EN40: Dynamics and Vibations Midtem Examination Thusday Mach 9 2017 School of Engineeing Bown Univesity NAME: Geneal Instuctions No collaboation of any kind is pemitted on this examination. You may bing 2 double sided pages of efeence notes. No othe mateial may be consulted Wite all you solutions in the space povided. No sheets should be added to the exam. Make diagams and sketches as clea as possible, and show all you deivations clealy. Incomplete solutions will eceive only patial cedit, even if the answe is coect. If you find you ae unable to complete pat of a question, poceed to the next pat. Please initial the statement below to show that you have ead it `By affixing my name to this pape, I affim that I have executed the examination in accodance with the Academic Hono Code of Bown Univesity. PLEASE WRITE YOUR NAME ABOVE ALSO! 1. (10 points) 2. (12 points) 3. (3 points) 4. (10 points) TOTAL (35 points)
1. A bottle with mass m ests on a table-cloth. The contact between them has fiction coefficient. At time t=0 the object and cloth ae both stationay. Fo time t>0, the cloth is pulled with constant acceleation acloth 5g the ight. Note that since g, slip must occu at the contact just afte t=0. acloth 1.1 Daw a fee body diagam showing the foces acting on the object on the figue below. j L i LAPHROAIG 18 h a cloth d/2 d/2 LAPHROAIG 18 [3 POINTS] 1.2 Find a fomula fo the hoizontal acceleation of the bottle. Assume that the bottle does not tip ove. 2
1.3 Find a fomula fo the hoizontal distance moved by the bottle as a function of time. 1.4 Find a fomula fo the distance moved by the cloth as a function of time 1.5 At time t=0 the bottle is a distance L fom the edge of the cloth. Calculate the distance that the bottle has moved in the i diection at the instant that the edge of the cloth eaches the edge of the bottle (L=0 at this instant). Expess you answe in tems of L and any othe paametes you think ae elevant. [3 POINTS] 3
2. The figue shows the tajectoy of a flying squiel, which is measued (in pola coodinates) to be (1 t) (metes) ( adians) 2 2(1 t) (Note that /2 at time t=0, and deceases as t inceases, so the squiel flies to the ight). 2.1 Find a fomula fo the velocity vecto of the squiel (in pola coodinates { e, e }) tems of time 4
2.2 Find a fomula fo the acceleation vecto in pola coodinates { e, e } as a function of time 2.3 Find unit vectos nomal and tangent to the path of the squiel AT TIME t=0, in the { e, e } basis. Choose the sign of n so that the n has a positive e component. 5
2.4 Daw the unit vectos { e, e } AT TIME t=0 on the figue povided below Tajectoy 2.5 Daw a fee body diagam showing the foces acting on the squiel at time t=0 on the figue povided. Include lift and dag foces, and any othe foces you think ae elevant. Tajectoy 6
2.6 If the squiel has mass 100 gams, calculate the magnitude of the lift and dag foce acting on the squiel at time t=0 (you can use the appoximation g 10 m / s and leave you answe in tems of - thee is no need to get a numbe fo the answe) 2 [4 POINTS] 7
3. The attactive foce between two atoms in a diatomic molecule is elated to the distance between them by 5 9 a a F 8F0 whee F 0 and a ae constants. (NB: the sping in the F F figue is used to indicate the atomic bond it does not obey the usual foce-v-length elation fo a linea sping) Show that the potential enegy of the inteatomic bond is (to within an abitay constant) 4 8 V ( ) F0 a 1 2 a a [3 POINTS] 8
4. The figue shows a staight-line collision between an ion (A) and a (neutal) diatomic molecule (B-C) The ion and each atom ae idealized as had sphees with mass m. The bond between atoms B and C has potential enegy given in poblem 3 4 8 V ( ) F0 a 1 2 a a Fo time t<0 atoms the bond between B and C has zeo foce (i.e. the distance between atoms =a); and the ion A is moving to the ight with speed V 0. The ion A collides with atom B at time t=0. The collision has estitution coefficient e=1. a A B C m m m a A B C m m m a A B C m m m NB: the sping in the figue is used to indicate the atomic bond it does not obey the usual foce-v-length elation fo a linea sping. Use the potential enegy elation fom poblem 3, not the potential enegy fo a sping!) 4.1 Calculate the velocity V 1 of atom B just afte the collision, in tems of V 0 (note that the bond between B and C exets no foces duing the collision, and ecall that e=1 and A and B have the same mass) V 0 V 1 t<0 Impact Just afte impact max A B C m m m Max stetch 9
4.2 What is the total linea momentum of the molecule with atoms (BC) just afte the impact, in tems of m and V 0? 4.3 What is the total kinetic enegy of the molecule with atoms (BC) just afte the impact, in tems of m and V 0? 10
4.4 Using enegy and momentum consevation, show that the maximum sepaation between atoms in molecule BC following the collision is 1 a 1/4 2 1 mv 0 1 2 Fa 0 (Hint: at the instant of maximum sepaation both atoms B and C have the same velocity). [4 POINTS] 4.5 Detemine the citical value of V 0 that will just beak the bond between the atoms (assume that the bond beaks if afte the collision). 11