center of mass of the charges Q1 and q2 a By integrating the above differential cross section obtain the net cross section for scattering to angles 9 gt 90 do I 2 09 gt90 6gt60 dQ d Hint When a solid angle integration extends over the full range of azimuthal angle and there is no dependence on the latter angle in the subintegral function then the suitable element of solid angle integration is dQ E 27r sin 9 d9 In the particular case it may be further useful to change the variable of integration from 9 to sin 9 2 b Find the net cross sections for Rutherford scattering of 77 MeV alpha particles from gold nuclei through angles 1 9 greater than 170 ll greater than 900 and Ill greater than 10 Express your cross sections both in m2 and in barns 1b 10 28 m2 100 fm2 represents approximately the size of uranium nucleus Data ma 4u Za 2 mAu 197u and ZAu 79 6247T60 14400Merm For ma ltlt mAu you can in practice ignore the difference between laboratory and center of mass reference frames 3 5 pts A rocket of mass m0 starts its engine in interstellar space Assuming constant speed u of the exhaust gas relative to the rocket at what fraction of the original mass is the rocket going to achieve maximal momentum r 10 pts Consider the problem of a rocket ascending vertically against gravity The rocket starts from rest and its initial mass is me The rocket7s fuel burns at a constant rate 04 and exhaust gas leaves the rocket at a constant speed u relative to the rocket A convenient characteristic of a rocket that is commonly used in place of 04 is the initial trust to weight ratio 7390 aumo g a From the expression for the velocity of the rocket as a function of the remaining mass m 117m07muln eliminate Oz and write the velocity in terms of u the mass ratio mom and To b Demonstrate that for the lift off to occur the rocket must be light enough so that To gt 1 A O V Integrate the velocity with respect to time to obtain elevation h of the rocket as a function of the remaining mass m Note that the integration with respect to time can be easily converted to integration with respect to mass exploiting the linear relation between the two variables Note further that fdx lnx z lnx 7 x Again eliminate 04 from your result and represent h in terms of To u g and the mass ratio mOm d Consider the case of Ariane 5 rocket with initial mass of mo 777 gtlt 105 kg During the initial stage 0 of the ight boosters are used that provide a thrust of om 129 X 107N and employ solid fuel with exhaust velocity of u 3010 ms At the end of stage 0 the rocket mass drops to m 223 gtlt 105 kg Find 7390 and mass ratio mOm Use those to determine the expected velocity and elevation of the rocket at the end of stage 0 U 10 pts A uniform rope of mass M and length L is hung off a small peg The rope can slide without friction over the peg A D V With z denoting the length of rope hanging to the right of the peg obtain an equation of motion for A C7 V Solve the equation of motion if an 0 stretch of the rope hangs to the right of the peg at t 0 and the rope is then at rest How does X the fate of the rope depend on whether 0 gt L2 or 0 lt L2 Find the time tf that it takes for the rope to fall off the peg How does this time depend on M A O V PHY321 Homework Set 3 1 5 pts An object is ejected straight up into the air at an initial velocity 00 E0 a b d Determine the time for reaching the maximal elevation when the object is subject to gravity alone Determine the time for reaching the maximal elevation when the object is subject to gravity combined with a retarding force of the form 71mm Demonstrate that your result from 1b agrees with lain the limit of k a 0 Note It is not suf cient just to put k 0 as this is likely to yield an ill de ned result Rather carefully employ Taylor expansion in your result from 1b7 assuming a small k to demonstrate the agreement when k a 0 On the basis of your expansion from 1c decide whether the retarding force ex tends or shortens the time to reach the maximal elevation 5 pts A woman of mass m sits on a train that coasts along the tracks at a constant speed u She gets up and begins to walk forward along the car at the speed 1 relative to the car a A D V What is the gain in kinetic energy of the woman for a passenger sitting on the train What is the gain in kinetic energy of the woman for an observer standing on a station How much work has the woman done in order to put herself into motion relative to the car How much additional work was done on the woman by the car while she was putting herself into motion Hint To make sure that your answer is correct7 take into account the action reaction law and the impulse momentum theorem As the woman pushes with her feet against the oor of the car7 the oor acts with a reaction force onto the feet That force increases the momentum of the woman relative to the car During the time when the force acts7 the woman is displaced relative to an outside observer7 because of the motion of the car If the woman started walking in a moving lightweight cart7 instead of the car of a train7 what might be the evidence of the work done by the cart7 for an outside observer 3 5 pts Determine the location of the center of mass of a uniform solid cone of base radius R and height H 4 10 pts A thin plate with uniform areal density 039 is bounded by two curves f and g over an x interval 17 where fx 2 a Derive an expression for the mass of the plate in terms of an integral involving the functions f and g b Derive an expression for the position X of the center of mass of the plate along the x axis7 in terms of integrals involving the f and 9 functions A O V Derive an expression for the position Y of the center of mass of the plate along the y axis7 in terms of analogous integrals d Find values for the center of mass coordi nates X and Y for a plate bounded by the functions f cosm and 9x 07 over the interval Find values for the center of mass coordi nates X and Y for a plate bounded by the functions f and g 2 over the interval 01 A D V U 5pts In testing a missile defense system7 a missile is red from the ground on a trajectory that would directly hit a bunker some distance away When the missile is at the top of the trajectory7 a laser light from the bunker ignites fuel in the missile and the missile disintegrates into two pieces7 one twice as massive as the other The pieces reach the ground nearly simultaneously7 60 m apart from each other a By how much does the larger piece miss the bunker Hint Consider motion of the center of mass b By how much does the smaller piece miss the bunker c How important is the information that the fuel ignited at the top of the trajec tory PHY321 Homework Set 4 1 5 pts A beaker lled with water is placed on a scale as shown With the beaker and water the dial of the scale is adjusted so that the scale reads zero No adjustments to the dial are made during the subsequent actions The dial is in grams a A cork of mass 20g and density of 025gcm3 is dropped into the beaker so that it oats there not touching the walls of the beaker What is the reading of the scale now Subsequently a rod of negligible diameter is used to push the cork entirely underwater without touching the walls What is the reading of the scale now EX plain your reasoning In the subsequent step in place of the rod a string is used to tie the cork to a hook at the bottom of the beaker keeping the cork submerged With negligible weight and Volume of the string what is the reading for the scale What is the tension in the string E 3 2 5 pts A sign in the shape of a rightiangle triangle with legs of a 50 cm and b 110 cm is made out of steel sheet with aerial density of U 120 gcm2 The sign is suspended with two Vertical wires from its corners so that the triangles legs are oriented in the horizontal and Vertical directions respectively as shown Determine the tensions T1 and T2 in the two wires 3 5 pts A chain of mass M and length L is suspended Vertically with its lower end touching the indicated scale The chain is released and falls onto the scale Determine the force read by the scale when the length x of the chain has fallen Neglect the size of individual links
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