1.3 Hence, calculate a formula for the force required to break the bond (i.e. the maximum value of F)

Transcription

1 EN40: Dynacs and Vbratons Hoework 4: Work, Energy and Lnear Moentu Due Frday March 6 th School of Engneerng Brown Unversty 1. The Rydberg potental s a sple odel of atoc nteractons. It specfes the potental energy of a bond between two atos n the for V = D(1 + ar)exp( ar) r = ( r / R) 1 where r s the dstance between atos; R s the separaton between atos at nu energy, and D and a are constants. The table below (fro Molecular Physcs, 106, 008, p.753) gves values for D, a and R for several bonds. 1.1 Plot the varaton of V wth r/r for the S-S bond (copare the curves for the three values of a n the paper). Use ev for the unts of energy, and 0.5<r/R<3 1. Fnd a forula for the force of attracton between the atos, n ters of D, a, R and ρ (f you use Mupad to do the dervatve be careful not to use captal D as a varable D eans a dervatve to upad). 1.3 Hence, calculate a forula for the force requred to break the bond (.e. the axu value of F) 1.4 Fnd a value for the strength of the S-S bond for the three possble choces of the value of a. You o can use unts of ev /A for the unts of force ths s a funny unt but often used n atostc calculatons.

2 . The Zero SR ZF1.5 electrc otorcycle has the followng specfcatons: Acceleraton fro 0 to 60ph n 3.3 sec. Curb weght of 188kg. Nonal Battery capacty (total energy stored n the battery): 11kWh (klo-watt-hours) Maxu engne power 50kW Range at 55ph (89 k/hr) 151 k Heght 56 ; wdth 77 Assue that ar resstance can be calculated fro the forula wth c a constant. FD = cv.1 Assung that ar resstance s the donant contrbuton to energy consupton durng steady cruse, use the gven range and battery capacty to calculate c. (fnd the total work done aganst ar drag n ters of c, and set ths equal to the battery capacty).. Assue that the power produced by the electrc otor s related to the vehcle s speed by P= 4 Pax (1 v/ v0 )( v/ v0 ). Assung that all the power developed by the otor s avalable to ncrease ts knetc energy durng acceleraton (neglect drag), show that the acceleraton satsfes 4Pax a= (1 v/ v0 ) v0 Hence calculate a forula for the speed as a functon of te, and use the gven power and te to reach 60ph to fnd a value for v 0. Assue a rder weght of 70 kg. Use Mupad to solve the equaton..3 Use the soluton to.1 and. to estate the axu possble speed of the otorcycle. 3. The Oron launch vehcle s placed n an ntal orbt whose pergee (closest to earth s surface) and apogee (furthest) have alttudes 185 k and 888 k above the earth s surface, respectvely. The velocty of the satellte at pergee s k/s. Take the earth s radus as 6370 k and the gravtatonal constant G = kg s. 3.1 Calculate the speed of the vehcle at the apogee of ths orbt 3. When the vehcle reaches ts pergee a rocket burn ncreases the speed of the rocket (wthout affectng ts alttude). Ths changes the orbt to -3k x 5800k whch has a 4.74 k/s velocty at apogee. Calculate the speed of the vehcle ust after the rocket burn. 3.3 Assung that the vehcle has a constant ass of kg, calculate the pulse exerted by the rocket burn

3 4. The fgure shows a sequence of ages recordng the poston of a sphercal pece of rock as t bounces off an nclned plane. The age (fro ths paper) s part of an experent to easure the resttuton coeffcent of collsons between rocks, whch s of nterest to geologsts and cvl engneers studyng rock-sldes). The scale s n c, the rock has ass 04.33g and the te nterval between fraes s 0.04s. t n α 4.1 Estate the angle of the slope α 4. Suppose the specen s dropped fro soe (unknown) ntal heght h at te t=0, and pacts the slope at te t 1 Wrte down a forula for the heght of the specen y above the pact pont as a functon of te, for t < t Assue that the frst two ages are taken at tes t = t0 and t = t0 + t, where t = 0.04 s. Use your soluton to 4. and the fgure to estate t 0. Hence use the heght of the frst age above the rap to fnd h 4.4 Use energy conservaton to deterne the speed of the specen ust before t pacts the rap. 4.5 Assue that the 5 th frae gves the axu heght of the rebound. Use ts poston to estate the horzontal and vertcal coponents of velocty after the pact. 4.6 Calculate the noral and tangental coponents of pulse exerted on the specen durng the pact Calculate the resttuton coeffcent for noral pact

4 3 v 3 1 v0 v 1 v 5. The fgure shows an approxate odel of an EN40 proect 1 ass launcher wth three asses. The botto and top asses 1, 3 are fxed. The goal of ths proble s to deterne the value of that wll axze the launch velocty. Instead of consderng the sprngs drectly, we approxate the nteracton between asses (and the lowest ass wth the ground) as perfectly elastc (e=1) rgd body collsons. The collsons occur n sequence: frst, ass 1 hts the ground; then colldes wth ass ; and fnally ass colldes wth 3 to launch t. The stack s dropped wth ass 1 a heght h above the ground. The dstance between the asses s very sall copared to h. 5.1 Calculate the velocty of the asses ust before the frst ass hts the ground, n ters of g and h. 5. Wrte down the veloctes of each ass after ass 1 has rebounded, but has not yet collded wth ass. 5.3 Calculate the velocty of ass after ts collson wth ass 1, but before ts collson wth ass 3, n ters of g, h and 1, 5.4 Calculate the velocty of ass 3 after ts collson wth ass, n ters of g, h and the asses. 5.5 Hence, fnd a forula for the value of ass that wll axze the launch velocty, n ters of 1, 3. Copare the values wth the predctons of the MATLAB code fro your proect.

5 6. The fgure shows a frctonless collson between two dentcal spheres wth ass and radus R. The resttuton coeffcent for the collson s e. The nubers (1), (), (3) show the sequence of the pc tures - (1) s before pact; () s pact, and (3) s after pact. At the nstant (1) partcle A has velocty v= V ( )/ and partcle B has velocty vb = V 6.1 Wrte down the total lnear oentu of the syste before the pact, n {,} coponents. V V (1) () B 6. Wrte down the total lnear oentu of the syste after pact, n ters of V 6.3 Explan why sphere B ust contnue to ove parallel to the drecton after pact 6.4 Wrte down coponent of velocty of sphere A after pact. x y (3) 6.5 Use oentu conservaton and the resttuton forula parallel to the drecton to fnd the veloctes of the two spheres after pact 6.6 Assung that e < ( 1) / ( + 1), fnd the dstances x,y at the nstant (3) (at ths nstant partcle B s located at the pont occuped by partcle A at the nstant of collson).