Introduction to and Andy Ruina and Rudra Pratap

Problems for Chapter 5 7495.21 A car accelerates to the right with constantacceleration starting from a stop. There is windresistance force proportional to the square ofthe speed of the car. Define all constants thatyou use.a) What is its position as a function oftime?b) What is the total force (sum of allforces) on the car as a function of time?c) How much power P is required of theengine to accelerate the car in this manner(as a function of time)?problem 5.21: Car.(Filename:s97p1.2)5.22 A ball of mass m is dropped verticallyfrom a height h. The only force acting on theball in its flight is gravity. The ball strikesthe ground with speed v − and after collisionit rebounds vertically with reduced speed v +directly proportional to the incoming speed,v + = ev − , where 0 < e < 1. What is the maximumheight the ball reaches after one bounce,in terms of h, e, and g. ∗a) Do this problem using linear momentumbalance and setting up and solvingthe related differential equations and“jump” conditions at collision.b) Do this problem again using energy balance.5.23 A ball is dropped from a height of h 0 =10 m onto a hard surface. After the first bounce,it reaches a height of h 1 = 6.4 m. What is thevertical coefficient of restitution, assuming itis decoupled from tangential motion? What isthe height of the second bounce, h 2 ?gproblem 5.23:h 2h 1h 0(Filename:Danef94s1q7)5.24 In problem 5.23, show that the number ofbounces goes to infinity in finite time, assumingthat the vertical coefficient is fixed. Findthe time in terms of the initial height h 0 , the coefficientof restitution, e, and the gravitationalconstant, g.5.2 Energy methods in 1D5.25 A small flat disk of mass m = 0.2 kg restson a horizontal surface. The disk is pushedwith an initial velocity v 0 = 2m/s. The diskslides on a straight path for 20 m before it comesto a complete stop. Assuming that there is aconstant friction force between the disk and thehorizontal surface and that it is the only energydissipative mechanism, find the friction forcebetween the two surfaces.5.26 A block of mass m = 0.5 kg is releasedfrom point A of an inclined (θ = 45 o ) frictionlesssurface. The mass slides down the inclinefor a distance of 2 m before hitting an obstacleat point C. Find the velocity of the block justbefore it hits the obstacle.AmBproblem 5.26:2 mθC(Filename:pfigure5.2.rp1)5.27 A particular mechanical system is modeledas a single degree of freedom mass-springsystem where the spring exhibits nonlinearcharacteristics. The potential energy of thespring is given by the function P(x) = 1 4 kx4 .Derive the equation of motion of the system.5.28 Consider a spring-mass system with m =2 kg and k = 5N/m. The mass is pulled tothe right a distance x = x 0 = 0.5 m from theunstretched position and released from rest. Atthe instant of release, no external forces act onthe mass other than the spring force and gravity.a) What is the initial potential and kineticenergy of the system?b) What is the potential and kinetic energyof the system as the mass passesthrough the static equilibrium (unstretchedspring) position?xlproblem 5.28:m(Filename:ch2.10.a)5.29 The power available to a very strong acceleratingcyclist is about 1 horsepower. Assumea rider starts from rest and uses thisconstant power. Assume a mass (bike +rider) of 150 lbm, a realistic drag force of.006 lbf/( ft/ s) 2 v 2 . Neglect other drag forces.(a) What is the peak speed of the cyclist?(b) Using analytic or numerical methodsmake a plot of speed vs. time.(c) What is the acceleration as t →∞inthis solution?(d) What is the acceleration as t → 0inyour solution?5.30 Given ˙v = − gR2r 2 , where g and R areconstants and v = drdt . Solve for v as a functionof r if v(r = R) = v 0 . [Hint: Use the chainrule of differentiation to eliminate t, i.e., dvdt =dvdr · drdt = dvdr · v. Orfind a related dynamicsproblem and use conservation of energy.]• • •Also see several problems in the harmonic oscillatorsection.5.3 The harmonic oscillatorThe first set of problems are entirely aboutthe harmonic oscillator governing differentialequation, with no mechanics content or context.5.31 Given that ẍ =−(1/s 2 )x, x(0) = 1m,and ẋ(0) = 0 find:a) x(π s) =?b) ẋ(π s) =?5.32 Given that ẍ + x = 0, x(0) = 1, andẋ(0) = 0, find the value of x at t = π/2 s.5.33 Given that ẍ + λ 2 x = C 0 , x(0) = x 0 , andẋ(0) = 0, find the value of x at t = π/λ s.• • •The next set of problems concern one mass connectedto one or more springs and possibly witha constant force applied.5.34 Consider a mass m on frictionless rollers.The mass is held in place by a spring with stiffnessk and rest length l. When the spring isrelaxed the position of the mass is x =0. Attimes t = 0 the mass is at x = d and is let gowith no velocity. The gravitational constant isg. In terms of the quantities above,a) What is the acceleration of the block att = 0 + ?b) What is the differential equation governingx(t)?c) What is the position of the mass at anarbitrary time t?d) What is the speed of the mass when itpasses through x =0?