EN40: Dynamics and Vibrations. Homework 4: Work, Energy and Linear Momentum Due Friday March 1 st


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1 EN40: Dynamcs and bratons Homework 4: Work, Energy and Lnear Momentum Due Frday March 1 st School of Engneerng Brown Unversty 1. The fgure (from ths publcaton) shows the energy per unt area requred to dsplace two atomc planes of Molybdynum by a dstance x from ther equlbrum postons. The authors report that: W ( x ) = 6.3Jm dw 19 3 ( x = 0) = N / m dx E 1.1 Calculate the values of the constants 0, d n the Unversal Bndng Energy relaton x(angstroms) x W = E E + exp( x / d d ) that wll ft ths data. Hence, estmate the force per unt area that wll cause the planes to separate (.e. the max force of attracton between the planes). 1. Instead of the UBER, the authors decded to ft ther calculated work of separaton usng the more elaborate functon (ther eq. (9)) of the form x x x x W= E0 E0( ) exp( x/ d) d d d d use Mupad to plot the attractve force as a functon of x for both the unversal bndng energy functon and the modfed formula (use the values of E 0, d from part 1.1. Why s ths correct?). Calculate the maxmum force predcted by the new formula.. The Tesla Model S electrc vehcle has the followng specfcatons: Acceleraton from 0 to 60mph n 4.sec. Curb weght of lbs. Battery capacty (total energy stored n the battery): 60kWh Range at 55mph 44 mles Heght 56 ; wdth 77 Assume that ar resstance can be calculated from the formula 1 FD = ρcdav 3 wth drag coeffcent C D, ar densty ρ = 1.kgm and proected frontal area A and v the speed
2 .1 Assumng that ar resstance s the domnant contrbuton to energy consumpton durng steady cruse, use the gven range and battery capacty to calculate the drag coeffcent.. Calculate the range of the vehcle at 70 mph..3 Assumng that the propulson system produces a constant power (.e. ndependent of velocty) estmate the power necessary to accelerate the vehcle to 60mph n 4. sec. You can neglect ar resstance to keep the calculaton smple. 3. The statc forcevdsplacement measurements for the three bows that you analyzed n Homework can be downloaded from ths webpage. 3.1 Use the MATLAB code that you wrote n Homework to calculate the knetc energy of one (or more) of the arrows ust after they leave the bow. Be sure to state whch bow! 3. The statc forcevdraw data for each bow are avalable as.csv fles on the webpage. By ntegratng the forcevdraw curve, plot a graph of the work done n drawng the bow(s) consdered n.1 as a functon of draw dstance d. You can use the MATLAB trapz functon to do the ntegral, or a method of your own desgn. There s no need to submt a copy of your MATLAB code. 3.3 Hence, calculate the dynamc effcency of each bow (the rato of the knetc energy of the arrow to the work done n drawng the bow. 4. The fgure shows a ball bouncng down a flght of stars. Assume that the ball travels wth constant horzontal speed, and that steady state condtons hold, so that the ball lands on each successve step wth the same velocty v= v0. The goal of ths problem s to calculate ths specal velocty, n terms of the resttuton coeffcent e and the heght of the step h. h v 0 v Wrte down the velocty vector ust after the bounce, n terms of, v 0 and e. 4. Use energy conservaton to fnd a formula relatng v 1 to v 0 and h. 4.3 Hence, show that v0 = gh / (1 e ) 4.4 Calculate the average vertcal velocty of the ball (t s easest to do ths by fndng the tme between two successve bounces). 4.5 Optonal (and qute hard) for extra credt: Suppose that the ball s launched at the top of a flght of stars wth velocty v0 = + v0. Show that t wll land on the nth step wth velocty
3 n ( n 1) 1 e vn = e v 0 + gh 1 e 5 The fgure shows a schematc dagram of a helcopter mpact test used by NASA. If you watch the move, you wll see that the mpact takes place n two stages: () the skds ht the ground, and get crushed; and () the man body of the helcopter hts the ground. The goal of ths problem s to estmate how much force the skds exert on the helcopter body as they are crushed. h θ t t Start Skds ht ground Body hts ground Sldng 5.1 The helcopter starts at a heght h above the ground. Use energy methods to fnd a formula for ts speed ust before the skds ht the ground. Hence, calculate the, components of velocty ust before the skds ht the ground, n terms of the angle θ (ths s ust trg ). 5. As the skds are crushed, they exert a constant vertcal force F on the helcopter body (horzontal force can be neglected). The cable exerts no force on the helcopter durng ths perod. Use Newton s laws and the constant acceleraton formulas to show that the, components of velocty of the helcopter body ust before t mpacts the ground are d Body Skds F v = gh cosθ g( hsn θ + d) Fd / m 5.3 After the skds are crushed the body of the helcopter hts the ground. Durng ths mpact, the helcopter body s subected to an mpulse µ IN+ IN where µ s the coeffcent of frcton. It remans n contact wth the ground after mpact. Show that IN = m g( hsn θ + d) Fd / m and hence deduce that the speed of the helcopter after the body hts the ground s t = gh cosθ µ g( hsn θ + d) Fd / m
4 5.4 Just after mpact the helcopter has a horzontal speed t. It then sldes over the ground for a tme t before comng to rest. Fnd a formula relatng the speed of the helcopter t at the start of the skd, to the frcton coeffcent µ and the tme t and g (you can use the straght lne moton formula or mpulsemomentum). 5.5 Use the results of (3) and (4) to show that ( ) F = mg(1 + hsn θ / d) ( mgh / d) (cos θ / µ ) t g / h 5.6 Calculate the force for the followng parameters: Slp tme t=sec Frcton coeffcent µ = 0.5 Drop heght h 1m Skd heght d 1m Mass 1300kg Cable angle at mpact θ = 30 o 6. The fgure shows a collson between two dentcal spheres wth mass m and radus R. The resttuton coeffcent for the collson s e. The numbers (1), (), (3) show the sequence of the pctures  (1) s before mpact; () s mpact, and (3) s after mpact. (1) B t n 6.1 Wrte down the total lnear momentum of the system before the mpact, n {,} components. A () 6. Fnd the velocty of sphere A before mpact n n,t components, and hence wrte down the total lnear momentum of the system n n,t coordnates. 6.3 Whch of the followng are conserved durng the mpact: (1) The total lnear momentum of the system n the drecton () The total lnear momentum of the system n the drecton (3) The total lnear momentum of the system n the n drecton (4) The total lnear momentum of the system n the t drecton (5) The lnear momentum of sphere A n the drecton? drecton? n drecton? t drecton? (6) The lnear momentum of sphere B n the drecton? drecton? n drecton? t drecton? R A x t y n (3) 4R B 6.4 Wrte down the veloctes of the two spheres n the t drecton after mpact (you don t need to do any calculatons!) 6.5Fnd a formula for the veloctes of the two spheres n the n drecton, n terms of and the resttuton coeffcent e.
5 6.6 Suppose that sphere B travels a dstance 4R. Calculate the coordnates x,y of sphere A relatve to the mpact pont n the {n,t} bass at the same nstant, n terms of e and R.
1.3 Hence, calculate a formula for the force required to break the bond (i.e. the maximum value of F)
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